Let $G$ be a subgroup of the symmetric group $S_6$ given by $G=\langle(123),(456),(23)(56)\rangle$. Show that $G$ has four normal subgroups of order 3.
I may be missing something, but I can only find two of them: $\langle(123)\rangle$ and $\langle (456)\rangle$. Suppose there is another one. It must be singly-generated, either by an element of the form $(abc)$ or $(abc)(def)$. The subgroup must be closed under conjugation with $(123),(456)$, etc. I always end up with more than three elements.