2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/229678We prove that the very simple lattices which consist of a largest, a smallest and $2n$ pairwise incomparable elements where $n$ is a positive integer can be realized as the lattices of intermediate subfactors of finite index and finite depth. Using the same techniques, we give a necessary and sufficient condition for subfactors coming from Loop groups of type $A$ at generic levels to be maximal.39 pages, latex, corrected proof of Cor. 5.23. To appear in Advance in MathematicsOperator AlgebrasMathematical PhysicsQuantum Algebratext