Classify all groups of order $4165=5(7^2)17$.
I've determined the following possibilities for each of the sylow subgroups
$r_5 = 1$
$r_7 = 1$ or $5(17)$
$r_{17} = 1$ or $5(7)$
I'm trying to show either the sylow $7$ subgroup or the sylow $17$ subgroup is normal so that I can create a subgroup of index $5$. Then I would use semi-direct product theorem. But maybe this is not necessary and maybe there is a simpler solution.
Using Sylow theory, we have a normal Sylow $5$-subgroup $N$. As Daniel noted, this gives an automorphism $G\to \operatorname{Aut}(N)$ where $G$ acts by conjugation. The map must be trivial, which means $gng^{-1}=n$ for all $g\in G$ and $n\in C_5$, i.e., $N\in Z(G)$.
Now $G/Z(G)$ must have order dividing $7^2\cdot 17$, and all of these groups are abelian (you can show this using Sylow theory). For any group, if $G/Z(G)$ is abelian, then $G$ is nilpotent.
Since $G$ is nilpotent, it is a product of its Sylow groups. Thus there are two possibilities: $G=C_5\times C_7\times C_7\times C_{17}=C_{595}\times C_7$, or $G=C_5\times C_{49}\times C_{17}=C_{4165}$