Number of conjugacy classes in a set of subgroups of $S_5$ which are isomorphic to klein's 4 group.

Question

Let, K be a set of subgroups of S₅ (symmetric group of 5 elements) that are isomorphic to the non-cyclic group of order 4. How many conjugacy classes are there in K?

I know that a non-cyclic group of order 4 is isomorphic to K₄. Now, do I have to find manually all the subgroups isomorphic to K₄ and their conjugate groups? Is there any intuitional shortcut?