What I did is list all permutations of S5, 120 in tatol. And then find 7 nonidentity elements of order a power of 2. With the identity element, I only need to check these 8 elements form a subgroup, then it's a Sylow 2-subgroup of S5. To find others, I just need to conjugate it with every elements of S5.
However, it's easy to say, but hard to do. Listing them all and conjugating them together are huge work to do.
So I keep wondering... except using Sylow's theorems about the orders and checking if they're normal or not (I've done them all), is there any efficient way to find all Sylow 2-subgroups of S5?