As is well-known (see, for example, a nice exposition by our own Qiaochu: https://qchu.wordpress.com/2012/11/07/fixed-points-of-random-permutations/) that the distribution of the number of fixed points of random permutations (so, uniformly chosen elements of $S_n$) is Poisson, for largish $n.$ The question is: what is known for proper subgroups of $S_n?$