How many $5$-element subgroups does $S_7$ have? 
How many $5$-element subgroups are there in $S_7$, the group of permutations on $7$ elements?

Let $H$ be a $5$-element subgroup of $S_7$. We have $\mbox{ord}(H) = 5$ and $5\mid 7!$. 
But I don't have any idea how can I find 5-element subgroups.

Here are some codes inspired by A. Konovalov's leading suggestions for doing the problem via GAP.
gap> G:=SymmetricGroup(7);
gap> ccs:=ConjugacyClassesSubgroups(G);;
gap> c5:=Filtered(ccs, c -> Size(Representative(c))=5);;
gap> Length(c5);
                                       1
gap> c5;
                      [ Group( [ (3,4,5,6,7) ] )^G ]
gap> Size(c5[1]);
                                      126

The conjugacy class of a 5-cycle contains 504 elements which is 4 times more:
gap> (3,4,5,6,7)^G;
gap> Size(last);
                                      504
gap> 504/126;
                                       4

