  We construct modular linear differential equations (MLDEs) w.r.t. subgroups
of the modular group whose solutions are Virasoro conformal blocks appearing in
the expansion of a crossing symmetric 4-point correlator on the sphere. This
uses a connection between crossing transformations and modular transformations.
We focus specifically on second order MLDEs with the cases of all identical and
pairwise identical operators in the correlator. The central charge, the
dimensions of the above operators and those of the intermediate ones are
expressed in terms of parameters that occur in such MLDEs. In doing so, the
$q$-expansions of the solutions to the MLDEs are compared with those of
Virasoro blocks; hence, Zamolodchikov's elliptic recursion formula provides an
important input. Using the actions of respective subgroups, bootstrap equations
involving the associated 3-point coefficients have been set up and solved as
well in terms of the MLDE parameters. We present explicit examples of MLDEs
corresponding to BPZ and novel non-BPZ equations, as well as unitary and
non-unitary CFTs.
