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If I know what the orders of the conjugacy classes of $S_4$ are, how can I use that information to show that $V_4=\{I, (12)(34), (13)(24),(14)(23)\}\unlhd S_4$?

One of our exam question asked to determine the orders of the conjugacy classes of $S_4$ and use that information to prove that $V_4 \unlhd S_4$. I was able to find all five of the conjugacy classes, but I couldn't see how does it follow from there that $V_4 \unlhd S_4$. How was I supposed to answer this?

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    A subgroup of $G$ is normal iff it is a union of conjugacy classes of $G$. – 2017-01-25

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A subgroup of a group is normal if and only if it is the union of conjugacy classes. This is because given an $h\in H$ where $H$ is normal, we know that for any $g\in G$, $g^{-1}hg \in H$. Thus, $C_h = \{g^{-1}hg : g\in G\} \subset H$. The reverse also follows from this.

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    If $C_h \subseteq H$ for all $h \in H$, this implies $\cup C_h \subseteq H$. Conversly, given any $h\in H$, $h = e^{-1}he \in C_h$ so that $H \subseteq \cup C_h$. – 2017-01-25