  The study of finite subgroups of a simple algebraic group $G$ reduces in a
sense to those which are almost simple. If an almost simple subgroup of $G$ has
a socle which is not isomorphic to a group of Lie type in the underlying
characteristic of $G$, then the subgroup is called non-generic. This paper
considers non-generic subgroups of simple algebraic groups of exceptional type
in arbitrary characteristic.
  A finite subgroup is called Lie primitive if it lies in no proper subgroup of
positive dimension. We prove here that many non-generic subgroup types,
including the alternating and symmetric groups $\text{Alt}_{n}$,
$\text{Sym}_{n}$ for $n \ge 10$, do not occur as Lie primitive subgroups of an
exceptional algebraic group.
  A subgroup of $G$ is called $G$-completely reducible if, whenever it lies in
a parabolic subgroup of $G$, it lies in a conjugate of the corresponding Levi
factor. Here, we derive a fairly short list of possible isomorphism types of
non-$G$-completely reducible, non-generic simple subgroups.
  As an intermediate result, for each simply connected $G$ of exceptional type,
and each non-generic finite simple group $H$ which embeds into $G/Z(G)$, we
derive a set of feasible characters, which restrict the possible composition
factors of $V \downarrow S$, whenever $S$ is a subgroup of $G$ with image $H$
in $G/Z(G)$, and $V$ is either the Lie algebra of $G$ or a non-trivial Weyl
module for $G$ of least dimension.
  This has implications for the subgroup structure of the finite groups of
exceptional Lie type. For instance, we show that for $n \ge 10$, $\text{Alt}_n$
and $\text{Sym}_n$, as well as numerous other almost simple groups, cannot
occur as a maximal subgroup of an almost simple group whose socle is a finite
simple group of exceptional Lie type.
