Let H ⊂ GL ( V ) be a connected complex reductive group where V is a finite-dimensional complex vector space. Let v ∈ V and let G = { g ∈ GL ( V ) ∣ g H v = H v } . Following Raïs we say that the orbit H v is characteristic for H if the identity component of G is H . If H is semisimple, we say that H v is semi-characteristic for H if the identity component of G is an extension of H by a torus. We classify the H -orbits which are not (semi)-characteristic in many cases.