The influence of numerical bounds for certain invariants of a group on its subgroup structure has been investigated by generations of group-theorists. Likewise, CLT groups, i.e. groups satisfying the converse of the Lagrange Theorem, have been subjected to longstanding enquiry. The present paper provides fascinating links between these two streams of research. Much ``footwork'' is done throughout the preliminary results (Section 3) leading to the main results. There are five invariants under consideration: the order of the centre and that of the commutator subgroup, the number of conjugacy classes, the sum of the degrees of the irreducible complex representations and the largest number of group elements that are inverted by a single automorphism. Section 4 consists of twenty-one estimates put in sometimes surprising theorems. Here is a random choice. Theorem 4.4: If \(|G|\) is odd and \(|{G\over Z(G)}|>{1\over 75}\), then \(G\) is a CLT group. Theorem 4.8: Let \(|G|\) be odd and suppose that \({1\over|G'|}>{1\over 25}\) (= 0.0400). Then \(G\) is supersoluble. Theorem 4.18: If twice the largest number of elements of \(G\) that are inverted by a single automorphism is larger than \(|G|\), then \(G\) is supersoluble. In many ways, the unique non-Abelian group of order 75, which is without a subgroup of order 15, plays a central role in the investigation.