  We identify all symmetry superalgebras $\mathfrak{g}$ of near horizon
geometries of black holes with a Killing horizon, assuming the solution is
smooth and that the spatial cross section of the event horizon is compact
without boundary. This includes all warped AdS$_2$ backgrounds with the most
general allowed fluxes in 10- and 11-dimensional supergravities. If the index
of a particular Dirac operator vanishes, we find that the even symmetry
subalgebra decomposes as $\mathfrak{g}_0=\mathfrak{sl}(2,\mathbb{R})\oplus
\mathfrak{t}_0$, where $\mathfrak{t}_0/\mathfrak{c}$ is the Lie algebra of a
group that acts transitively and effectively on spheres, and $\mathfrak{c}$ is
the center of $\mathfrak{g}$. If the Dirac operator index does not vanish, then
the symmetry superalgebra is nilpotent with one even generator. We also
demonstrate that there are no near horizon geometries, and also therefore no
warped AdS$_2$ backgrounds, in 10- and 11-dimensions that preserve more than 16
supersymmetries.
