In this paper, the quadratic regularized and standard fuzzy c-means clustering algorithms (qFCM and sFCM) are generalized with respect to hard c-means (HCM) regularization. First, qFCM is generalized from quadratic regularization to power regularization. The relation between this generalization and sFCM is then compared to the relation between other pairs of methods from the perspective of HCM regularization, and, based on this comparison, sFCM is generalized through the addition of a fuzzification parameter. In this process, we see that other methods can be constructed by combining HCM and a regularization term that can either be weighted by data-cluster dissimilarity or not. Furthermore, we see numerically that the existence or nonexistence of this weighting determines the property of these methods' classification rules for an extremely large datum. We also note that the problem of non-convergence in some methods can be avoided through further modification.