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To solve the above problem you have to know the following theorem

Theorem : Let n>=2 and σ ∈ Sn be a cycle. Then σ is a k-cycle if and only if order of σ is k.

So, to find the number of subgroups of order 2 in S4  you have to find the number of 2-cycles in S4.

Now, the number of 2-cycles in S4 is 4C2 = 6.

So, the number of subgroups of order 2 in S4 is 6. (Answer)
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