We prove an upper bound for the number of primes p ≤ x in an arithmetic progression 1 (mod Q) that are exceptional in the sense that $\mathbb{Z}{*}_p$ has no generator in the interval [1,B]. As a consequence we prove that if $Q>exp\left[c\right(logp)/(logB\left)\right(loglogp\left)\right]$ with a sufficiently large absolute constant c, then there exists a prime q dividing Q such that ${\nu }_q\left(or{d}_{p}b\right)={\nu }_q(p-1)$ for some positive integer b ≤ B. Moreover we estimate the number of such q’s under suitable conditions.