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 | C2×C5×C7×D4 | C2 |
| hypercenter | C2×D4 | D35 |
| hypocenter | C5×C7 | H(5,𝔽2) |
| LCS | G ▷ C2×C5×C7 ▷ C5×C7 | |
| UCS | C1 ◁ C2 ◁ C2×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C5×C7×D4 | C2 | semidirect |
| 2. C2×C22⋊D35 | C2 | semidirect |
| 3. #10587989 | C2 | semidirect |
| 4. D4×D35 | C2 | semidirect |
| 5. #10587987 | C2 | semidirect |
| 6. C23×C5×C7 | C22 | semidirect |
| 7. C22×D35 | C22 | semidirect |
| 8. C5×C7×D4 | C22 | semidirect |
| 9. C2×C4×C5×C7 | C22 | semidirect |
| 10. C22⋊D35 | C22 | semidirect |
| 11. C22⋊D35 | C22 | semidirect |
| 12. C2×Dic35 | C22 | semidirect |
| 13. D140 | C22 | semidirect |
| 14. C4×D35 | C22 | semidirect |
| 15. Dic70 | C22 | semidirect |
| 25. C2×C7×D4 | D5 | semidirect |
| 30. C2×C5×D4 | D7 | semidirect |
| 39. C23×C7 | C2×D5 | semidirect |
| 41. C7×D4 | C2×D5 | semidirect |
| 42. C2×C4×C7 | C2×D5 | semidirect |
| 49. C23×C5 | C2×D7 | semidirect |
| 51. C5×D4 | C2×D7 | semidirect |
| 52. C2×C4×C5 | C2×D7 | semidirect |
| 59. C5×C7 | H(5,𝔽2) | semidirect |
| 79. C2×D4 | D35 | semidirect |
| 89. C23 | C2×D35 | semidirect |
| 94. D4 | C2×D35 | semidirect |
| 97. C2×C4 | C2×D35 | semidirect |
| 98. C7 | #4144 | semidirect |
| 99. C5 | #6548 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C5×C7×D4 | 2 |
| C2×C22⋊D35 | 2 |
| #10587989 | 2 |
| D4×D35 | 2 |
| #10587987 | 2 |
| #6548 | 5 |
| #4144 | 7 |