Q:

Non-normal subgroups in $G$, with $|G|$ having a prime divisor smaller than 5

I have just proved that if 5 is the smallest prime dividing the order of a finite group $G$, then any subgroup o index 5 in $G$ is normal. Could anyone give me an example of a non-normal subgroup of index 5 in a group $G$ for which $|G|$ has a prime divisor smaller than 5?

A:

Would the dihedral group $D_5$ of order ten fit the bill? None of its five subgroups of order $2$ are normal. – Jyrki Lahtonen♦ 7 hours ago 

