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A method is obtained for counting the normal subgroups N of a noneuclidean crystallographic group À without reflections, with a given finite quotient group ÀaN; this has applications to the enumeration of regular coverings of orbifolds. The method, which involves Mo« bius inversion and character theory, is also applied to count normal surface subgroups and non-normal subgroups of finite index in À .doi:10.7146/math.scand.a-13930 fatcat:myp4wyhbrbazbbvwkbdri4khcy