Finite Group DatabasebetaFinite Group DatabasebetaOrder 1120.
332 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 332 conjugacy classes of subgroups — too many to draw. Only its 170 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C5 | C23×D7 |
| commutator, cosocle | C2×C7 | C24×C5 |
| Frattini | C2 | C23×C5×D7 |
| Fitting | C2×C5×C7×D4 | C2 |
| socle | C2×C5×C7 | C24 |
| hypercenter | C2×C5×D4 | D7 |
| hypocenter | C7 | C5×H(5,𝔽2) |
| LCS | G ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C5×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×C5×C22⋊D7 | C2 | semidirect |
| 3. C5×#1159 | C2 | semidirect |
| 4. C5×D4×D7 | C2 | semidirect |
| 5. C5×#1157 | C2 | semidirect |
| 6. C23×C5×C7 | C22 | semidirect |
| 7. C22×C5×D7 | C22 | semidirect |
| 8. C5×C7×D4 | C22 | semidirect |
| 9. C2×C4×C5×C7 | C22 | semidirect |
| 10. C5×C22⋊D7 | C22 | semidirect |
| 11. C5×C22⋊D7 | C22 | semidirect |
| 12. C2×C5×Dic7 | C22 | semidirect |
| 13. C5×D28 | C22 | semidirect |
| 14. C4×C5×D7 | C22 | semidirect |
| 15. C5×Dic14 | C22 | semidirect |
| 16. #6548 | C5 | direct (N × H) |
| 25. C2×C7×D4 | C2×C5 | semidirect |
| 26. C2×C22⋊D7 | C2×C5 | semidirect |
| 27. #1159 | C2×C5 | semidirect |
| 28. D4×D7 | C2×C5 | semidirect |
| 29. #1157 | C2×C5 | semidirect |
| 34. C2×C5×D4 | D7 | semidirect |
| 39. C23×C7 | C22×C5 | semidirect |
| 40. C22×D7 | C22×C5 | semidirect |
| 41. C7×D4 | C22×C5 | semidirect |
| 42. C2×C4×C7 | C22×C5 | semidirect |
| 43. C22⋊D7 | C22×C5 | semidirect |
| 44. C22⋊D7 | C22×C5 | semidirect |
| 45. C2×Dic7 | C22×C5 | semidirect |
| 46. D28 | C22×C5 | semidirect |
| 47. C4×D7 | C22×C5 | semidirect |
| 48. Dic14 | C22×C5 | semidirect |
| 50. C23×C5 | C2×D7 | semidirect |
| 55. C5×D4 | C2×D7 | semidirect |
| 58. C2×C4×C5 | C2×D7 | semidirect |
| 59. C5×C7 | H(5,𝔽2) | semidirect |
| 79. C2×D4 | C5×D7 | semidirect |
| 89. C23 | C2×C5×D7 | semidirect |
| 94. D4 | C2×C5×D7 | semidirect |
| 97. C2×C4 | C2×C5×D7 | semidirect |
| 98. C7 | C5×H(5,𝔽2) | semidirect |
| 99. C5 | #6548 | direct (N × H) |
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×C5×C22⋊D7 | 2 |
| C5×#1159 | 2 |
| C5×D4×D7 | 2 |
| C5×#1157 | 2 |
| #6548 | 5 |
| C5×H(5,𝔽2) | 7 |