Nilpotent implies every maximal subgroup is normal

Property-theoretic statement
As am implication of group properties: The property of being a fact about::nilpotent group is stronger than the property of being a fact about::group in which every maximal subgroup is normal.

As an implication of subgroup properties: In a nilpotent group, the property of being a maximal subgroup is stronger than the property of being a normal subgroup.

Verbal statement
In a nilpotent group, every fact about::maximal subgroup is normal.

More on nilpotent groups
It's further true that since the maximal subgroup is normal, it is maximal normal, so the quotient is simple. Since a quotient of a nilpotent group is nilpotent, and the only simple nilpotent groups are abelian, we conclude that every maximal subgroup is normal with abelian quotient (specifically, the quotient is a cyclic group of prime order).

Subgroups of small index in other types of groups

 * Maximal subgroup has prime power index in finite solvable group
 * Subgroup of index equal to least prime divisor of group order is normal

Subgroups of small order

 * Minimal normal implies central in nilpotent
 * Normal of order equal to least prime divisor of group order implies central

Facts used

 * 1) uses::Nilpotent implies normalizer condition: Any nilpotent group satisfies the normalizer condition: it has no proper self-normalizing subgroup.
 * 2) uses::Normalizer condition implies every maximal subgroup is normal

Proof
The proof follows by combining facts (1) and (2).