Relationship between cyclotomic polynomial modulo a prime and its associated quotient group?

Question

Suppose $\mathbb{Z}[X]/\Phi_m(x)$ where $\Phi_m(x)$ is a $m$-th cyclotomic polynomial .

And it can be factorized over modulo $p$ as follows

$\Phi_m(x)\equiv F_1(X)F_2(X)\ldots F_k(X) \mod p$

Is there any relationship between $F_i(X)$ ($i=1,\ldots,k$) and $\mathbb{Z}_m^*/\langle p \rangle$ ??