Let $\boldsymbol{ G}$ be an algebraic group of exceptional Lie type in characteristic $p$, $G=\boldsymbol{ G}^{ \sigma}$ its fixed-point subgroup under the action of a Steinberg endomorphism $\sigma$, and $\overline{ G}$ an almost simple group with socle $G$. A maximal subgroup $M<\overline{ G}$ is called non-generic if it is almost simple and its socle is not isomorphic to a group of Lie type in characteristic $p$. A finite subgroup $H$ of $\boldsymbol{ G}$ is Lie primitive if it does not lie in any proper closed positive-dimensional subgroup of $\boldsymbol{ G}$; it is Lie imprimitive if it lies in a positive-dimensional subgroup $\boldsymbol{ X}$ of $\boldsymbol{ G}$; it is strongly imprimitive if $\boldsymbol{ X}$ can be chosen to be stable under the action of $N_{ \mathrm{ Aut}\boldsymbol{ G}}(H)$, where $\mathrm{ Aut}\boldsymbol{ G}$ is the group generated by inner, diagonal, graph, and field automorphisms of $\boldsymbol{ G}$. We study the possible embeddings of a non-generic primitive simple group $H$ in the adjoint algebraic group $\boldsymbol{ G}$ in characteristic coprime to $|H|$ when $(\boldsymbol{ G},H)$ is one of $(\boldsymbol{ F}_4,\mathrm{ PSL}_2(25))$, $(\boldsymbol{ F}_4,\mathrm{ PSL}_2(27))$, $(\boldsymbol{ E}_7,\mathrm{ PSL}_2(29))$, $(\boldsymbol{ E}_7,\mathrm{ PSL}_2(37))$. In particular, we construct copies of $H$ in $G$ over a suitable finite field $k$, and use them to deduce the number of conjugacy classes of $H$ in $G$ and $\boldsymbol{ G}$, and whether $N_{ \overline{ G}}(H)$ is a maximal subgroup of $\overline{ G}$. We also study the case of $H\simeq\mathrm{ Alt}_6\simeq\mathrm{ PSL}_2(9)$ when $\boldsymbol{ G}$ is one of $\boldsymbol{ F}_4$ and $\boldsymbol{ E}_6$ in characteristic coprime to $|H|$, and show that in such cases $H$ is a strongly imprimitive subgroup of $\boldsymbol{ G}$; in particular, $N_{ \overline{ G}}(H)$ is not a maximal subgroup of $\overline{ G}$.
Categories: math.GR