Finite Group DatabasebetaFinite Group DatabasebetaOrder 2016.
136 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 24 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C42×C7 | C2×C32 |
| Frattini | C22 | A4×AGL(1,𝔽7) |
| socle | C22×C7 | C2×C3×A4 |
| LCS | G ▷ C42×C7 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. ΓL(1,𝔽8)×C42⋊C3 | C2 | semidirect |
| 2. D7⋊(C42⋊C3) | C3 | semidirect |
| 3. D7×C42⋊C3 | C3 | semidirect |
| 4. C42×AGL(1,𝔽7) | C3 | semidirect |
| 7. C7⋊(C42⋊C3) | C2×C3 | semidirect |
| 8. C7×C42⋊C3 | C2×C3 | semidirect |
| 9. C42×ΓL(1,𝔽8) | C2×C3 | semidirect |
| 13. C42×D7 | C32 | semidirect |
| 26. C42×C7 | C2×C32 | semidirect |
| 53. C42⋊C3 | AGL(1,𝔽7) | direct (N × H) |
| 58. AGL(1,𝔽7) | C42⋊C3 | direct (N × H) |
| 79. ΓL(1,𝔽8) | C2×C42⋊C3 | semidirect |
| 83. C42 | C3×AGL(1,𝔽7) | semidirect |
| 86. D7 | C3×C42⋊C3 | semidirect |
| 104. C7 | C2×C3×C42⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| ΓL(1,𝔽8)×C42⋊C3 | 2 |
| D7⋊(C42⋊C3) | 3 |
| D7×C42⋊C3 | 3 |
| C42×AGL(1,𝔽7) | 3 |
| A4×AGL(1,𝔽7) | 4 |
| C2×C3×C42⋊C3 | 7 |