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Let $G$ and $\tilde G$ be connected complex reductive Lie groups, $G$semisimple. Let $\Lambda^+$ be the monoid of dominant weights for a positiveroot system $\Delta^+$, and let $l(w)$ be the length of a Weyl group element$w$. Let $V_\lambda$ denote an irreducible $G$-module of highest weight$\lambda\in\Lambda^+$. For any closed embedding $\iota:\tilde G\subset G$, weconsider  Property (A): $\quad\forall\lambda\in\Lambda^+,\exists q\in\mathbb{N}$ suchthat $V_{q\lambda}^{\tilde G}\ne0$.  A necessary condition for (A) is for $G$ to have no simple factors to which$G$ projects surjectively. We show that this condition is sufficient if $\tildeG$ is of type ${\bf A}_1$ or ${\bf E}_8$.  We define and study an integral invariant of a root system,$\ell_G=\min\{\ell^\lambda:\lambda\in\Lambda^+\setminus\{0\}\}$, where$\ell^\lambda=\min\{l(w):w\lambda\notin{\rm Cone}(\Delta^+)\}$. We derive thefollowing sufficient condition for (A), independent of $\iota$: $$ \ell_G -\#\tilde\Delta^+ > 0 \;\Longrightarrow\; (A). $$ We compute $\ell_G$ andrelated data for all simple $G$, except ${\bf E}_8$, where we obtain lower andupper bounds. We consider a stronger property (A-$k$) defined in terms ofGeometric Invariant Theory, related to extreme values of codimensions ofunstable loci, and derive a sufficient condition in the form $\ell_G -\#\tilde\Delta^+ > k$. The invariant $\ell_G$ proves too week to handle$G=SL_n$ and we employ a companion $\ell_G^{\rm sd}$ to infer (A-$k$) for alarger class of subgroups. We derive corollaries on Mori-theoretic propertiesof GIT-quotients.

On reductive subgroups of reductive groups having invariants in almost all representations