The Sylow Theorems · Hard problem

A group has order $351=3^3\cdot13$. Sylow III leaves $n_{13}\in\{1,27\}$ (neither forced) and, separately, $n_3\in\{1,13\}$ (neither forced either). If the Sylow $13$-subgroup is NOT normal, what is $n_3$, the number of Sylow $3$-subgroups?