Finite Group DatabasebetaFinite Group DatabasebetaOrder 2352.
224 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D7×ΓL(1,𝔽8) |
| commutator, socle, cosocle | C2×C72 | C23×C3 |
| Fitting | C72×D4 | C2×C3 |
| hypercenter | D4 | D7×ΓL(1,𝔽8) |
| hypocenter | C72 | C2×C3×D4 |
| LCS | G ▷ C2×C72 ▷ C72 | |
| UCS | C1 ◁ C2 ◁ D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×D7×ΓL(1,𝔽8) | C2 | semidirect |
| 2. C7×D4×ΓL(1,𝔽8) | C2 | semidirect |
| 3. ΓL(1,𝔽8)×C22⋊D7 | C2 | semidirect |
| 4. ΓL(1,𝔽8)×D28 | C2 | semidirect |
| 5. C4×D7×ΓL(1,𝔽8) | C2 | semidirect |
| 6. C7×D4×D7 | C3 | semidirect |
| 7. C22×C7×ΓL(1,𝔽8) | C22 | semidirect |
| 10. C2×D7×ΓL(1,𝔽8) | C22 | semidirect |
| 11. C4×C7×ΓL(1,𝔽8) | C22 | semidirect |
| 12. ΓL(1,𝔽8)×Dic7 | C22 | semidirect |
| 13. C22×C7×D7 | C2×C3 | semidirect |
| 14. C72×D4 | C2×C3 | semidirect |
| 15. C7×C22⋊D7 | C2×C3 | semidirect |
| 16. C7×D28 | C2×C3 | semidirect |
| 17. C4×C7×D7 | C2×C3 | semidirect |
| 23. D7×ΓL(1,𝔽8) | D4 | direct (N × H) |
| 24. C22×C72 | C22×C3 | semidirect |
| 27. C2×C7×D7 | C22×C3 | semidirect |
| 28. C4×C72 | C22×C3 | semidirect |
| 29. C7×Dic7 | C22×C3 | semidirect |
| 38. D4×ΓL(1,𝔽8) | D7 | direct (N × H) |
| 40. C7×ΓL(1,𝔽8) | C2×D4 | semidirect |
| 42. D4×D7 | ΓL(1,𝔽8) | direct (N × H) |
| 46. C7×D7 | C3×D4 | semidirect |
| 54. C22×ΓL(1,𝔽8) | C2×D7 | semidirect |
| 58. C4×ΓL(1,𝔽8) | C2×D7 | semidirect |
| 60. C22×D7 | C2×ΓL(1,𝔽8) | semidirect |
| 64. C7×D4 | C2×ΓL(1,𝔽8) | semidirect |
| 65. C7×D4 | C3×D7 | semidirect |
| 67. C22⋊D7 | C2×ΓL(1,𝔽8) | semidirect |
| 68. D28 | C2×ΓL(1,𝔽8) | semidirect |
| 69. C4×D7 | C2×ΓL(1,𝔽8) | semidirect |
| 70. C72 | C2×C3×D4 | semidirect |
| 84. C22×C7 | C22×ΓL(1,𝔽8) | semidirect |
| 85. C22×C7 | C2×C3×D7 | semidirect |
| 88. C2×D7 | C22×ΓL(1,𝔽8) | semidirect |
| 91. C4×C7 | C22×ΓL(1,𝔽8) | semidirect |
| 92. C4×C7 | C2×C3×D7 | semidirect |
| 93. Dic7 | C22×ΓL(1,𝔽8) | semidirect |
| 100. ΓL(1,𝔽8) | D4×D7 | direct (N × H) |
| 111. D7 | D4×ΓL(1,𝔽8) | direct (N × H) |
| 121. D4 | D7×ΓL(1,𝔽8) | direct (N × H) |
| 124. C7 | C2×D4×ΓL(1,𝔽8) | semidirect |
| 125. C7 | C3×D4×D7 | semidirect |
| 133. C22 | C2×D7×ΓL(1,𝔽8) | semidirect |
| 135. C4 | C2×D7×ΓL(1,𝔽8) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×D7×ΓL(1,𝔽8) | 2 |
| C7×D4×ΓL(1,𝔽8) | 2 |
| ΓL(1,𝔽8)×C22⋊D7 | 2 |
| ΓL(1,𝔽8)×D28 | 2 |
| C4×D7×ΓL(1,𝔽8) | 2 |
| C7×D4×D7 | 3 |
| C3×D4×D7 | 7 |
| C2×D4×ΓL(1,𝔽8) | 7 |