  The geometrically frustrated spin-1/2 Ising-Heisenberg model on triangulated
Husimi lattices is exactly solved by combining the generalized star-triangle
transformation with the method of exact recursion relations. The ground-state
and finite-temperature phase diagrams are rigorously calculated along with both
sublattice magnetizations of the Ising and Heisenberg spins. It is evidenced
that the Ising-Heisenberg model on triangulated Husimi lattices with two or
three inter-connected triangles-in-triangles units displays in a highly
frustrated region a quantum disorder irrespective of temperature, whereas the
same model on triangulated Husimi lattices with a greater connectivity of
triangles-in-triangles units exhibits at low enough temperatures an outstanding
quantum order due to the order-by-disorder mechanism. The quantum reduction of
both sublattice magnetizations in the peculiar quantum ordered state gradually
diminishes with increasing the coordination number of underlying Husimi
lattice.
