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The smallest open case for classifying Steiner triple systems is order 21. ASteiner triple system of order 21, an STS(21), can have subsystems of orders 7and 9, and it is known that there are 12,661,527,336 isomorphism classes ofSTS(21)s with sub-STS(9)s. Here, the classification of STS(21)s with subsystemsis completed by settling the case of STS(21)s with sub-STS(7)s. There are116,635,963,205,551 isomorphism classes of such systems. An estimation of thenumber of isomorphism classes of STS(21)s is given.

Steiner Triple Systems of Order 21 with Subsystems