I come from the question here: Safe primes subgroup in Diffie–Hellman key exchange
Where the accepted answer states that there are only 4 possible outcomes for the order of a subgroup when using a safe prime in the Diffie-Hellman Key Exchange: (1, 2, q, p-1) where q is the Sophie Germaine prime (i.e. (p-1)/2) and p is the safe prime.
The reason for these 4 outcomes seem to be due to Lagrange's Theorem (order of subgroups must divide order of the group), but I am wondering why all values between 1 and p-1 used as the generator will produce either q or p-1 as the order. For order=1, the generator has to be the identity (1), but for order=2, why can't other values between 1 and p-1 acting as a generator produce order=2 in their subgroup? How do we see this?