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Auto-calibration is the recovery of the full camera geometry and Euclidean scene structure from several images of an unknown 3D scene, using rigidity constraints and partial knowledge of the camera intrinsic parameters. It fails for certain special classes of camera motion. This paper derives necessary and sufficient conditions for unique auto-calibration, for several practically important cases where some of the intrinsic parameters are known (e.g. skew, aspect ratio) and others can vary (e.g. focal length). We introduce a novel subgroup condition on the camera calibration matrix, which helps to systematize this sort of auto-calibration problem. We show that for subgroup constraints, criticality is independent of the exact values of the intrinsic parameters and depends only on the camera motion. We study such critical motions for arbitrary numbers of images under the following constraints: vanishing skew, known aspect ratio and full internal calibration modulo unknown focal lengths. We give explicit, geometric descriptions for most of the singular cases. For example, in the case of unknown focal lengths, the only critical motions are: (i) arbitrary rotations about the optical axis and translations, (ii) arbitrary rotations about at most two centres, (iii) forward-looking motions along an ellipse and/or a corresponding hyperbola in an orthogonal plane. Some practically important special cases are also analyzed in more detail.