Order of cyclic groups of residue classes Mod p (Reference)

Question

Given a prime $p$ and a non-trivial residue class $r$ (not equal to $0,1,p-1$), what is the order $d$ of the cyclic group $C$ with generator $r$ such that:
$$ C= \{ x\mid x=r^k(\hbox{mod } p) \} ? $$

Answer 1

That is a hard question.

It is not even known when the order of $2$ is $p-1$. See Artin's conjecture on primitive roots.