Every finite group has a composition series
My question is, how can it be guaranteed that there exists a maximal normal subgroup of $G$ of order $< G$?
Another argument: If the set $S$ of normal subgroups of $G$ is $\emptyset$ then $G$ is simple and there is the composition series $1 \subset G $.
Otherwise every chain in $S$ ordered with respect to inclusion has an upper-bound, so we can apply Zorn lemma and there is a maximal element.