All of the following $p_{i}$ $q$ and $k_{i}$ are prime number, $n_{i} ,m_{i}$ are pre-assigned integers. Consider the product of cyclic groups $\prod_{1}^{n} \mathbb Z^{p_{i}^{n_{i}}}$ then we asked the question:

How many subgroups of order $\prod_{1}^{n} p_{i}^{m_{i}}$ are there in the finite product of cyclic group?$(m_{i} \leqslant n_{i})$

For the initial problem, it's relatively easier,

How many subgroups of order $2^k$ in the product $\mathbb Z^{2^p} \times > \mathbb Z^{2^q}$? $(k \leqslant p \times q)$

It's not considered as a homework problem, I guess. Back ground