Determining all Sylow $p$-subgroups of $S_n$ up to isomorphism?
Given two permutation groups H on n points and K on m points, there is a permutation group called H × K that acts on n + m points. The group is abstractly the direct product of the two groups, and the action is very simple: an element like $(h,k)$ acts on the first n points exactly like h did on its n points, and on the last m points exactly like k did on its m points.
For instance the Sylow 3-subgroup of Sym(6) is $\langle (1,2,3) \rangle \times \langle (4,5,6) \rangle$.