Does this set can say more about group?

Question

Let $G$ be a finite group, set $\Omega=\{H\leq G\mid N_G(H)=H\}$

It is easy to observe followings;

• $|\Omega|=1$ if and only if $G$ is nilpotent as normalizer grows in nilpotent groups.
• For $H\in \Omega$, $Z(G)\leq H$
• $K=\bigcap\limits_{H\leq \Omega} H$ is a characteristic group of $G$ including $Z(G)$.
• $N_G(P)\in \Omega$ for all sylow-$p$ subgroup of $G$.
• It also includes all nonnormal maximal subgroups of $G$.

My question is that "can we say anthing about cardinality of $\Omega$ when $G$ is solvable?" And any further observation would be appriciated.