Cyclic subgroups of same order of a non cyclic group

Question

Suppose $G$ is a group that is NOT cyclic. Let $H$ and $K$ be its cyclic subgroups of the same order.

Then is it true that: Either $H = K$ or $H \cap K = \{e\}$.

If yes, how?

Answer 1

Consider the quaternion group $H_8=\{\pm1,\pm i,\pm j,\pm k\}$. Try $H=\left<i\right>$ and $K=\left<j\right>$.