Finite Group DatabasebetaFinite Group DatabasebetaOrder 22050.
320 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 320 conjugacy classes of subgroups — too many to draw. Only its 52 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 | #20301302 |
| commutator, cosocle, hypocenter | C52×C72 | C2×C32 |
| Frattini | C1 | G |
| Fitting, socle | C3×C52×C72 | C2×C3 |
| LCS | G ▷ C52×C72 | |
| UCS | C1 ◁ C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×C52⋊#3861 | C2 | semidirect |
| 2. C3×C52×C7⋊D7 | C3 | semidirect |
| 3. #20301302 | C3 | direct (N × H) |
| 4. C52⋊#3861 | C2×C3 | semidirect |
| 5. C3×C52×C72 | C2×C3 | semidirect |
| 9. C52×C7⋊D7 | C32 | semidirect |
| 14. C52×C72 | C2×C32 | semidirect |
| 23. C3×C52×C7 | AGL(1,𝔽7) | semidirect |
| 24. C3×C52×C7 | AGL(1,𝔽7) | semidirect |
| 25. C3×C52×C7 | AGL(1,𝔽7) | semidirect |
| 35. C3×C7⋊D7 | C52⋊C3 | semidirect |
| 42. C52×C7 | C3×AGL(1,𝔽7) | semidirect |
| 43. C52×C7 | C3×AGL(1,𝔽7) | semidirect |
| 44. C52×C7 | C3×AGL(1,𝔽7) | semidirect |
| 47. C3×C72 | C2×C52⋊C3 | semidirect |
| 55. C7⋊D7 | C3×C52⋊C3 | semidirect |
| 56. C3×C52 | #64424 | semidirect |
| 65. C72 | C2×C3×C52⋊C3 | semidirect |
| 76. C52 | C3×#64424 | semidirect |
| 77. C3×C7 | D7⋊(C52⋊C3) | semidirect |
| 78. C3×C7 | D7⋊(C52⋊C3) | semidirect |
| 79. C3×C7 | D7⋊(C52⋊C3) | semidirect |
| 90. C7 | C3×D7⋊(C52⋊C3) | semidirect |
| 91. C7 | C3×D7⋊(C52⋊C3) | semidirect |
| 92. C7 | C3×D7⋊(C52⋊C3) | semidirect |
| 97. C3 | #20301302 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C52⋊#3861 | 2 |
| C3×C52×C7⋊D7 | 3 |
| #20301302 | 3 |
| C3×D7⋊(C52⋊C3) | 7 |
| C3×#64424 | 25 |