You have to use properties of [math]S_7[/math]. In particular, every [math]s \in S_7[/math] can be written as the product of disjoint cycles, and then the order of s is the lcm of the cycle lengths.
Example: [math]s=(1 2 3)(5 6)[/math]. Since the cycles are disjoint, we have [math]o(s)=lcm(3,2)=6[/math].
Now consider any element [math]s \in S_7[/math]. If we wanted the order of s to be 20 (for example), we'd only be allowed to use cycle lenghts 1, 2, 4, 5, 10, 20. Since 10 and 20 are too large (we're in [math]S_7[/math] after all), that means we can only use cycle lengths 2, 4 and 5 (not counting 1 as a cycle length). But out of those, we'd have to use both 4 and 5 (to obtain a lcm of 20) - but we can't, since the cycles have to be disjoint and [math]4+5>7[/math]. So there is no element of order 20 in [math]S_7[/math]. You do the rest.