Reference for the symmetric group $S_n$ and all its subgroups for each $n$?

Question

Is there any reference that contains a list of all the subgroups contained in $S_n$ for at least up to $n = 10$? If they contained the cycle-graphs and other group properties as well even better.

Something akin to OEIS but for groups would be great.

Answer 1

I would strongly recommend using GAP or Magma to do this. It will give you the answer is a useful form! For example, $S_{10}$ has a total of 29594446 subgroups split up into 1593 conjugacy classes. In GAP:

gap> C := ConjugacyClassesSubgroups(SymmetricGroup(10));;
gap> Length(C);
1593
gap> Sum(List(C,x->Size(x)));
29594446

Or Magma:

> C:=Subgroups(Sym(10));
> #C;
1593
> &+[g`length:g in C];
29594446

This takes just a few seconds in both cases.