The intersection is obviously normal. There are only finitely many conjugates (why?). The intersection of 2 subgroups of finite index is of finite index, and by induction the intersection of finitely many subgroups of finite index has finite index -- needs proof, I guess. So the intersection of the finitely many conjugates is of finite index.
Until I see some evidence of effort on your part, this is the last response I will make to a question. This is the third question of yours I've answered with no input from you.