Subgroup whose index of intersection with any subgroup divides the index of that subgroup

Definition
A subgroup $$H$$ of a finite group $$G$$ is termed a subgroup whose index of intersection with any subgroup divides the index of that subgroup if, for any subgroup $$K$$ of $$G$$, the index $$[H:H \cap K]$$ divides the index $$[G:K]$$.

Note that by the product formula, we always have $$[H:H \cap K] \le [G:K]$$, and divisibility holds if the product of subgroups $$HK$$ is a subgroup of $$G$$.

Stronger properties

 * Weaker than::Normal subgroup of finite group
 * Weaker than::Permutable subgroup of finite group
 * Weaker than::Subnormal subgroup of finite group
 * Weaker than::Subgroup of finite abelian group
 * Weaker than::Subgroup of finite nilpotent group