230 space groups: the exact 230, 219, and 65 counts

In Euclidean space $\mathbb{R}^3$, let $\mathcal C$ be the collection of subgroups $G$ of the group of bijective affine isometries satisfying both of these conditions: for every $x\in\mathbb{R}^3$ and every real $\varepsilon>0$, the set $\{g\in G:\operatorname{dist}(g(x),x)\leq\varepsilon\}$ is finite; and there exist three linearly independent vectors $v_1,v_2,v_3$ such that, for each $i$, some $g\in G$ satisfies $g(x)=x+v_i$ for every $x\in\mathbb{R}^3$. For $G,H\in\mathcal C$, write $G\sim H$ when there exists an invertible affine map $\varphi:\mathbb{R}^3\to\mathbb{R}^3$ such that $\{\varphi\circ g\circ\varphi^{-1}:g\in G\}=H$, viewing both sides as sets of affine maps, and write $G\sim_+H$ when such a $\varphi$ can be chosen with strictly positive determinant of its linear part. Let $\mathcal C_+$ consist of those $G\in\mathcal C$ for which every $g\in G$ has strictly positive determinant of its linear part. The theorem asserts simultaneously that the collection of distinct subsets $\{\{H\in\mathcal C:G\sim_+H\}:G\in\mathcal C\}$ has cardinality $230$, the collection $\{\{H\in\mathcal C:G\sim H\}:G\in\mathcal C\}$ has cardinality $219$, and the collection $\{\{H\in\mathcal C_+:G\sim_+H\}:G\in\mathcal C_+\}$ has cardinality $65$, where each cardinality is taken in the extended natural numbers, with every infinite set assigned cardinality $\infty$.