How to prove/disprove $H=\{(1),(123),(132)\}$ is normal in $G=A_5$?

Question

I can't figure out how to do this without manually computing all of the possibilities. Here are some relevant theorems I know:

$H$ is normal in $G$ iff:

1. $\quad\forall g\in G, gHg^{-1}=H$
2. $\quad\forall g\in G, gH=Hg$
3. $\quad\forall g\in G, gHg^{-1}\subseteq H$

Answer 1

As a tip, maybe you can construct an homeomorphism that has $H$ as its kernel and then H must be normal in $A_5$ because you know (or can easily proof) the kernel is always a normal subgroup of the domain.

Edit: Way to disprove it.

Disprove is easy, take an element in $A_5$ for example $(12345)$ now, for H to be normal $(12345)(1)(12345)=(12345)(12345)=(13524)$ should be in $H$ (because $(1)\in H$) but is obvious that $(13524)\notin H$ and then $H$ is not normal in $A_5$.