Showing that $H$ is a normal subroup of $A_4$ Consider the subgroup

$H$={$(1),(12)(34),(13)(24),(14)(23)$}

How would I show that $H$ is a normal subgroup of $A_4$?

If I perform a few computations such as:
$$(132)^{-1}(12)(34)(132)=(123)(12)(34)(123)=(14)(23)$$
$$(243)^{-1}(12)(34)(243)=(234)(12)(34)(243)=(13)(24)$$
$$(124)^{-1}(12)(34)(124)=(142)(12)(34)(124)=(12)(34)$$

,then we see that an element of the form $\beta^{-1}\alpha\beta$, where $\beta \in A_4$ and $\alpha \in H$, remains in $H$.

But because it's tedious to show all the other cases, how would one prove this without showing all the cases?
Hint: Conjugation preserves the cycle type (this spares you quite some computation) and here you even only have one non-trivial cycle type.
The elements of $H$ are exactly the solutions in $A_4$ to the equation $x^2=e$. Therefore, if $H$ is a sugbgroup of $A_4$, then it is a characteristic subgroup.