Subgroup defined up to isoclinism

Definition
A subgroup of a group is termed a subgroup defined up to isoclinism if it satisfies the following condition: It either contains the center or is contained in the derived subgroup (or both).

The significance of this is that any isoclinism between groups establishes a bijection between the subgroups defined up to isoclinism of one group and the subgroups defined up to isoclinism of the other group.

Examples

 * In an abelian group, the only subgroups defined up to isoclinism are the trivial subgroup and the whole group.
 * In a centerless group, every subgroup is a subgroup defined up to isoclinism.
 * In a perfect group, every subgroup is a subgroup defined up to isoclinism.