Upper join of characteristic subgroups

Definition
A subgroup $$H$$ of a group $$G$$ is termed an upper join of characteristic subgroups if there exist subgroups $$K_i,i \in I$$ of $$G$$ containing $$H$$ (where $$I$$ is an indexing set) such that:


 * $$H$$ is a defining ingredient::characteristic subgroup inside $$K_i$$.
 * The join of the $$K_i$$s is $$G$$.

Note that $$H$$ need not be a characteristic subgroup of $$G$$ because characteristicity is not upper join-closed.

Stronger properties

 * Weaker than::Characteristic subgroup
 * Weaker than::Commutator subgroup of direct factor
 * Weaker than::Perfect normal subgroup

Weaker properties

 * Stronger than::Normal subgroup