Cosets and Lagrange's Theorem · Hard problem

In $S_6$, let $g=(123456)$ and $H=\langle g\rangle$, order $6$. Since $H\cap A_6$ is a subgroup of $H$, its order divides $6$. Using the fact that a $k$-cycle is an even permutation exactly when $k$ is odd, how many elements does $H\cap A_6$ have?