Core-free and permutable implies subdirect product of finite nilpotent groups

Definition
Suppose $$H$$ is a fact about::core-free permutable subgroup of a group $$G$$. Then, $$H$$ is isomorphic to a fact about::subdirect product of fact about::finite nilpotent groups.

Related facts

 * Core-free permutable subnormal implies solvable of length at most one less than subnormal depth
 * Core-free permutable subgroup of finite group implies nilpotent

Applications

 * Perfect and permutable implies normal