We consider the problem of detecting an unknown signal that lies in a union of subspaces (UoS) and that is observed in additive white Gaussian noise with unknown variance. The main contribution of this paper is the derivation of a detector that can accommodate a union made of nested subspaces. This detector includes the generalized likelihood ratio test (GLRT) as a special case when the subspace dimensions are all identical. It relies on the framework of multifamily likelihood ratio tests (MFLRT) and is shown by numerical examples to achieve better performance than existing detectors.