Finite Group DatabasebetaFinite Group DatabasebetaOrder 2352.
552 conjugacy classes of subgroups; only the 187 normal subgroups are shown.
This group has 552 conjugacy classes of subgroups — too many to draw. Only its 187 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D7×AGL(1,𝔽7) |
| commutator, cosocle, hypocenter | C72 | C24×C3 |
| Frattini | C1 | G |
| Fitting, socle | C22×C72 | C22×C3 |
| LCS | G ▷ C72 | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×D7×ΓL(1,𝔽8) | C2 | semidirect |
| 2. C22×ΓL(1,𝔽8)⋊D7 | C2 | semidirect |
| 3. C22×C7×AGL(1,𝔽7) | C2 | semidirect |
| 4. C2×D7×AGL(1,𝔽7) | C2 | direct (N × H) |
| 5. C22×D72 | C3 | semidirect |
| 6. C22×C7×ΓL(1,𝔽8) | C22 | semidirect |
| 7. C2×ΓL(1,𝔽8)⋊D7 | C22 | semidirect |
| 8. C2×D7×ΓL(1,𝔽8) | C22 | semidirect |
| 9. C2×C7×AGL(1,𝔽7) | C22 | semidirect |
| 10. D7×AGL(1,𝔽7) | C22 | direct (N × H) |
| 11. C22×C7⋊D7 | C2×C3 | semidirect |
| 12. C22×C7×D7 | C2×C3 | semidirect |
| 13. C22×C7×D7 | C2×C3 | semidirect |
| 14. C2×D72 | C2×C3 | semidirect |
| 17. C2×C7×ΓL(1,𝔽8) | C23 | semidirect |
| 18. ΓL(1,𝔽8)⋊D7 | C23 | semidirect |
| 19. D7×ΓL(1,𝔽8) | C23 | semidirect |
| 20. C7×AGL(1,𝔽7) | C23 | semidirect |
| 21. C22×C72 | C22×C3 | semidirect |
| 22. C2×C7⋊D7 | C22×C3 | semidirect |
| 23. C2×C7×D7 | C22×C3 | semidirect |
| 24. C2×C7×D7 | C22×C3 | semidirect |
| 25. D72 | C22×C3 | semidirect |
| 33. C22×AGL(1,𝔽7) | D7 | direct (N × H) |
| 34. C7×ΓL(1,𝔽8) | C24 | semidirect |
| 37. C2×C72 | C23×C3 | semidirect |
| 38. C7⋊D7 | C23×C3 | semidirect |
| 39. C7×D7 | C23×C3 | semidirect |
| 40. C7×D7 | C23×C3 | semidirect |
| 47. C22×ΓL(1,𝔽8) | C2×D7 | semidirect |
| 50. C2×AGL(1,𝔽7) | C2×D7 | direct (N × H) |
| 58. C22×D7 | AGL(1,𝔽7) | direct (N × H) |
| 59. C22×D7 | C3×D7 | semidirect |
| 60. C72 | C24×C3 | semidirect |
| 67. C2×ΓL(1,𝔽8) | C22×D7 | semidirect |
| 69. AGL(1,𝔽7) | C22×D7 | direct (N × H) |
| 73. C22×C7 | C2×AGL(1,𝔽7) | semidirect |
| 74. C22×C7 | C2×C3×D7 | semidirect |
| 80. C2×D7 | C2×AGL(1,𝔽7) | direct (N × H) |
| 81. C2×D7 | C2×C3×D7 | semidirect |
| 87. ΓL(1,𝔽8) | C23×D7 | semidirect |
| 92. C2×C7 | C22×AGL(1,𝔽7) | semidirect |
| 93. C2×C7 | C22×C3×D7 | semidirect |
| 97. D7 | C22×AGL(1,𝔽7) | direct (N × H) |
| 98. D7 | C22×C3×D7 | semidirect |
| 109. C7 | C23×AGL(1,𝔽7) | semidirect |
| 110. C7 | C23×C3×D7 | semidirect |
| 119. C22 | D7×AGL(1,𝔽7) | direct (N × H) |
| 124. C2 | C2×D7×AGL(1,𝔽7) | direct (N × H) |
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 |
| C22×ΓL(1,𝔽8)⋊D7 | 2 |
| C22×C7×AGL(1,𝔽7) | 2 |
| C2×D7×AGL(1,𝔽7) | 2 |
| C22×D72 | 3 |
| C23×C3×D7 | 7 |
| C23×AGL(1,𝔽7) | 7 |