Maximal subgroups of finite nilpotent groups are normal of prime index

Given: The hypotheses and objects in the Statement.

A subgroup $M<G$ is maximal proper when there is no subgroup $H$ with $M<H<G$. Equivalently, every subgroup containing $M$ is either $M$ or $G$. The word maximal refers to inclusion among proper subgroups, not to cardinality. (Maximal proper subgroups).

Every proper subgroup of a finite nilpotent group is properly contained in its normalizer. (Every proper subgroup of a finite nilpotent group is properly contained in its normalizer).

Let $G$ be a finite group and let $p$ be prime. If $p\mid \mathrm{\mid }G\mathrm{\mid }$, then $G$ contains an element of order $p$. (Cauchy's theorem: if a prime $p$ divides $\mathrm{\mid }G\mathrm{\mid }$, then $G$ has an element of order $p$).

Every subgroup and every quotient of a nilpotent group is nilpotent. Every finite direct product of nilpotent groups is nilpotent; the class of a subgroup or quotient is at most the class of the original group, and the class of a nonempty finite product is at most the maximum of the factor classes. The empty product is the trivial group of class zero. (Subgroups, quotients, and finite direct products of nilpotent groups are nilpotent).