Let $ K_1,K_2 $ be conjugate subgroups of a matrix group $ G $. That is $ K_1=gK_2g^{-1} $ for some $ g \in G $. Suppose that $ K_1\cap K_2 $ is very large in the sense that it is an irreducible subgroup of $ G $. Can we conclude that $ K_1=K_2 $?
For example take $ G=SU(2) $ and suppose that $ K_1,K_2 $ are conjugate (closed) subgroups of $ SU(2) $ and that $ K_1\cap K_2 $ is an irreducible subgroup. Can we conclude that $ K_1=K_2 $?
Something that got me thinking about this is that if $ K_1 \cap K_2 $ is big in the sense that $ K_1 \cong N_G(K_1\cap K_2) \cong K_2 $ then I think equality holds and moreover $ K_1 = N_G(K_1\cap K_2) = K_2 $