Elements of an specific order in a non cyclic non abelian group

Question

I've got a non cyclic non abelian group G = < (1 2 3 4 5),(2 5)(3 4) > which is a subgroups of S_5 and the order of G is 10.
And my question is: Is there any element of order 4 in G?

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What I have done is:
Firstly, if H is a subgroup of G, the order of H has to be 1,2,5 or 10. But now, can I suppose that H is going to be cyclic? or maybe H is not cyclic? I dont't know what else to do.

Answer 1

By Lagrange's theorem, the order of every element $g$ of some group $G$ (which equals the order of the cyclic subgroup $\langle g \rangle$ of $G$ generated by $g$) divides the order of $G$. Therefore, a group of order $10$ cannot possibly contain an element of order $4$.