$|A_5| = 60 = 2^2\times 3\times 5$.
The subgroup $H = \{1, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)\}$ is a Sylow 2-subgroup of $A_5$, $(1 2 3)$ is a Sylow 3-subgroups of $A_5$, and $(1 2 3 4 5)$ is a Sylow 5-subgroup of $A_5$. How should I find all $p$-Sylow subgroups and prove they are all the $p$-Sylow subgroups?