The normal subgroups of the finite symmetric groups are well understood. If n ≤ 2, S n has at most 2 elements, and so has no nontrivial proper subgroups. The alternating group of degree n is always a normal subgroup, a proper one for n ≥ 2 and nontrivial for n ≥ 3 ; for n ≥ 3 it is in fact the only nontrivial proper normal subgroup of S n, except when n = 4 where there is one additional such normal subgroup, which is isomorphic to the Klein four group.