Enumeration of Steiner triple systems with subsystems

A Steiner triple system of order υ, an STS(v), is a set of 3-element subsets, called blocks, of a v-element set of points, such that every pair of distinct points occurs in exactly one block. A subsystem of order w in an STS(v), a sub-STS(w), is a subset of blocks that forms an STS(w). Constructive and nonconstructive techniques for enumerating up to isomorphism the STS(v) that admit at least one sub-STS(w) are presented here for general parameters v and w. The techniques are further applied to show that the number of isomorphism classes of STS(21)s with at least one sub-STS(9) is 12661527336 and of STS(27)s with a sub-STS(13) is 1356574942538935943268083236.