Divisibility-closed subgroup of nilpotent group

Definition
A subgroup $$H$$ of a group $$G$$ is termed a divisibility-closed subgroup of nilpotent group if $$G$$ is a nilpotent group and $$H$$ is a divisibility-closed subgroup of $$G$$.

Weaker properties
{| class="sortable" border="1" ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
 * Stronger than::powering-invariant subgroup of nilpotent group || || || ||
 * Stronger than::powering-invariant subgroup of nilpotent group || || || ||