Let $p$ be a prime of the form $5k-4$ and define $f_p(x)=(x^k+x)modp$.

Let $C(p)$ be the number of values $0\le x<p$ such that $f_p^{(m)}(x)=x$ for some positive integer $m$, that is, $x$ can be obtained by iteratively applying $f_p$ on itself starting at $x$.

For example, $C(11)=7$, due to $x=0,1,2,3,8,9,10$.

Let $S(N)$ be the sum of $C(p)$ for all primes of the form $5k-4$ not exceeding $N$. For example $S(100)=127$.

Find $S({10}^8)$.