**Question:** What is the most frequent order of subgroups of $S\_n$? 
More precisely: Let $a\_k$ be the number of subgroups of $S\_n$ with order $k$. What is the maximum of $a\_k$?


This question came up during a discussion of the open problem of determining the size of the largest antichain in the subgroup lattice of $S\_n$.



Which subgroup order of the symmetric group is the most frequent?
This is already sort of in the comments, but just to say it clearly:


In the paper Enumerating Finite Groups of Given Order
Author(s): L. Pyber
Source: Annals of Mathematics, Second Series, Vol. 137, No. 1 (Jan., 1993), pp. 203-220


Pyber proves that the number of subgroups of $S\_n$ is bounded below by $2^{(1/16 + o(1))n^2}.$ He shows this by noting that $S\_n$ contains $C\_2^{[\frac{n}2]},$ and the latter has the specified number of subgroups. He then conjectures that the lower bound is, in fact, sharp. In other words, modulo this conjecture, at least morally, the maximally present order of a subgroup is, indeed, a power of two, and the actual power of two is presumably $2^{[[\frac{n}2]/2]}.$ It seems that Pyber's conjecture is still open.


