All of the following ${p_{i},q_{i}}$are prime numbers, ${n,m,k}$ 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**
My tutor once stated this problem as some simple application of Sylow Thm and found herself stuck there...