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

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.

• here's a resource for, e.g., $n=6$: groupprops.subwiki.org/wiki/Symmetric_group:S6 If this is what you are looking for, you can check whether higher $n$ cases are dealt with as well. – user810157 Aug 27 '20 at 6:43

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
> &+[glength:g in C];
29594446
`

This takes just a few seconds in both cases.

• Is there a way to create a Hasse Diagram of the subgroups in GAP? Or graph the subgroup lattice? – ReverseFlow Aug 27 '20 at 21:58
• Yes, you can use the GAP function $\mathsf{LatticeSubgroups}$ - I just tried it, and it took less than 30 seconds for $S_{10}$. You will have to look at manual for details on the functionality that this provides. – Derek Holt Aug 28 '20 at 10:50