Modular subgroup

Symbol-free definition
A subgroup of a group is termed a modular subgroup if it is a modular element in the lattice of subgroups.

Definition with symbols
A subgroup $$A$$ of a group $$G$$ is termed a modular subgroup if for any subgroups $$B$$ and $$C$$ of $$G$$ such that $$A \le C$$:

$$ \langle A, B \cap C \rangle = \langle A,B \rangle \cap C$$

Metaproperties
The whole group is clearly a modular subgroup of itself. So is the trivial subgroup.

Suppose $$H \le K \le G$$ such that $$H$$ is modular in $$G$$. Then, clearly, $$H$$ must be a modular element with respect to all choices of subgroups in $$G$$, and hence, in particular, in $$K$$.

Thus, $$H$$ is also modular in $$K$$.