  The prime graph $\Gamma(G)$ of a finite group $G$ (also known as the
Gruenberg-Kegel graph) has as its vertices the prime divisors of $|G|$, and
$p\text-q$ is an edge in $\Gamma(G)$ if and only if $G$ has an element of order
$pq$. Since their inception in the 1970s these graphs have been studied
extensively; however, completely classifying the possible prime graphs for
larger families of groups remains a difficult problem. For solvable groups such
a classification was found in 2015. In this paper we go beyond solvable groups
for the first time and characterize prime graphs of a more general class of
groups we call pseudo-solvable. These are groups whose composition factors are
either cyclic or $A_5$. The classification is based on two conditions: the
vertices $\{2,3,5\}$ form a triangle in $\overline\Gamma(G)$ or $\{p,3,5\}$
form a triangle for some prime $p\neq 2$.
