Let $G$ be compact, connected Lie group, $K$ a closed, connected subgroup, and $T_K$ a maximal torus in $K$. As far as normalizers go, one can show that $N_G(K) \leq K \cdot N_G(T_K)$ using conjugacy of maximal tori in $K$.
Does the opposite containment $N_G(T_K) \leq N_G(K)$ hold? If not, what is the obvious counterexample I am missing?