A group that has a finite number of subgroups is finite

Question

I have to show that a group that has a finite number of subgroups is finite. For starters, i'm not sure why this is true, i was thinking what if i have 2 subgroups, one that is infinite and the other one that might or not be finite, that means that the group isn't finite, or is my consideration wrong?

Answer 1

For each $g\in G$, you have the group $\langle g\rangle$. If either of these is infinite, it is isomorphic to $\mathbb Z$ and hence has infinitely many subgroup. So if we exclude this, $G$ is the union of finitely many finite sets.

Answer 2

Consider only the cyclic subgroups. No one of them can be infinite, because an infinite cyclic group has infinitely many subgroups. So every cyclic subgroup is finite, and the group is the finite set-theoretic union of these finite cyclic subgroups.