Finite Group DatabasebetaFinite Group DatabasebetaOrder 1120.
332 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 332 conjugacy classes of subgroups — too many to draw. Only its 119 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C23×D35 |
| commutator, socle, cosocle | C2×C5×C7 | C24 |
| Fitting | C5×C7×#41 | C2 |
| hypercenter | #41 | D35 |
| hypocenter | C5×C7 | H(5,𝔽2) |
| LCS | G ▷ C2×C5×C7 ▷ C5×C7 | |
| UCS | C1 ◁ C2 ◁ #41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C5×C7×#41 | C2 | semidirect |
| 2. #10587991 | C2 | semidirect |
| 3. D4×D35 | C2 | semidirect |
| 4. #10587987 | C2 | semidirect |
| 5. C2×D140 | C2 | semidirect |
| 6. C22×D35 | C22 | semidirect |
| 7. C5×C7×Q8 | C22 | semidirect |
| 8. C5×C7×D4 | C22 | semidirect |
| 9. C2×C4×C5×C7 | C22 | semidirect |
| 10. C22⋊D35 | C22 | semidirect |
| 11. D140 | C22 | semidirect |
| 12. D140 | C22 | semidirect |
| 13. C4×D35 | C22 | semidirect |
| 14. Dic70 | C22 | semidirect |
| 23. C7×#41 | D5 | semidirect |
| 28. C5×#41 | D7 | semidirect |
| 37. C7×Q8 | C2×D5 | semidirect |
| 38. C7×D4 | C2×D5 | semidirect |
| 39. C2×C4×C7 | C2×D5 | semidirect |
| 46. C5×Q8 | C2×D7 | semidirect |
| 47. C5×D4 | C2×D7 | semidirect |
| 48. C2×C4×C5 | C2×D7 | semidirect |
| 54. C5×C7 | H(5,𝔽2) | semidirect |
| 70. #41 | D35 | semidirect |
| 81. Q8 | C2×D35 | semidirect |
| 85. D4 | C2×D35 | semidirect |
| 87. C2×C4 | C2×D35 | semidirect |
| 88. C7 | #4149 | semidirect |
| 89. C5 | #6553 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×C7×#41 | 2 |
| #10587991 | 2 |
| D4×D35 | 2 |
| #10587987 | 2 |
| C2×D140 | 2 |
| #6553 | 5 |
| #4149 | 7 |