I have to prove that a group of order $340$ has a cyclic normal subgroup of order $85$. Because $$340=17\times5\times2^2$$ and due to the third Sylow theorem that there is only one $17$-Sylow subgroup and one $5$-Sylow subgroup, so they are normal. So the group of order $85$ is also normal.
The problem is that, I know that, it has to be isomorphic to $\Bbb{Z}_{85}$ or $\Bbb{Z}_{17} \times \Bbb{Z}_5$ and I don't know how to prove that it's the first one. Any advice? Thanks