Finite Group DatabasebetaFinite Group DatabasebetaOrder 22932.
120 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 120 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | ΓL(1,𝔽8)×D273 |
| commutator, hypocenter | C3×C72×C13 | C3×C4 |
| Fitting, socle, cosocle | C2×C3×C72×C13 | C2×C3 |
| prime radical | C2×C3×C7×C13×ΓL(1,𝔽8) | C2 |
| LCS | G ▷ C3×C72×C13 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C7×Dic273 | C3 | semidirect |
| 4. C3×C7×C13×ΓL(1,𝔽8) | C4 | semidirect |
| 11. C3×C72×C13 | C3×C4 | semidirect |
| 13. C7×C13×ΓL(1,𝔽8) | Dic3 | semidirect |
| 18. Dic273 | ΓL(1,𝔽8) | direct (N × H) |
| 24. C3×C13×ΓL(1,𝔽8) | Dic7 | semidirect |
| 25. C72×C13 | C3×Dic3 | semidirect |
| 36. C3×C7×ΓL(1,𝔽8) | Dic13 | semidirect |
| 45. C3×C7×C13 | C4×ΓL(1,𝔽8) | semidirect |
| 46. C3×C7×C13 | C3×Dic7 | semidirect |
| 48. C13×ΓL(1,𝔽8) | Dic21 | semidirect |
| 58. C3×C72 | C3×Dic13 | semidirect |
| 60. C7×ΓL(1,𝔽8) | Dic39 | semidirect |
| 66. C7×C13 | Dic3×ΓL(1,𝔽8) | semidirect |
| 67. C7×C13 | C3×Dic21 | semidirect |
| 76. C3×ΓL(1,𝔽8) | Dic91 | semidirect |
| 78. C72 | C3×Dic39 | semidirect |
| 88. C3×C13 | ΓL(1,𝔽8)×Dic7 | semidirect |
| 96. C3×C7 | ΓL(1,𝔽8)×Dic13 | semidirect |
| 97. C3×C7 | C3×Dic91 | semidirect |
| 99. ΓL(1,𝔽8) | Dic273 | direct (N × H) |
| 104. C13 | ΓL(1,𝔽8)×Dic21 | semidirect |
| 109. C7 | ΓL(1,𝔽8)×Dic39 | semidirect |
| 110. C7 | C3×Dic273 | semidirect |
| 117. C3 | ΓL(1,𝔽8)×Dic91 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×C7×C13×ΓL(1,𝔽8) | 2 |
| C7×Dic273 | 3 |
| ΓL(1,𝔽8)×Dic91 | 3 |
| C3×Dic273 | 7 |
| ΓL(1,𝔽8)×Dic39 | 7 |
| ΓL(1,𝔽8)×Dic21 | 13 |