Finite Group DatabasebetaFinite Group DatabasebetaOrder 4704.
540 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 540 conjugacy classes of subgroups — too many to draw. Only its 96 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×ΓL(1,𝔽8)⋊D7 |
| commutator, socle, cosocle | C22×C72 | C23×C3 |
| Fitting | C2×C72×D4 | C2×C3 |
| hypercenter | C2×D4 | ΓL(1,𝔽8)⋊D7 |
| hypocenter | C72 | C3×#127 |
| LCS | G ▷ C22×C72 ▷ C72 | |
| UCS | C1 ◁ C22 ◁ C2×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C4×ΓL(1,𝔽8)⋊Dic7 | C2 | semidirect |
| 2. C2×#11924202 | C2 | semidirect |
| 3. C2×C4⋊(ΓL(1,𝔽8)⋊D7) | C2 | semidirect |
| 4. C2×C7×D4×ΓL(1,𝔽8) | C2 | semidirect |
| 5. #13096573 | C3 | semidirect |
| 13. C2×C72×D4 | C2×C3 | semidirect |
| 14. C2×C22⋊(C7⋊D7) | C2×C3 | semidirect |
| 15. C2×C4⋊(C7⋊D7) | C2×C3 | semidirect |
| 16. C4×C7⋊Dic7 | C2×C3 | semidirect |
| 24. C4×C7×ΓL(1,𝔽8) | D4 | semidirect |
| 25. ΓL(1,𝔽8)⋊Dic7 | D4 | semidirect |
| 36. C2×D4×ΓL(1,𝔽8) | D7 | semidirect |
| 53. C4×C72 | C3×D4 | semidirect |
| 54. C7⋊Dic7 | C3×D4 | semidirect |
| 69. C7×ΓL(1,𝔽8) | #127 | semidirect |
| 71. C2×C7×D4 | AGL(1,𝔽7) | semidirect |
| 72. C2×C7×D4 | C3×D7 | semidirect |
| 99. C4×ΓL(1,𝔽8) | C22⋊D7 | semidirect |
| 122. C72 | C3×#127 | semidirect |
| 152. C4×C7 | (C22⋊D7)⋊C3 | semidirect |
| 153. C4×C7 | C3×C22⋊D7 | semidirect |
| 165. ΓL(1,𝔽8) | #6503 | semidirect |
| 168. C2×D4 | ΓL(1,𝔽8)⋊D7 | semidirect |
| 197. C7 | #6503⋊C3 | semidirect |
| 198. C7 | C3×#6503 | semidirect |
| 209. C4 | #11924202 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×ΓL(1,𝔽8)⋊Dic7 | 2 |
| C2×#11924202 | 2 |
| C2×C4⋊(ΓL(1,𝔽8)⋊D7) | 2 |
| C2×C7×D4×ΓL(1,𝔽8) | 2 |
| #13096573 | 3 |
| C3×#6503 | 7 |
| #6503⋊C3 | 7 |