TITLE: Enumerating all subgroups of the symmetric group
QUESTION [41 upvotes]: Is there an efficient way to enumerate the unique subgroups of the symmetric group? Naïvely, for the symmetric group $S_n$ of order $\left | S_n \right | = n!$, there are $2^{n!}$ subsets of the group members that could potentially form a subgroup. In addition, many of these subgroups are going to be isomorphic to each other. I feel that the question has an easy answer in terms of the conjugacy classes, but I don't see how. If the answer generalizes to all finite groups, please elaborate!

REPLY [4 votes]: If you are interested in classifying and identifying the possible subgroups of $S_n$ Wilson's Finite Simple Groups has an analysis in Chapter 2 - covering Intransitive subgroups, transitive imprimitive subgroups, primitive wreath products, affine subgroups, subgroups of diagonal type and almost simple groups. From this classification you can see that there is some complexity involved, and that counting will be hard.