If $G = \langle x \rangle$ and $H$ is a subgroup of $G$ is $H\unlhd G$?

Question

If $G = \langle x \rangle$ and $H \leq G$, is $H\unlhd G$?

So I know that the group generated by $x$ is known as the generator, and it makes $G$ cyclic iff $G = \langle x \rangle$. If $H \leq G$, can we assume that $H\unlhd G$?

Very lost here on how to prove if a subgroup is normal.

Answer 1

A subgroup $H\leq G$ is normal if $aH=Ha$ for all $a\in G$.

One way to prove a subgroup is normal is to use the normal subgroup test:

If $H\leq G$ and $aHa^{-1}\subseteq H$ for all $a\in G$ then $H\trianglelefteq G$.

It is not difficult to show any subgroup of an Abelian group is normal. (Proof). Your question is a special case of that fact since any cyclic group is Abelian.