Is the following true ?

Every solvable transitive subgroup $G\subset S_p$ (the symmetric group on $p$ letters, where $p$ is a prime) contains a unique subgroup $C$ of order $p$ and is contained in the normaliser $N$ of $C$ in $S_p$. The quotient $G/C$ is cyclic of order dividing $p-1$. If $G$ is not cyclic, then it has exactly $p$ subgroups of index $p$.

I need such a result for an arithmetic application. A reference or a short argument will be appreciated.