Subgroup of order $10$ must be normal?

Question

Let $G$ be a group such that $H \le G $ and $o(H) = 10$.

Does there exist a group $G$ with a subgroup $H$ such that $H$ is not normal subgroup of $G$?

my intuition is that there is no such a group $G$, but I'm not sure how to prove it. Any help please?

Answer 1

What about $G=S_{10}$ and $H=\bigl\langle(1\ \ 2\ \ 3\ \ 4\ \ 5\ \ 6\ \ 7\ \ 8\ \ 9\ \ 10)\bigr\rangle$?