Intersection of normal subgroup and subgroup?

Can a subgroup $H$ of a group $G$ and a normal subgroup $N$ of $G$ have a non-empty intersection, say $I$ not equal to the group containing only the identity? In case, yes what would be an example? What properties (normality in, subgroup of) would $I$ have?

Yes they can, for example $N = A_n$ and $H = (123)$ in $G = S_n$.Here, $N \cap H = H$.
In general, the intersection will have no particular property except being a subgroup of $N$ and $H$. For example, taking $N=G$ you see that there is no reason that $I$ is normal.
Take any non trivial cyclic group $C_{p^r}$, $r>1$. All its subgroups are linearly ordered.
• Well $C_2$ is not the trivial group, is abelian but fails to enlighten me here. – Rudi_Birnbaum Aug 13 '16 at 11:03