the orders of subgroup products and coset products

A sect is a subset of a group given by the product of a finite number of subgroups. it is generally not a direct product nor even a subgroup of the group. for finite groups, the orders of sects are their basic invariants. in this paper we describe properties of the orders of sects, such as divisibility and inequalities, which give constraints on the possible values of the orders of sects. we further consider clans, which are subsets of groups given by products of finite numbers of cosets. we also describe properties of their orders.