We are given $H = \{(1),(13),(24),(12)(34),(13)(24),(14)(23),(1234),(1432)\}$ is a subgroup of $S_4$. Also assume $K = \{(1),(13)(24)\}$ is a normal subgroup of $H$. Show $H/K$ isomorphic to $Z_2\oplus Z_2$.

This is just a practice question (not assignment). So I have tried finding $H/K$ explicitly.

$H/K = \{\{(1),(13)(24)\},\{(13),(24)\},\{(14)(23),(12)(34)\},\{(1234),(1423)\}\}$. We know there are only $2$ groups of order $4$. One of the elements in $H/K$ we see is $(1234)K$, doesn't this element have a order of $4$, making $H/K$ cyclic and hence not isomorphic to $Z_2\oplus Z_2$?