Jordan-Holder theorem
Factor an integer two different ways and you always get the same primes; the Jordan-Hölder theorem is the group-theoretic version of that uniqueness. No matter how you slice a group into a composition series, the simple pieces you end up with are always the same — only their order along the chain can change.
Precisely, if a group G has two composition series 1 = G_0 ⊴ ... ⊴ G_n = G and 1 = H_0 ⊴ ... ⊴ H_m = G, then n = m and there is a permutation matching the composition factors so that G_{i+1}/G_i is isomorphic to H_{σ(i)+1}/H_{σ(i)} for each i. In short, the multiset of composition factors is an invariant of G. The standard proof uses the Schreier refinement theorem, which says any two subnormal series have refinements that are equivalent.
This makes the composition factors a genuine invariant, the analogue of the prime factorization of a number — but with a crucial difference: the factors do not determine the group. Two non-isomorphic groups can share the same composition factors (for example Z/4Z and Z/2Z x Z/2Z both have factors Z/2Z, Z/2Z), because how the simple pieces are assembled — the extension problem — is extra data the factors do not record.
The symmetric group S_4 has composition factors Z/2Z, Z/3Z, Z/2Z, Z/2Z (via the chain 1 ⊴ V_4 ⊴ A_4 ⊴ S_4 refined), whereas S_5 has factors Z/2Z and the simple group A_5. Because A_5 is nonabelian simple, S_5 is not solvable — and Jordan-Hölder guarantees this factor list is unambiguous.
Composition factors distinguish solvable S_4 from unsolvable S_5.