For q a prime-power, let IP ( q ) {\text {IP}}(q) denote the miquelian inversive plane of order q. The classification of certain translation planes of order q 2 {q^2} , called subregular, has been reduced to the classification of sets of disjoint circles in IP ( q ) {\text {IP}}(q) . While R. H. Bruck has extensively studied triples of disjoint circles, this paper is concerned with sets of four or more circles in IP ( q ) {\text {IP}}(q) . In a previous paper, the author has shown (for odd q) that the number of quadruples of disjoint circles in IP ( q ) {\text {IP}}(q) is asymptotic to q 12 / 1536 {q^{12}}/1536 . Hence a judicious approach to the classification problem is to study “interesting” quadruples. In general, let C 1 , … , C n {C_1}, \ldots ,{C_n} be a nonlinear set of n disjoint circles in IP ( q ) {\text {IP}}(q) . Let H be the subgroup of the collineation group of IP ( q ) {\text {IP}}(q) composed of collineations that permute the C i {C_i} ’s among themselves, and let K be that subgroup composed of collineations fixing each of the C i ′ s {C_i}’s . An interesting set of n disjoint circles would be one for which K = 1 K = 1 . It is shown that K = 1 K = 1 if and only if ( ∗ ) { (i) there does not exist a circle D orthogonal to each of the given n circles, and (ii) we do not have one circle in our set orthogonal to each of the other n − 1 circles . \begin{equation}\tag {$*$} \left \{ {\begin {array}{*{20}{c}} {{\text {(i)}}\;{\text {there}}\;{\text {does}}\;{\text {not}}\;{\text {exist}}\;{\text {a}}\;{\text {circle}}\;D\;{\text {orthogonal}}\;{\text {to}}} \\ {{\text {each}}\;{\text {of}}\;{\text {the}}\;{\text {given}}\;n\;{\text {circles,}}\;{\text {and}}} \\ {{\text {(ii)}}\;{\text {we}}\;{\text {do}}\;{\text {not}}\;{\text {have}}\;{\text {one}}\;{\text {circle}}\;{\text {in}}\;{\text {our}}\;{\text {set}}\;{\text {orthogonal}}} \\ {{\text {to}}\;{\text {each}}\;{\text {of}}\;{\text {the}}\;{\text {other}}\;n - 1\;{\text {circles}}{\text {.}}} \\ \end{array} } \right .\end{equation} When n = 4 n = 4 and under mild restrictions on q, an algorithm is developed that finds all nonlinear quadruples of disjoint circles satisfying the orthogonality conditions ( ∗ ) ( \ast ) and having nontrivial group H. Given such a quadruple, the algorithm determines exactly what group H is acting. It is also shown that most quadruples in IP ( q ) {\text {IP}}(q) , for large q, do indeed satisfy the conditions ( ∗ ) ( \ast ) . In addition, the cases when n = 5 , 6 , n = 5,6, or 7 are explored to a lesser degree.