Every finite group has a composition series. let Ni+1 be a maximal proper normal subgroup in Ni. If the factor group had a normal subgroup, it would map back, by correspondence, to a larger normal subgroup in Ni.
A finite solvable group always has a solvable series with prime cyclic factors. If the factor group F is not prime, and p is a prime factor of |F|, find a subgroup Zp of F by Macay's theorem. Note that Zp is normal in the abelian group F, and take the preimage, giving an intermediate normal subgroup between Ni and Ni+1. Refine the series until all factor groups have prime order.