Is there an elegant method to show the alternating group $A_5$ as disjoint union of its Sylow subgroups and the trivial element?

Question

Per the discussion in the comments of this question the alternating group $A_5$ is apparently the disjoint union of its Sylow subgroups and the trivial element. I know it can be done explicitly by writing down all the subgroups in terms of their cycle notation. But is there a more elegant way to see this without explicit decompositions?