Finite Group DatabasebetaFinite Group DatabasebetaOrder 1008.
140 conjugacy classes of subgroups; only the 22 normal subgroups are shown.
This group has 140 conjugacy classes of subgroups — too many to draw. Only its 22 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C7×A4 | C22×C3 |
| Fitting, socle | C22×C7 | C2×C3×D3 |
| LCS | G ▷ C7×A4 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. A4×AGL(1,𝔽7) | C2 | semidirect |
| 2. A4⋊AGL(1,𝔽7) | C2 | semidirect |
| 3. ΓL(1,𝔽8)×S4 | C2 | semidirect |
| 4. D7×S4 | C3 | semidirect |
| 6. A4×ΓL(1,𝔽8) | C22 | semidirect |
| 9. A4×D7 | C2×C3 | semidirect |
| 11. C22×AGL(1,𝔽7) | D3 | semidirect |
| 12. C7⋊S4 | C2×C3 | semidirect |
| 13. C7×S4 | C2×C3 | semidirect |
| 25. C7×A4 | C22×C3 | semidirect |
| 27. C22×ΓL(1,𝔽8) | C2×D3 | semidirect |
| 41. C22×D7 | C3×D3 | semidirect |
| 57. AGL(1,𝔽7) | S4 | direct (N × H) |
| 61. C22×C7 | C2×C3×D3 | semidirect |
| 74. S4 | AGL(1,𝔽7) | direct (N × H) |
| 82. ΓL(1,𝔽8) | C2×S4 | semidirect |
| 91. D7 | C3×S4 | semidirect |
| 101. A4 | C2×AGL(1,𝔽7) | semidirect |
| 112. C7 | C2×C3×S4 | semidirect |
| 128. C22 | D3×AGL(1,𝔽7) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| A4×AGL(1,𝔽7) | 2 |
| A4⋊AGL(1,𝔽7) | 2 |
| ΓL(1,𝔽8)×S4 | 2 |
| D7×S4 | 3 |
| D4×AGL(1,𝔽7) | 3 |
| D3×AGL(1,𝔽7) | 4 |
| C2×C3×S4 | 7 |