Optimal system of Lie algebra subalgebras of the point symmetries group for nonlinear heat equation without source

In this paper we construct an optimal system of subalgebras for the nine-dimension Lie algebra of infinitesimal operators for a point symmetries group of a nonlinear heat equation with isotropic heat conductivity tensor and with a power dependence of the temperature. The results are presented as a lemma and a theorem. It is proven that up to transformations of internal automorphisms and some discrete automorphisms, there are 117 dissimilar subalgebras classes of various dimensions.