# Intersection of all non normal.subgroups of a group

Let $$G$$ be a group and collect all the non-normal subgroups of $$G$$. Then what will the intersection of all those be? Normal?

The examples that I took, like $$S_3, S_4, D_4$$, every time the trivial subgroup is coming and that is normal.

But is there any theorem to support this?