Finite Group DatabasebetaFinite Group DatabasebetaOrder 22050.
114 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 114 conjugacy classes of subgroups — too many to draw. Only its 39 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 | C3×C72×C25 | C2×C3 |
| Frattini | C5 | ΓL(1,𝔽8)×D105 |
| socle | C3×C5×C72 | C3×D5 |
| LCS | G ▷ C3×C72×C25 | |
| UCS | C1 | |
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. C7×D525 | C3 | semidirect |
| 6. C7×ΓL(1,𝔽8)×C25 | D3 | semidirect |
| 7. C3×C72×C25 | C2×C3 | semidirect |
| 13. C3×ΓL(1,𝔽8)×C25 | D7 | semidirect |
| 16. C72×C25 | C3×D3 | semidirect |
| 17. D525 | ΓL(1,𝔽8) | direct (N × H) |
| 30. C3×C7×C25 | C2×ΓL(1,𝔽8) | semidirect |
| 31. C3×C7×C25 | C3×D7 | semidirect |
| 33. ΓL(1,𝔽8)×C25 | D21 | semidirect |
| 36. C3×C7×ΓL(1,𝔽8) | D25 | semidirect |
| 50. C7×C25 | D3×ΓL(1,𝔽8) | semidirect |
| 51. C7×C25 | C3×D21 | semidirect |
| 54. C3×C72 | C3×D25 | semidirect |
| 56. C7×ΓL(1,𝔽8) | D75 | semidirect |
| 70. C3×C25 | D7×ΓL(1,𝔽8) | semidirect |
| 74. C3×ΓL(1,𝔽8) | D175 | semidirect |
| 76. C72 | C3×D75 | semidirect |
| 87. C25 | ΓL(1,𝔽8)×D21 | semidirect |
| 91. C3×C7 | ΓL(1,𝔽8)×D25 | semidirect |
| 92. C3×C7 | C3×D175 | semidirect |
| 94. ΓL(1,𝔽8) | D525 | direct (N × H) |
| 104. C7 | ΓL(1,𝔽8)×D75 | semidirect |
| 105. C7 | C3×D525 | semidirect |
| 111. C3 | ΓL(1,𝔽8)×D175 | semidirect |
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 |
| C7×D525 | 3 |
| ΓL(1,𝔽8)×D175 | 3 |
| ΓL(1,𝔽8)×D105 | 5 |
| C3×D525 | 7 |
| ΓL(1,𝔽8)×D75 | 7 |