Finite Group DatabasebetaFinite Group DatabasebetaOrder 22050.
152 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 152 conjugacy classes of subgroups — too many to draw. Only its 28 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C7 | C3⋊#11760384 |
| commutator, cosocle, hypocenter | C3×C52×C7 | C2×C3×C7 |
| Frattini | C1 | G |
| Fitting, socle | C3×C52×C72 | C2×C3 |
| LCS | G ▷ C3×C52×C7 | |
| UCS | C1 ◁ C7 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×C72×C52⋊C3 | C2 | semidirect |
| 2. C7×C3⋊(C5⋊D35) | C3 | semidirect |
| 5. C72×C52⋊C3 | D3 | semidirect |
| 6. C3×C52×C72 | C2×C3 | semidirect |
| 7. C3⋊#11760384 | C7 | direct (N × H) |
| 11. C3×C7×C52⋊C3 | D7 | semidirect |
| 12. C3×C7×C52⋊C3 | C2×C7 | semidirect |
| 14. C52×C72 | C3×D3 | semidirect |
| 15. C3⋊(C5⋊D35) | C3×C7 | semidirect |
| 22. C3×C52×C7 | C3×D7 | semidirect |
| 23. C3×C52×C7 | C2×C3×C7 | semidirect |
| 28. C7×C52⋊C3 | D3×C7 | semidirect |
| 29. C7×C52⋊C3 | D21 | semidirect |
| 38. C3×C52⋊C3 | C7×D7 | semidirect |
| 42. C52×C7 | C3×D3×C7 | semidirect |
| 43. C52×C7 | C3×D21 | semidirect |
| 48. C3×C72 | (C5⋊D5)⋊C3 | semidirect |
| 55. C3×C52 | C3×C7×D7 | semidirect |
| 57. C52⋊C3 | C7×D21 | semidirect |
| 64. C72 | C3⋊((C5⋊D5)⋊C3) | semidirect |
| 73. C52 | C3×C7×D21 | semidirect |
| 81. C3×C7 | C7×(C5⋊D5)⋊C3 | semidirect |
| 82. C3×C7 | #11760384 | semidirect |
| 90. C7 | C7×C3⋊((C5⋊D5)⋊C3) | semidirect |
| 91. C7 | C3⋊#11760384 | direct (N × H) |
| 97. C3 | C7×#11760384 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C72×C52⋊C3 | 2 |
| C7×C3⋊(C5⋊D35) | 3 |
| C7×#11760384 | 3 |
| C3⋊#11760384 | 7 |
| C7×C3⋊((C5⋊D5)⋊C3) | 7 |
| C3×C7×D21 | 25 |