I know some facts such as: Nilpotent groups are solvable, $p$-groups are nilpotent, a finite group whose order is a product of distinct primes is solvable, and finite groups are nilpotent if and only if it is a finite product of solvable groups.
Are there any other major relationships between these concepts that one should be aware of?
Also, how do you compute a transfer? Could you provide a simple example or two?