Finite simple groups of order $p+1$

Question

$\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\PSU{PSU}$Cross-post from MSE. There are some very interesting comments on the original post if you want to go check it out.

Are there any well known patterns about which finite simple groups have order $ p+1 $ for $ p $ a prime?

Here is a list of all non-cyclic simple groups of order up to 100,000 and whether they have order p+1 (there are 31 such groups, 16 have order $ p+1 $)

• $ \PSL_2(5) $, $p=59$

• $ \PSL_2(7) $, $p=167$

• $ \PSL_2(9) $, $p=359$

• $ \PSL_2(8) $, $p=503$

• $ \PSL_2(11) $, $p=659$

• $ \PSL_2(13) $, $p=1091$

• $ \PSL_2(17) $, $p=2447$

• $ A_7 $, $2519$ not prime

• $ \PSL_2(19) $, $3419$ not prime

• $ \PSL_2(16) $, $p=4079$

• $ \PSL_3(3) $, $5615$ not prime

• $ PSU_3(3) $, $p=6047$

• $ \PSL_2(23) $, $6071$ not prime

• $ \PSL_2(25) $, $7799$ not prime

• $ M_{11} $, $p=7919$

• $ \PSL_2(27) $, $9827$ not prime

• $ \PSL_2(29) $, $12{,}179$ not prime

• $ \PSL_2(31) $, $p=14{,}879$

• $ \PSL_4(2) $, $20{,}159$ not prime

• $ \PSL_3(4) $, $20{,}159$ not prime

• $ \PSL_2(37) $, $p=25{,}307$

• $ \PSU_4(2) $, $p=25{,}919$

• $ \operatorname{Suz}(8) $, $29{,}119$ not prime

• $ \PSL_2(32) $, $32{,}735$ not prime

• $ \PSL_2(41) $, $p=34{,}439$

• $ \PSL_2(43) $, $p=39{,}731$

• $ \PSL_2(47) $, $51{,}887$ not prime

• $ \PSL_2(49) $, $58{,}799$ not prime

• $ \PSU_3(4) $, $62{,}399$ not prime

• $ \PSL_2(53) $, $p=74{,}411$

• $ M_{12} $, $95{,}039$ not prime.