Conjugate-comparable subgroup

Definition
A subgroup of a group is termed a conjugate-comparable subgroup if it is comparable with each of its defining ingredient::conjugate subgroups, in other words, every conjugate subgroup to it either contains it or is contained in it.

Collapse to normality

 * Any finite subgroup that is conjugate-comparable is normal.
 * Any subgroup of finite index that is conjugate-comparable is normal.
 * Any subgroup of a slender group, Artinian group, or periodic group that is conjugate-comparable is normal.

Facts

 * Automorph-comparable of normal implies conjugate-comparable
 * Left residual of conjugate-comparable by normal is automorph-comparable