Every two finite solvable groups of the same order have the same composition factors

Question

I proved the following statement:

The composition factors of every finite solvable group are isomorphic to cylic groups of prime order.

I want to use this result to prove that every two finite solvable groups of the same order have the same composition factors.

If I knew that for every prime number in the decomposition of the order of the groups there is a composition factor that is cyclic of that order I would obviously be finished. However, I'm not even sure if this statement is true and I don't have a good idea on how to proceed in case the statement isn't true.

Help will be appreciated! :)

Answer 1

If $1=G_0\lhd G_1\lhd \ldots \lhd G_n=G$, then $|G|=\prod_{i=1}^n [G_{i}:G_{i-1}]$.