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Derek Holt
Derek Holt
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{* 0^^54230^^1352, 1^^18841^^2654, 2^^10122^^2753, 3^^5113^^1816, 4^^2584^^902, 5^^1315^^363, 6^^536^^121, 7^^257^^29, 8^^128^^9,... 9 *}

and it was similar for $S_{50} \times S_{9950}$ but a bit different for $S_5 \times S_{9995}$:.

{* 0^^2665, 1^^3218, 2^^1933, 3^^1149, 4^^338, 5^^201, 6^^76, 7^^33, 8^^6,...*}

{* 0^^8349, 1^^4350^^5301, 2^^1391^^2000, 3^^502^^1293, 4^^153^^733, 5^^24^^368, 6^^25^^167, 906^^70, 927^^40, 93^^48^^19, 94^^79^^5, 
   95^^34, 96^^53, 97^^69, 98^^138, 99^^36, 100^^101, 101^^41, 102^^1310^^2, 103^^4,11^^2 ...*}.

which is not very Poisson-like.

 {* 0^^82930^^4440, 1^^6701^^3394, 2^^1772^^1143, 3^^2083^^574, 4^^874^^204, 5^^355^^48, 6^^896^^105, 7^^487^^42, 8^^188^^8, 9^^5
    10^^17,... 11^^13, 12^^4, 14, 15^^4, 16, 21, 28 *}

{* 0^^3092, 1^^5400, 3^^1195, 7^^2200^^2867, 15^^651^^5803, 31^^263^^1271, 637^^58, 12715 *}

{* 0^^34870^^3478, 1^^39031^^3915, 2^^19042^^1916, 3^^5983^^575, 4^^714^^77, 5^^315^^32, 13^^512, 13^^2, 14, 15^^3 *}

 {* 0^^4126, 1^^41330^^4152, 3^^14721^^4244, 7^^2293^^1393, 15^^287^^197, 31^^915^^13, 63^^331 *}

 {* 0^^51290^^5092, 1^^16751^^1762, 2^^21312^^2137, 3^^6363^^650, 4^^2764^^221, 5^^795^^81, 8^^368^^29, 9^^129^^4, 10^^1210^^14, 11^^3,
   11^^3, 12^^312, 16^^316^^4, 17^^317, 18, 57 *}

{* 0^^57450^^5711, 1^^1541^^171, 2^^34702^^3467, 3^^883^^91, 4^^3724^^350, 6^^1676^^202, 1212^^3, 22^^322^^5 *}

  {* 0^^76420^^8916, 32^^22^^165, 484^^196, 64^^76^^165, 80^^28^^126, 96^^610^^113, 10812^^87, 112^^414^^56, 116^^216^^42, 118, 
 120^^3, 
    124^^218^^32, 126^^220^^27, 128^^5322^^19, 144^^324^^14, 15226^^15, 15628^^9, 160^^1030^^6, 16832, 176^^434^^4, 18036, 18238, 40^^3, 
   184^^3, 188, 190^^2,50^^2 192^^31,...*}

{* 0^^5423, 1^^1884, 2^^1012, 3^^511, 4^^258, 5^^131, 6^^53, 7^^25, 8^^12,...*}

and it was similar for $S_{50} \times S_{9950}$ but a bit different for $S_5 \times S_{9995}$:

{* 0^^2665, 1^^3218, 2^^1933, 3^^1149, 4^^338, 5^^201, 6^^76, 7^^33, 8^^6,...*}

{* 0^^8349, 1^^435, 2^^139, 3^^50, 4^^15, 5^^2, 6^^2, 90, 92, 93^^4, 94^^7, 
   95^^34, 96^^53, 97^^69, 98^^138, 99^^36, 100^^101, 101^^41, 102^^13, 103^^4, ...*}

which is not very Poisson-like.

{* 0^^8293, 1^^670, 2^^177, 3^^208, 4^^87, 5^^35, 6^^89, 7^^48, 8^^18, 9^^5,...*}

{* 0^^3092, 1^^5400, 3^^1195, 7^^220, 15^^65, 31^^26, 63, 127 *}

{* 0^^3487, 1^^3903, 2^^1904, 3^^598, 4^^71, 5^^31, 13^^5, 14 *}

{* 0^^4126, 1^^4133, 3^^1472, 7^^229, 15^^28, 31^^9, 63^^3 *}

{* 0^^5129, 1^^1675, 2^^2131, 3^^636, 4^^276, 5^^79, 8^^36, 9^^12, 10^^12, 
   11^^3, 12^^3, 16^^3, 17^^3, 18, 57 *}

{* 0^^5745, 1^^154, 2^^3470, 3^^88, 4^^372, 6^^167, 12, 22^^3 *}

{* 0^^7642, 32^^2, 48, 64^^7, 80^^2, 96^^6, 108, 112^^4, 116^^2, 118, 120^^3, 
    124^^2, 126^^2, 128^^53, 144^^3, 152, 156, 160^^10, 168, 176^^4, 180, 182, 
   184^^3, 188, 190^^2, 192^^31,...*}

{* 0^^1352, 1^^2654, 2^^2753, 3^^1816, 4^^902, 5^^363, 6^^121, 7^^29, 8^^9, 9 *}

and it was similar for $S_{50} \times S_{9950}$.

{* 0^^5301, 1^^2000, 2^^1293, 3^^733, 4^^368, 5^^167, 6^^70, 7^^40, 8^^19, 9^^5,
   10^^2, 11^^2 *}.

 {* 0^^4440, 1^^3394, 2^^1143, 3^^574, 4^^204, 5^^48, 6^^105, 7^^42, 8^^8, 
    10^^17, 11^^13, 12^^4, 14, 15^^4, 16, 21, 28 *}

{* 0^^2867, 1^^5803, 3^^1271, 7^^58, 15 *}

{* 0^^3478, 1^^3915, 2^^1916, 3^^575, 4^^77, 5^^32, 12, 13^^2, 14, 15^^3 *}

 {* 0^^4152, 1^^4244, 3^^1393, 7^^197, 15^^13, 31 *}

 {* 0^^5092, 1^^1762, 2^^2137, 3^^650, 4^^221, 5^^81, 8^^29, 9^^4, 10^^14, 11^^3,
    12, 16^^4, 17, 18 *}

{* 0^^5711, 1^^171, 2^^3467, 3^^91, 4^^350, 6^^202, 12^^3, 22^^5 *}

  {* 0^^8916, 2^^165, 4^^196, 6^^165, 8^^126, 10^^113, 12^^87, 14^^56, 16^^42,  
     18^^32, 20^^27, 22^^19, 24^^14, 26^^15, 28^^9, 30^^6, 32, 34^^4, 36, 38, 40^^3, 
     50^^2 *}
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Derek Holt
Derek Holt
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You asked for $p$-groups. First a Sylow $2$-subgroup of $S_{8192}$ of order $2^{8191}$:

{* 0^^7642, 32^^2, 48, 64^^7, 80^^2, 96^^6, 108, 112^^4, 116^^2, 118, 120^^3, 
   124^^2, 126^^2, 128^^53, 144^^3, 152, 156, 160^^10, 168, 176^^4, 180, 182, 
   184^^3, 188, 190^^2, 192^^31,...*}

It's hard to see what is going on there.

And a Sylow $5$-subgroup of $S_{15625}$ of order $5^{3906}$.

{* 0^^8724, 100^^2, 250^^3, 340^^3, 345, 350^^4, 370^^2, 375^^15, 420, 435, 445,
   450, 470^^3, 475^^10, 490^^2, 495^^7, 500^^52, 540, 545, 550, 565^^2, 570, 
   575^^5, 595^^4, 600^^19, 615^^2, 620^^12, 625^^53,...*}

You asked for $p$-groups. First a Sylow $2$-subgroup of $S_{8192}$ of order $2^{8191}$:

{* 0^^7642, 32^^2, 48, 64^^7, 80^^2, 96^^6, 108, 112^^4, 116^^2, 118, 120^^3, 
   124^^2, 126^^2, 128^^53, 144^^3, 152, 156, 160^^10, 168, 176^^4, 180, 182, 
   184^^3, 188, 190^^2, 192^^31,...*}

It's hard to see what is going on there.

And a Sylow $5$-subgroup of $S_{15625}$ of order $5^{3906}$.

{* 0^^8724, 100^^2, 250^^3, 340^^3, 345, 350^^4, 370^^2, 375^^15, 420, 435, 445,
   450, 470^^3, 475^^10, 490^^2, 495^^7, 500^^52, 540, 545, 550, 565^^2, 570, 
   575^^5, 595^^4, 600^^19, 615^^2, 620^^12, 625^^53,...*}
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Derek Holt
Derek Holt
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I generally agree with Geoff that the question is too broad, and that you should tell us what types of families of subgroups you are most interested in.

I don't know of any proven results in this direction, but in case it is of any use in formulating conjectures, here are a few Sunday afternoon experimental results (done in Magma) on maximal subgroups of $S_n$ for $n$ about $10000$. For each example I took a sample of $10000$ random elements. For $S_{10000}$ itself, I got

{* 0^^3649, 1^^3681, 2^^1845, 3^^642, 4^^147, 5^^29, 6^^5, 7^^2 *}

For almost all examples I was finding that there were more fixed-point-free elements than in the symmetric group. The intransitive maximals are of the form $S_m \times S_n$ of degree $m+n$. I expect you could calculate the distributions theoretically. For $S_{5000} \times S_{5000}$, I got

{* 0^^5423, 1^^1884, 2^^1012, 3^^511, 4^^258, 5^^131, 6^^53, 7^^25, 8^^12,...*}

and it was similar for $S_{50} \times S_{9950}$ but a bit different for $S_5 \times S_{9995}$:

{* 0^^2665, 1^^3218, 2^^1933, 3^^1149, 4^^338, 5^^201, 6^^76, 7^^33, 8^^6,...*}

For transitive maximals, the large imprimitive groups have the form $S_n \wr S_m$ of degree $mn$. For $S_{100} \wr S_{100}$ I got

{* 0^^8349, 1^^435, 2^^139, 3^^50, 4^^15, 5^^2, 6^^2, 90, 92, 93^^4, 94^^7, 
   95^^34, 96^^53, 97^^69, 98^^138, 99^^36, 100^^101, 101^^41, 102^^13, 103^^4, ...*}

which is not very Poisson-like.

Turning to primitive groups, we have examples like $S_{140}$ acting on the $9730$ unordered pairs. For this, I got:

{* 0^^8293, 1^^670, 2^^177, 3^^208, 4^^87, 5^^35, 6^^89, 7^^48, 8^^18, 9^^5,...*}

and similarly for other large primitives like primitive wreath product actions.

Classical groups look more intersting, and I wasn't detecting much pattern. Here are a few examples. ${\rm PSL}(13,2)$, degree $8191$:

{* 0^^3092, 1^^5400, 3^^1195, 7^^220, 15^^65, 31^^26, 63, 127 *}

${\rm PSL}(5,11)$, degree $16105$:

{* 0^^3487, 1^^3903, 2^^1904, 3^^598, 4^^71, 5^^31, 13^^5, 14 *}

${\rm PSp}(14,2)$, degree $16383$:

{* 0^^4126, 1^^4133, 3^^1472, 7^^229, 15^^28, 31^^9, 63^^3 *}

${\rm PSp}(6,7)$, degree $19608$ ($2$ more popular than $1$ fixed point):

{* 0^^5129, 1^^1675, 2^^2131, 3^^636, 4^^276, 5^^79, 8^^36, 9^^12, 10^^12, 
   11^^3, 12^^3, 16^^3, 17^^3, 18, 57 *}

${\rm P \Omega}^+(6,9)$, degree $7462$ (even stronger preference for 2, and even numbers):

{* 0^^5745, 1^^154, 2^^3470, 3^^88, 4^^372, 6^^167, 12, 22^^3 *}