Suppose $\{p_i\}$ are prime numbers, and $\{n_i ,m_i\}$ are pre-assigned integers. Consider the product of cyclic groups $G = \prod_{i=1}^{n} (\mathbb Z/p_{i}^{n_{i}}\mathbb Z)$.

How many subgroups of order $\prod_{i=1}^{n} p_{i}^{m_{i}}$ are there in $G$ (where $m_i \leqslant n_i$)?

Here is a problem that is relatively easier:

How many subgroups of order $2^k$ are there in the product $\mathbb Z/2^p\mathbb Z \times > \mathbb Z/2^q \mathbb Z$ (where $k \leqslant p + q$)?