= Finite-index characteristic soluble subgroup
Every <virtually soluble group> $K$ has a soluble <characteristic subgroup> of finite index. First take a soluble <normal subgroup> $K_0$ of finite index by the <subgroup core> construction. Choose a soluble normal subgroup $R\supseteq K_0$ maximizing its image size in the finite quotient $K/K_0$. For every soluble normal subgroup $S$, the product $RS$ is soluble, because its quotient by $R$ is a quotient of $S$. Maximality forces $S\leq R$. Hence $R$ is the unique largest soluble normal subgroup and is invariant under all <automorphisms>. This avoids incorrectly intersecting infinitely many conjugates when proving closure under <group extensions>.
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