Finite non-nilpotent and every proper subgroup is nilpotent implies not simple

Statement
Suppose $$G$$ is a finite group that is not a nilpotent group, but every proper subgroup of $$G$$ is a  (and in particular, a  ). Then, $$G$$ cannot be a simple group.

Facts used

 * 1) uses::Nilpotent implies normalizer condition: In a nilpotent group, every proper subgroup is properly contained in its normalizer.
 * 2) uses::Finite and any two maximal subgroups intersect trivially implies not simple non-abelian

Proof
Given: A finite non-nilpotent group $$G$$ such that every proper subgroup of $$G$$ is nilpotent.

To prove: $$G$$ is not simple.

Proof: We assume that $$G$$ is simple, and derive a contradiction. Let $$n$$ be the number of elements of $$G$$.