Finite Group DatabasebetaFinite Group DatabasebetaOrder 22050.
168 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 168 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | ΓL(1,𝔽8)⋊D175 |
| commutator, cosocle, hypocenter | C72×C25 | C2×C32 |
| Frattini | C5 | C3×C5⋊(ΓL(1,𝔽8)⋊D7) |
| Fitting | C3×C72×C25 | C2×C3 |
| socle | C3×C5×C72 | C3×D5 |
| LCS | G ▷ C72×C25 | |
| UCS | C1 ◁ C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×C7×ΓL(1,𝔽8)×C25 | C2 | semidirect |
| 2. C3×C7⋊D175 | C3 | semidirect |
| 3. ΓL(1,𝔽8)⋊D175 | C3 | direct (N × H) |
| 5. C7×ΓL(1,𝔽8)×C25 | C2×C3 | semidirect |
| 6. C3×C72×C25 | C2×C3 | semidirect |
| 9. C7⋊D175 | C32 | semidirect |
| 12. C3×ΓL(1,𝔽8)×C25 | D7 | semidirect |
| 15. C72×C25 | C2×C32 | semidirect |
| 28. C3×C7×C25 | AGL(1,𝔽7) | semidirect |
| 29. C3×C7×C25 | C3×D7 | semidirect |
| 30. ΓL(1,𝔽8)×C25 | C3×D7 | semidirect |
| 33. C3×C7×ΓL(1,𝔽8) | D25 | semidirect |
| 49. C7×C25 | C3×AGL(1,𝔽7) | semidirect |
| 50. C7×C25 | C32×D7 | semidirect |
| 53. C3×C72 | C3×D25 | semidirect |
| 54. C7×ΓL(1,𝔽8) | C3×D25 | semidirect |
| 65. C3×C25 | ΓL(1,𝔽8)⋊D7 | semidirect |
| 70. C3×ΓL(1,𝔽8) | D175 | semidirect |
| 72. C72 | C32×D25 | semidirect |
| 84. C25 | C3×ΓL(1,𝔽8)⋊D7 | semidirect |
| 87. C3×C7 | ΓL(1,𝔽8)⋊D25 | semidirect |
| 88. C3×C7 | C3×D175 | semidirect |
| 89. ΓL(1,𝔽8) | C3×D175 | semidirect |
| 99. C7 | C3×ΓL(1,𝔽8)⋊D25 | semidirect |
| 100. C7 | C32×D175 | semidirect |
| 105. C3 | ΓL(1,𝔽8)⋊D175 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C7×ΓL(1,𝔽8)×C25 | 2 |
| C3×C7⋊D175 | 3 |
| ΓL(1,𝔽8)⋊D175 | 3 |
| C3×C5⋊(ΓL(1,𝔽8)⋊D7) | 5 |
| C32×D175 | 7 |
| C3×ΓL(1,𝔽8)⋊D25 | 7 |