Isomorph-conjugate subgroup

Symbol-free definition
A subgroup of a group is termed isomorph-conjugate if any subgroup isomorphic to that subgroup is conjugate to it in the whole group.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed isomorph-conjugate if whenever there is a subgroup $$K$$ isomorphic to $$H$$, $$H$$ and $$K$$ are conjugate subgroups in $$G$$.

In terms of the relation implication operator
The property of being isomorph-conjugate can be viewed in terms of the relation implication operator with the relation on the left being that of a subgroup pair being isomorphic and the relation on the right being that of a subgroup pair being conjugate in the whole group.

Stronger properties

 * Isomorph-free subgroup
 * Order-dominating subgroup
 * Order-unique subgroup
 * Order-conjugate subgroup
 * Sylow subgroup:
 * Intermediately isomorph-conjugate subgroup

Weaker properties

 * Automorph-conjugate subgroup
 * Isomorph-automorphic subgroup

Conjunction with other properties

 * Any normal isomorph-conjugate subgroup is isomorph-free.

Metaproperties
An isomorph-conjugate subgroup of an isomorph-conjugate subgroup is not necessarily isomorph-conjugate.

An intersection of two isomorph-conjugate subgroups of a group need not be isomorph-conjugate.

Testing
This subgroup property can be tested using GAP code, though there is no direct built-in GAP function for it. The GAP code is available at GAP:IsIsomorphConjugateSubgroup, and is invoked as follows:

IsIsomorphConjugateSubgroup(group,subgroup);