Let $$G=\mathbb{Z}/p_1^{e_1}\times\cdots\times\mathbb{Z}/p_n^{e_n}$$ be any finite abelian group.

What are $G$'s subgroups? I can get many subgroups by grouping the factors and multiplying them by constants, for example: If $$G=\mathbb{Z}/3\times \mathbb{Z}/9\times \mathbb{Z}/4\times \mathbb{Z}/8,$$ then I can take $$H=3(\mathbb{Z}/3\times \mathbb{Z}/9)\times 2(\mathbb{Z}/4) \times \mathbb{Z}/8.$$ Do I get all subgroups that way? (I'm interested in all subgroups, not just up-to-isomorphism).

Which are the subgroups $H$ in $G$ for which $G/H$ is primary cyclic?

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