Normal Subgroups and Quotient Groups · Challenge problem

In $S_4$, let $H = \langle(1\,2),(3\,4)\rangle = \{e,(1\,2),(3\,4),(1\,2)(3\,4)\}$ (order $4$) and let $N = \{e,(1\,2)(3\,4),(1\,3)(2\,4),(1\,4)(2\,3)\}$, the normal Klein four-subgroup. Using the Second Isomorphism Theorem, find $|HN/N|$.