Finite Group DatabasebetaFinite Group DatabasebetaOrder 672.
260 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 63 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×AGL(1,𝔽7) |
| commutator, socle, cosocle | C22×C7 | C23×C3 |
| Fitting | C2×C7×D4 | C2×C3 |
| hypercenter | C2×D4 | AGL(1,𝔽7) |
| hypocenter | C7 | C3×#120 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C2×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×AGL(1,𝔽7) | C2 | semidirect |
| 2. C2×D4×ΓL(1,𝔽8) | C2 | semidirect |
| 3. C2×(C22⋊D7)⋊C3 | C2 | semidirect |
| 4. (C22⋊Dic7)⋊C3 | C2 | semidirect |
| 5. (C2×AGL(1,𝔽7))⋊C4 | C2 | semidirect |
| 6. #6500 | C3 | semidirect |
| 8. C23×ΓL(1,𝔽8) | C22 | semidirect |
| 13. C2×C4×ΓL(1,𝔽8) | C22 | semidirect |
| 15. C2×ΓL(1,𝔽8)⋊C4 | C22 | semidirect |
| 16. C23×D7 | C2×C3 | semidirect |
| 17. C2×C7×D4 | C2×C3 | semidirect |
| 18. C2×C22⋊D7 | C2×C3 | semidirect |
| 19. C22⋊Dic7 | C2×C3 | semidirect |
| 20. (C2×D7)⋊C4 | C2×C3 | semidirect |
| 25. C22×ΓL(1,𝔽8) | D4 | semidirect |
| 30. C2×AGL(1,𝔽7) | D4 | semidirect |
| 34. C23×C7 | C22×C3 | semidirect |
| 39. C2×C4×C7 | C22×C3 | semidirect |
| 41. C2×Dic7 | C22×C3 | semidirect |
| 56. C22×C7 | C3×D4 | semidirect |
| 61. C2×D7 | C3×D4 | semidirect |
| 73. ΓL(1,𝔽8) | #120 | semidirect |
| 76. C2×D4 | AGL(1,𝔽7) | semidirect |
| 99. C23 | C2×AGL(1,𝔽7) | semidirect |
| 103. C2×C4 | C2×AGL(1,𝔽7) | semidirect |
| 104. C7 | C3×#120 | semidirect |
| 117. C22 | (C22⋊D7)⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×AGL(1,𝔽7) | 2 |
| C2×D4×ΓL(1,𝔽8) | 2 |
| C2×(C22⋊D7)⋊C3 | 2 |
| (C22⋊Dic7)⋊C3 | 2 |
| (C2×AGL(1,𝔽7))⋊C4 | 2 |
| #6500 | 3 |
| C3×#120 | 7 |