How many subgroups does $\mathbb{Z}_{16}\times\mathbb{Z}_{30}$ have? How many of them are cyclic?
First, I would like to know, if this answer for the first part of the question is correct:
We know that $\mathbb{Z}_{16}\times\mathbb{Z}_{30}\cong \mathbb{Z}_{16}\times\mathbb{Z}_{2}\times\mathbb{Z}_{3}\times\mathbb{Z}_{5}$ and that if the order of groups are coprime, the subgroups of the direct product are products of subgroups of the groups. So I can count the subgroups of $\mathbb{Z}_{16}\times\mathbb{Z}_{2}$, $\mathbb{Z}_{3}$, $\mathbb{Z}_{5}$ separately and multiply. The last two have just the 2 trivial subgroups.
I started counting the subgroups of $\mathbb{Z}_{16}\times\mathbb{Z}_{2}$ using Goursat's Lemma. I found 3 subquotients of $\mathbb{Z_2}$, two of them are zero and one is $\mathbb{Z}_2$. For $\mathbb{Z}_{16}$ there are 15 subquotients which are isomorphic to one of $\{\{0\} (\text{appearing 5 times}),\mathbb{Z}_2 (\text{appearing 4 times}),\mathbb{Z}_4,\mathbb{Z}_8,\mathbb{Z}_{16}\}$. Now, I need to find isomorphism between the subquotients of $\mathbb{Z}_2$ and $\mathbb{Z}_{16}$. The only possible ones are the zero map between the zero groups and the identity on $\mathbb{Z}_2$. So there are $2\cdot5+1\cdot4=14$ subgroups of $\mathbb{Z}_{16}\times\mathbb{Z}_{2}$?
So in total there are $2\cdot2\cdot 14=56$ subgroups of $\mathbb{Z}_{16}\times\mathbb{Z}_{30}$?
Now, it comes to counting how many of them are cyclic. I have no idea how to approach this. I don't know anything about the cyclic subgroups of a product of two groups... I can found some by hand (These which comes from a product of cyclic subgroups of $\mathbb{Z}_{16}$ and $\mathbb{Z}_{30}$ which have comprime order (I found 23)). But I guess that are not all? Or if so, how do I know that?