Subgroup of cyclic normal subgroup is normal

Let $\displaystyle K \triangleleft G$ where $\displaystyle K$ is cyclic. Show that every subgroup of $\displaystyle K$ is normal in $\displaystyle G$.

Been working on this for awhile and i know that every subgroup of a cyclic subgroup will be cyclic and correspondingly abelian and that we must show $\displaystyle gxg^{-1} \in H \; x\in H, \forall g \in G$, where $\displaystyle x=h^i, \; i \in \mathbb{Z}$ with h the generator of the subgroup of K. Totally stuck though on where to go after this.