Motivation:
I have co-authored a package for sagemath to compute with combinatorial species, also known as sequences of group actions of the symmetric groups. In an effort to find good tests for that package, it occurred to me that the set of subgroups of the symmetric group $\mathfrak S_n$ itself carries an action of $\mathfrak S_n$, by conjugation. The orbits of this action are, by definition, the conjugacy classes of subgroups of $\mathfrak S_n$.
Question:
what is known about the molecular decomposition of this species? Slightly less ambitious: what is known about the corresponding symmetric group character?
Experiments:
I computed the first few decompositions into transitive actions. In species notation, they are as follows for $1\leq n\leq 6$: \begin{align*} &X \\ &2 E_2 \\ &3 E_3 + X E_2 \\ &4 E_4 + 4 P_4 + E_2^2 + 2 X E_3 \\ &3 E_5 + 6 E_3 E_2 + 4 X P_4 + 3 X E_4 + 3 \{((1,2,3,4), (1,2)(3,5))\} \\ &3 E_6 + 8 E_4 E_2 + 12 E_2 P_4 + 3 E_3^2 + 2 X E_5 + 3 P_6 + 3 X E_2 E_3 \\ &+ 3 X \{((1,2,3,4), (1,2)(3,5))\} \\ &+ 3 \{((1,2)(5,6), (1,2)(3,5), (1,2,4), (1,3)(2,5)(4,6))\} \\ &+ 6 \{((1,2,4)(3,6), (1,3)(2,5)(4,6))\}\\ &+ 2 \{((2,4)(3,5), (1,2)(5,6), (1,3,2,4))\}\\ &+ 8 \{((1,4)(2,3), (1,6,3,2,5,4))\} \\ & 3 E_7 + 6 E_5 E_2 + 19 E_3 P_4 + 12 E_4 E_3 + 8 X E_2 P_4 + 4 E_2^2 E_3 + 3 X P_6 + 4 X E_4 E_2 + X E_3^2 + 2 X E_6 \\ & + 8 X \{((1,4)(2,3), (1,6,3,2,5,4))\} \\ & + 3 X \{((1,2)(5,6), (1,2)(3,5), (1,2,4), (1,3)(2,5)(4,6))\} \\ & + 8 E_2 \{((1,2,3,4), (1,2)(3,5))\} \\ & + 4 \{((2,3,5)(4,7,6), (1,2,4)(3,6,5), (1,3,6,7,5,2))\} \\ & + 6 X \{((1,2,4)(3,6), (1,3)(2,5)(4,6))\} \\ & + \{((2,4,3), (1,3,2,4)(6,7), (2,4,3)(5,7,6))\} \\ & + \{((3,4)(6,7), (1,2)(6,7), (2,3)(5,6))\} \\ & + 2 X \{((2,4)(3,5), (1,2)(5,6), (1,3,2,4))\} \\ & + \{((1,3,2)(5,7,6), (1,5,4)(2,6,7))\} \end{align*}
Here, $X$ is the singleton species (corresponding to the trivial action of $\mathfrak S_1$), $E_n$ is the species of sets of size $n$ (corresponding to the trivial action of $\mathfrak S_n$), $P_n$ is the species of polygons, etc. For the remaining species, the generators of the stabilizer group of some fixed structure are given. For example, $\{((1,2,3,4), (1,2)(3,5))\}$ means that there is a representative of the species structures which is invariant when permuting the labels with the two given permutations.
The character, in terms of Schur functions, is as follows: \begin{align*} &s_\emptyset\\ &s_{1}\\ &2 s_{2}\\ &s_{2,1} + 4 s_{3}\\ &5 s_{2,2} + 3 s_{3,1} + 11 s_{4}\\ &7 s_{2,2,1} + 10 s_{3,2} + 13 s_{4,1} + 19 s_{5}\\ &6 s_{2,1,1,1,1} + 3 s_{2,2,1,1} + 34 s_{2,2,2} + 3 s_{3,1,1,1} + 21 s_{3,2,1} + 6 s_{3,3} + 3 s_{4,1,1} + 64 s_{4,2} + 34 s_{5,1} + 56 s_{6}\\ & 3 s_{1,1,1,1,1,1,1} + 8 s_{2,1,1,1,1,1} + 8 s_{2,2,1,1,1} + 43 s_{2,2,2,1} + 13 s_{3,1,1,1,1} + 23 s_{3,2,1,1} + 74 s_{3,2,2} + 17 s_{3,3,1} + 7 s_{4,1,1,1} + 95 s_{4,2,1} + 92 s_{4,3} + 17 s_{5,1,1} + 135 s_{5,2} + 105 s_{6,1} + 96 s_{7} \end{align*}
Subgroups of $\mathfrak S_n$ are counted in https://oeis.org/A005432, conjugacy classes of subgroups in https://oeis.org/A000638. If I understand correctly, the number of distinct molecular species in the molecular decomposition is https://oeis.org/A091071.
Warning:
the package that produced this decomposition is still a bit experimental. It is quite possible that it is not correct. I only checked by hand for $n=1,2,3$.
