Subgroup of coprime order with automorphism group is contained in center of group
$N/Z(G)\leq G/Z(G)$. If $N\not\leq Z(G)$, what can you say about the order of $N/Z(G)$?
$N/Z(G)\leq G/Z(G)$. If $N\not\leq Z(G)$, what can you say about the order of $N/Z(G)$?