(See lower down for the definitions of these groups.) If a group of direct isometries "H" has a subgroup "L" of index 2, then, apart from the corresponding group containing inversion there is also a corresponding group that contains indirect isometries but no inversion: where isometry ( "A", "I" ) is identified with "A". An example would be "C"4 for "H" and "S"4 for "M". Thus "M" is obtained from "H" by inverting the isometries in . This group "M" is as abstract group isomorphic with "H". Conversely, for all isometry groups that contain indirect isometries but no inversion we can obtain a rotation group by inverting the indirect isometries. This is clarifying when categorizing isometry groups, see below. In 2D the cyclic group of "k"-fold rotations "Ck" is for every positive integer "k" a normal subgroup of O(2,R) and SO(2,R). Accordingly, in 3D, for every axis the cyclic group of "k"-fold rotations about that axis is a normal subgroup of the group of all rotations about that axis. Since any subgroup of index two is normal, the group of rotations ("Cn") is normal both in the group obtained by adding reflections in planes through the axis ("Cnv") and in the group obtained by adding a reflection plane perpendicular to the axis ("Cnh"). The isometries of R3 that leave the origin fixed, forming the group O(3,R), can be categorized as follows:  The 4th and 5th in particular, and in a wider sense the 6th also, are called improper rotations. See also the similar overview including translations. When comparing the symmetry type of two objects, the origin is chosen for each separately, i.e. they need not have the same center. Moreover, two objects are considered to be of the same symmetry type if their symmetry groups are conjugate subgroups of O(3) (two subgroups "H"1, "H"2 of a group "G" are "conjugate", if there exists "g" ∈ "G" such that "H"1 = "g"−1"H"2"g" ). For example, two 3D objects have the same symmetry type: In the case of multiple mirror planes and/or axes of rotation, two symmetry groups are of the same symmetry type if and only if there is a rotation mapping the whole structure of the first symmetry group to that of the second.