Cyclic subgroups and their cyclic groups

Question

If a subgroup $\langle a\rangle$ is cyclic, then is the group that contains this cyclic subgroup also cyclic?

Answer 1

No.
A group generated by one element is always cyclic.
Take for example $S_3$ and its cyclic subgroup $\langle(12)\rangle$

Answer 2

No. In fact, any group whatsoever is a union of cyclic subgroups.