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In this paper we study the orbits of the Mathieu group      M 24 {M_{24}}       on sets of  n  points,      1 ≦  n ≦  12 1 \leqq n \leqq 12      . For      n ≧  6 , M 24 n \geqq 6,{M_{24}}       is not transitive on these sets, so we may classify the sets into types corresponding to the orbits of      M 24 {M_{24}}       and then show how to construct a set of each type from smaller sets. We determine the stabilizer of a set of each type and describe its representation on the 24 points. From the conclusions, the class of subgroups which are maximal among the intransitives of      M 24 {M_{24}}       can be read off. This work forms the first part of a study which yields, in particular, a complete list of the primitive representations of      M 24 {M_{24}}      .

On subgroups of 𝑀₂₄. I. Stabilizers of subsets