Direct product of subgroups of factors in direct product decomposition into cyclic groups

Definition
Suppose $$G$$ is a finitely generated abelian group. A subgroup $$H$$ of $$G$$ is termed a direct product of subgroups of factors in direct product decomposition into cyclic groups if there exists a decomposition of $$G$$ as an internal direct product of cyclic groups:

$$G = G_1 \times G_2 \times \dots \times G_n$$

and subgroups $$H_i \le G_i$$ such that:

$$H = H_1 \times H_2 \times \dots \times H_n$$