Ranks of homotopy groups of homogeneous spaces

A simple way to evaluate the ranks of homotopy groups πj(M) is indicated for homogeneous spaces of the form M = G/H, where G is a compact connected Lie group and H is a connected regular subgroup or a subgroup of maximal rank inG. A classification of the spaces whose Onishchik ranks are equal to 3 is obtained. The transitive actions on the products of homogeneous spaces of the form G/H are also described, where G and H are simple and H is a subgroup of corank 1 in G and the defect of the space G/H is equal to 1.