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.