Is my thinking correct when asked to show that a group $G$ of order $35^3$ is solvable, I first show that by the sylow theorems there exists a sylow $p$-subgroup of order $5^3$ and another unique sylow $p$-subgroup of order $7^3$. Then since these two unique sylow $p$-subgroups compose the group and are solvable then the entire group is solvable.
What are the techniques of showing a group is solvable with the sylow theorems?
I'm having trouble proving this one and another where the group is of order $80$.
Thank you for the help.