Centralizer-closed subgroup property

This article defines a subgroup metaproperty: a property that can be evaluated to true/false for any subgroup property
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Definition

A subgroup property $\alpha$ is termed centralizer-closed if the following is true: whenever a subgroup $H$ of a group $G$ satisfies property $\alpha$, so does the centralizer $C_G(H)$.

Examples

Examples of subgroup properties that are centralizer-closed

	Quick phrase
Automorph-conjugate subgroup
C-closed normal subgroup
Central factor	factor in central product product with centralizer is whole group quotient action by outer automorphisms is trivial every inner automorphism restricts to an inner automorphism
Characteristic subgroup	invariant under all automorphisms automorphism-invariant strongly normal normal under outer automorphisms
Characteristic subgroup of group of prime power order
Finite direct power-closed characteristic subgroup
Local powering-invariant subgroup
Normal subgroup	invariant under inner automorphisms, self-conjugate subgroup same left and right cosets kernel of a homomorphism subgroup that is a union of conjugacy classes
Powering-invariant subgroup

Examples of subgroup properties that are not centralizer-closed

	Quick phrase
Subnormal subgroup
Transitively normal subgroup