  Electrons exposed to a two-dimensional (2D) periodic potential and a uniform,
perpendicular magnetic field exhibit a fractal, self-similiar energy spectrum
known as the Hofstadter butterfly. Recently, related high-temperature quantum
oscillations (Brown-Zak oscillations) were discovered in graphene moir\'{e}
systems, whose origin lie in the repetitive occurrence of extended
minibands/magnetic Bloch states at rational fractions of magnetic flux per unit
cell giving rise to an increase in band conductivity. In this work, we report
on the experimental observation of band conductivity oscillations in an
electrostatically defined and gate-tunable graphene superlattice, which are
governed both by the internal structure of the Hofstadter butterfly (Brown-Zak
oscillations) and by a commensurability relation between the cyclotron radius
of electrons and the superlattice period (Weiss oscillations). We obtain a
complete, unified description of band conductivity oscillations in
two-dimensional superlattices, yielding a detailed match between theory and
experiment.
