TITLE: Determine the number of subgroups of a cyclic group
QUESTION [7 upvotes]: Can we actually tell the number of subgroups of a particular cyclic group without actually listing them?

REPLY [2 votes]: If $G=\langle g \rangle $ is a finite cyclic group of order $n$, then any subgroup of $G$ has the form $S=\langle g^d\rangle$ for a divisor $d\mid n$. Different values of $d$ give different sizes so there is just one subgroup of G having a given size. Hence there are $\tau(n)$ different subgroups.
Each non-trivial subgroup of an infinite cyclic group has two generators, which are inverses of each other. Fixing one generator $g$ of the whole group, we can write each subgroup in the form $⟨g^n⟩$ for a unique $n ≥ 0$. So the number of subgroups is countably infinite.