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The Landau problem and harmonic oscillator in the plane share a Hilbert space that carries the structure of Dirac's remarkable s o (2 , 3) representation. We show that the orthosymplectic algebra o s p (1 | 4) is the spectrum generating algebra for the Landau problem and, hence, for the 2D isotropic harmonic oscillator. The 2D harmonic oscillator is in duality with the 2D quantum Coulomb–Kepler systems, with the o s p (1 | 4) symmetry broken down to the conformal symmetry s o (2 , 3) . The even s o (2 , 3) submodule (coined Rac) generated from the ground state of zero angular momentum is identified with the Hilbert space of a 2D hydrogen atom. An odd element of the superalgebra o s p (1 | 4) creates a pseudo-vacuum with intrinsic angular momentum 1/2 from the vacuum. The odd s o (2 , 3) -submodule (coined Di) built upon the pseudo-vacuum is the Hilbert space of a magnetized 2D hydrogen atom: a quantum system of a dyon and an electron. Thus, the Hilbert space of the Landau problem is a direct sum of two massless unitary s o (2 , 3) representations, namely, the Di and Rac singletons introduced by Flato and Fronsdal. [ABSTRACT FROM AUTHOR]