I find it a lot easier to use cycle notation when working with the symmetric group. example is the permutation that takes 1 to 2, 2 to 4, 4 to 7 and 7 to 1. The unmentioned numbers are left the same.
Your coset it has size two and . So you should expect 3 cosets each of size 2. Basically you just gotta multiply them out and see what happens.
Here are the left cosets of H
Right cosets of H found similarly
Compare these cosets and see that the last two do not match up, so these are not the same. In particular this tells you that H is infact not a normal subgroup of because