  We study mean-field states resulting from the pairing of electrons in
time-reversal broken fractal Hofstadter bands, which arise in two-dimensional
lattices where the unit cell traps magnetic flux $\Phi = (p/q)\Phi_0$
comparable to the flux quantum $\Phi_0 = h/e$. It is established that the
dimension and degeneracy of the irreducible representations of the magnetic
translation group (MTG) furnished by the charge 2e pairing fields have
different properties from those furnished by single particle Bloch states, and
in particular are shown to depend on the parity of the denominator $q$. We
explore this symmetry analysis to formulate a Ginzburg-Landau theory describing
the thermodynamic properties of Hofstadter superconductors at arbitrary
rational flux $\Phi = (p/q)\Phi_0$ in terms of a multicomponent order parameter
that describes the finite momentum pairing of electrons across different Fermi
surface patches. This phenomenological theory leads to a rich phase diagram
characterized by different symmetry breaking patterns of the MTG, which can be
interpreted as distinct classes of vortex lattices. A class of
$\mathbb{Z}_q$-symmetric Hofstadter SCs is identified, in which the MTG breaks
down to a $\mathbb{Z}_q$ subgroup. We study the topological properties of such
$\mathbb{Z}_q$-symmetric Hofstadter SCs and show that the parity of the Chern
numbers is fixed by the parity of $q$. We identify the conditions for the
realization of Bogoliubov Fermi surfaces in the presence of parity and MTG
symmetries, establishing a novel topological invariant capturing the existence
of such charge-neutral gapless excitations. Our findings, which could bear
relevance to the description of re-entrant superconductivity in moir\'e systems
in the Hofstadter regime, establish Hofstadter SC as a fertile setting to
explore symmetry broken and topological orders.
