Finite N-group is solvable or almost simple

Statement
Suppose $$G$$ is a finite N-group, i.e., $$G$$ is a finite group that is also a uses property satisfaction of::N-group, i.e., the normalizer of any nontrivial solvable subgroup of $$G$$ is solvable. Then, $$G$$ is either a solvable group (or equivalently, a  proves property satisfaction of::finite solvable group) or an  proves property satisfaction of::almost simple group.

The definition of almost simple that we will use here is: a group is almost simple if it has a centralizer-free non-abelian simple normal subgroup.

Facts used

 * 1) uses::Solvability is subgroup-closed
 * 2) uses::Minimal normal implies characteristically simple
 * 3) uses::Equivalence of definitions of finite characteristically simple group

Proof
Given: A finite N-group $$G$$.

To prove: $$G$$ is either solvable or almost simple.

Proof: