finite group of odd order

let G be a group has a finite number of elemenys, half of them have order 2 and the other half order different than 2, show that the subgroup H of the elements of order other than 2 has an odd number of elements and that it is a comutative subgroup.

so far i know how to prove that a group has an odd number of elements iif it has no element of order 2 and since H has index 2 in G, it must be normal in G because its left cosets will be equal to its right cosets since it has only 2, itself and the other one.

any help will be greatly apreciated.