Subisomorph-containing subgroup

Definition
A subgroup $$H$$ of a group $$G$$ is termed subisomorph-containing if whenever $$K$$ is a subgroup of $$G$$ and $$L$$ is a subgroup of $$H$$ such that $$K$$ and $$L$$ are isomorphic, then $$L$$ is also a subgroup of $$H$$.

In groups with specific properties

 * Finite group and periodic group: In a finite group and more generally in a periodic group, the notion of subisomorph-containing subgroup coincides with the notions of subhomomorph-containing subgroup and variety-containing subgroup.
 * Group of prime power order: For groups of prime power order, subisomorph-containing subgroups must be omega subgroups of group of prime power order, though the converse, while true for regular p-groups, is not always true.

Stronger properties

 * Weaker than::Variety-containing subgroup
 * Weaker than::Subhomomorph-containing subgroup:

Weaker properties

 * Stronger than::Isomorph-containing subgroup
 * Stronger than::Injective endomorphism-invariant subgroup
 * Stronger than::Intermediately injective endomorphism-invariant subgroup
 * Stronger than::Intermediately characteristic subgroup
 * Stronger than::Characteristic subgroup