Let $text{Mod}(S_g)$ be the mapping class group of the closed orientable surface $S_g$ of genus $ggeq 2$. In this paper, we derive necessary and sufficient conditions under which two torsion elements in $text{Mod}(S_g)$ will have conjugates that generate a finite split nonabelian metacyclic subgroup of $text{Mod}(S_g)$. As applications of the main result, we give a complete characterization of the finite dihedral and the generalized quaternionic subgroups of $text{Mod}(S_g)$ up to a certain equivalence that we will call weak conjugacy. Furthermore, we show that any finite-order mapping class whose corresponding orbifold is a sphere, has a conjugate that lifts under certain finite-sheeted regular cyclic covers of $S_g$. Moreover, for $g geq 5$, we show the existence of an infinite dihedral subgroup of $text{Mod}(S_g)$ that is generated by the hyperelliptic involution and a root of a bounding pair map of degree $3$. Finally, we provide a complete classification of the weak conjugacy classes of the non-abelian finite split metacyclic subgroups of $text{Mod}(S_3)$ and $text{Mod}(S_5)$.
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