Finite Group DatabasebetaFinite Group DatabasebetaOrder 2352.
224 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 61 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 | C2×D7×AGL(1,𝔽7) |
| commutator, socle, cosocle | C2×C72 | C23×C3 |
| Fitting | C22×C72 | C22×C3 |
| hypercenter | C22 | D7×AGL(1,𝔽7) |
| hypocenter | C72 | C2×C3×D4 |
| LCS | G ▷ C2×C72 ▷ C72 | |
| UCS | C1 ◁ C2 ◁ 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. C2×D7×AGL(1,𝔽7) | C2 | semidirect |
| 3. #11924202 | C2 | semidirect |
| 4. C7×(C22⋊D7)⋊C3 | C2 | semidirect |
| 5. #11924170 | C2 | semidirect |
| 6. #11924169 | C2 | semidirect |
| 7. D7×ΓL(1,𝔽8)⋊C4 | C2 | semidirect |
| 8. D7×C22⋊D7 | C3 | semidirect |
| 9. C22×C7×ΓL(1,𝔽8) | C22 | semidirect |
| 10. C2×ΓL(1,𝔽8)⋊D7 | C22 | semidirect |
| 13. C2×D7×ΓL(1,𝔽8) | C22 | semidirect |
| 14. C2×C7×AGL(1,𝔽7) | C22 | semidirect |
| 16. ΓL(1,𝔽8)⋊Dic7 | C22 | semidirect |
| 17. C7×ΓL(1,𝔽8)⋊C4 | C22 | semidirect |
| 18. C22×C7×D7 | C2×C3 | semidirect |
| 19. C2×D72 | C2×C3 | semidirect |
| 20. C22⋊(C7⋊D7) | C2×C3 | semidirect |
| 21. C7×C22⋊D7 | C2×C3 | semidirect |
| 22. #87641 | C2×C3 | semidirect |
| 23. #87640 | C2×C3 | semidirect |
| 24. D7×Dic7 | C2×C3 | semidirect |
| 31. D7×ΓL(1,𝔽8) | D4 | semidirect |
| 33. C22×C72 | C22×C3 | semidirect |
| 34. C2×C7⋊D7 | C22×C3 | semidirect |
| 37. C2×C7×D7 | C22×C3 | semidirect |
| 38. C2×C7×D7 | C22×C3 | semidirect |
| 40. C7⋊Dic7 | C22×C3 | semidirect |
| 41. C7×Dic7 | C22×C3 | semidirect |
| 54. (C22⋊D7)⋊C3 | D7 | direct (N × H) |
| 56. C7×ΓL(1,𝔽8) | C2×D4 | semidirect |
| 64. C7×D7 | C3×D4 | semidirect |
| 75. C22×ΓL(1,𝔽8) | C2×D7 | semidirect |
| 78. C2×AGL(1,𝔽7) | C2×D7 | semidirect |
| 82. ΓL(1,𝔽8)⋊C4 | C2×D7 | semidirect |
| 86. C22×D7 | AGL(1,𝔽7) | semidirect |
| 94. C22⋊D7 | C3×D7 | semidirect |
| 98. C72 | C2×C3×D4 | semidirect |
| 117. C22×C7 | C2×AGL(1,𝔽7) | semidirect |
| 118. C22×C7 | C2×C3×D7 | semidirect |
| 126. C2×D7 | C2×AGL(1,𝔽7) | semidirect |
| 127. C2×D7 | C2×C3×D7 | semidirect |
| 132. Dic7 | C2×C3×D7 | semidirect |
| 141. ΓL(1,𝔽8) | D4×D7 | semidirect |
| 157. D7 | (C22⋊D7)⋊C3 | direct (N × H) |
| 175. C7 | C2×(C22⋊D7)⋊C3 | semidirect |
| 176. C7 | C3×D4×D7 | semidirect |
| 189. C22 | D7×AGL(1,𝔽7) | 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 |
| C2×D7×AGL(1,𝔽7) | 2 |
| #11924202 | 2 |
| C7×(C22⋊D7)⋊C3 | 2 |
| #11924170 | 2 |
| #11924169 | 2 |
| D7×ΓL(1,𝔽8)⋊C4 | 2 |
| D7×C22⋊D7 | 3 |
| C3×D4×D7 | 7 |
| C2×(C22⋊D7)⋊C3 | 7 |