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?
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
Alternatively, if you like non-solvable (even simple) groups: Consider the alternating group $A_5$ of order $60$; it contains $5$ subgroups of order $12$ (and thus of index $5$), namely, the groups isomorphic to the alternating group $A_4$. And none of these can be normal, since $A_5$ is simple. (If you don't want to use the simplicity, it is very easy to show non-normality otherwise, too).