What I did is list all permutations of S_{5}, 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 S_{5}. To find others, I just need to conjugate it with every elements of S_{5. }

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 S_{5}?