The monster sporadic group is the automorphism group of a central charge c=24 chiral conformal field theory (CFT). In addition to its c=24 stress tensor T(z), this theory contains many other conformal vectors of smaller central charge; for example, it admits 48 commuting c=12 conformal vectors whose sum is T(z). Such decompositions of the stress tensor allow one to construct new CFTs from the monster CFT in a manner analogous to the Goddard-Kent-Olive (GKO) coset method for affine Lie algebras. We use this procedure to produce the first evidence for the existence of a number of CFTs with sporadic symmetry groups. To this end, we employ a variety of techniques, including Hecke operators, modular linear differential equations, and Rademacher sums, to compute the characters of these CFTs. Our examples include (extensions of) nine of the sporadic groups appearing as subquotients of the monster, as well as the simple groups 2E6(2) and F4(2) of Lie type.