  We consider several transformation groups of a locally conformally K\"ahler
manifold and discuss their inter-relations. Among other results, we prove that
all conformal vector fields on a compact Vaisman manifold which is neither
locally conformally hyperk\"ahler nor a diagonal Hopf manifold are Killing,
holomorphic and that all affine vector fields with respect to the minimal Weyl
connection of a locally conformally K\"ahler manifold which is neither
Weyl-reducible nor locally conformally hyperk\"ahler are holomorphic and
conformal
