Abstract: For a maximal subgroup M of a finite group G ,a θ subgroup for M is any subgroup C of G such that ... and Core G(M∩C) is maximal among normal subgroups of G properly contained in C . This concept is motivated by the concept of θ pairs which was introduced by N .P.Mukherjee and P.Bhattacharya. The present paper represents an attempt to carry the studies on θ pairs further.We study intrinsic properties of a given family of maximal subgroups and its associate θ subgroups which imply a group to be solvable, π solvable and supersolvable. All of these results are effective improvements of known relative results on θ pairs.