If there is a subgroup of order k inside a group of order n, then k divides n.
Thus, in our group, 7 divides the order.
We can also deduce that the order is not even as follows: Suppose it were even. Then by Cauchy's theorem (if p prime divides n, then G has an element of order p), our group has an element of order 2. But an element of order 2 is a g such that g^2 = e. This contradicts our assumption (it is its own inverse).
I'm guessing 35 was the only of the choices that was odd and a multiple of 7.