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I'm trying to find an example of a group $G$ such that $|G| = 120$, and a non-normal subgroup $H$ within it. Of course, my first instinct is to let $G = S_5$, but this doesn't work because all subgroups of $S_5$ are normal. Help would be greatly appreciated!

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Oh, dear: not at all! The only non-trivial normal subgroup $\,S_5\,$ has is $\,A_5\,$...**all** its other non-trivial subgroups are non-normal – DonAntonio Dec 16 '12 at 2:42
If I take the all the permutations with cycle type 5, say {(12345), (12354),(12435),...}, why is this not normal? – Card Flower Dec 16 '12 at 2:51
It's a cyclic Sylow sub group of A_5 of order 5, and all the above are conjugate so the conjugacy class is not of order 1, so not normal – gnometorule Dec 16 '12 at 2:57
@CardFlower, conjugation classes are not usually subgroups! – DonAntonio Dec 16 '12 at 3:03
And oups. I meant <(12345)> obviously, and its 6 (i think) conjugacy classes. – gnometorule Dec 16 '12 at 3:06

$S_3$ has $6$ elements and a few non-normal subgroups.

Let $H$ be one of the subgroups that is not normal in $S_3$.

Consider next the group $G := S_3 \times \mathbb{Z}_{20}$, which has $120$ elements.

Then $H \times\{0\}$ is a non-normal subgroup of $G$.

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