Subgroups of Symmetric Group $S_4$ and Isomorphism

Question

During my Algebra class we were given this exercise to solve at home, but I couldn't find any solution and I also did not really get the one our teacher gave us. So, the text was:

Given G = S₄ = Sym({1, 2, 3, 4}). Then:

1. Is there a subgroup of G of order 5? Motivate your answer.
2. Give and example of two subgroups H₁ and H₂ so that |H₁| = |H₂| and H₁ $\ncong$ H₂ (H₁ not isomorphic to H₂)

Part 1 of the question is fairly simple in fact thanks to Lagrange Theorem on subgroups, but the problems arise with Part 2. The given solution is:

Let's consider as an example:

1. H₁ = { id, (1,2), (3,4), (1,2)(3,4) }
2. H₂ = { id, (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) }

In fact, H₁ elements are all of order 2 (apart from id = identity ) while (1,2,3,4) $\in$ H₂ has order 4. Thus H₁ $\ncong$ H₂ ( H₁ not isomorphic with H₂ )

I've tried my best, but given the fact that our teacher sometimes is very messy I still can not figure out why this is a valid proof, nor the steps needed to find it. This is my first time posting, so sorry in advance for any mistake. Thank you for your time!

Answer 1

(1,2)(3,4) is a different permutation from (1,2,3,4), and the correct group for $H_2$ is {id,(1,2,3,4),(1,3)(2,4),(1,4,3,2)}.