Finite Group DatabasebetaFinite Group DatabasebetaOrder 1120.
332 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 332 conjugacy classes of subgroups — too many to draw. Only its 108 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 | C22×D5×D7 |
| commutator, socle, cosocle | C2×C5×C7 | C24 |
| Fitting | C23×C5×C7 | C22 |
| hypercenter | C23 | D5×D7 |
| hypocenter | C5×C7 | H(5,𝔽2) |
| LCS | G ▷ C2×C5×C7 ▷ C5×C7 | |
| UCS | C1 ◁ C2 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C22⋊D35 | C2 | semidirect |
| 2. C2×C7×C22⋊D5 | C2 | semidirect |
| 3. C2×C5×C22⋊D7 | C2 | semidirect |
| 4. #10587963 | C2 | semidirect |
| 5. D5×C22⋊D7 | C2 | semidirect |
| 6. D7×C22⋊D5 | C2 | semidirect |
| 7. #10587955 | C2 | semidirect |
| 8. #10587953 | C2 | semidirect |
| 9. #10587952 | C2 | semidirect |
| 10. C23×C5×C7 | C22 | semidirect |
| 11. C22×D35 | C22 | semidirect |
| 12. C22×C7×D5 | C22 | semidirect |
| 13. C22×C5×D7 | C22 | semidirect |
| 14. C2×D5×D7 | C22 | semidirect |
| 15. C22⋊D35 | C22 | semidirect |
| 16. C22⋊D35 | C22 | semidirect |
| 17. C2×Dic35 | C22 | semidirect |
| 18. C7×C22⋊D5 | C22 | semidirect |
| 19. C7×C22⋊D5 | C22 | semidirect |
| 20. C2×C7×Dic5 | C22 | semidirect |
| 21. C5×C22⋊D7 | C22 | semidirect |
| 22. C5×C22⋊D7 | C22 | semidirect |
| 23. C2×C5×Dic7 | C22 | semidirect |
| 24. #63308 | C22 | semidirect |
| 25. #63307 | C22 | semidirect |
| 26. #63306 | C22 | semidirect |
| 27. #63305 | C22 | semidirect |
| 28. #63304 | C22 | semidirect |
| 29. D5×Dic7 | C22 | semidirect |
| 30. D7×Dic5 | C22 | semidirect |
| 50. C2×C22⋊D7 | D5 | semidirect |
| 59. C2×C22⋊D5 | D7 | semidirect |
| 73. C23×C7 | C2×D5 | semidirect |
| 76. C22×D7 | C2×D5 | semidirect |
| 84. C22⋊D7 | C2×D5 | semidirect |
| 85. C22⋊D7 | C2×D5 | semidirect |
| 88. C2×Dic7 | C2×D5 | semidirect |
| 94. C23×C5 | C2×D7 | semidirect |
| 97. C22×D5 | C2×D7 | semidirect |
| 105. C22⋊D5 | C2×D7 | semidirect |
| 106. C22⋊D5 | C2×D7 | semidirect |
| 109. C2×Dic5 | C2×D7 | semidirect |
| 114. C5×C7 | H(5,𝔽2) | semidirect |
| 171. C23 | D5×D7 | semidirect |
| 188. C7 | #4144 | semidirect |
| 189. C5 | #6548 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C22⋊D35 | 2 |
| C2×C7×C22⋊D5 | 2 |
| C2×C5×C22⋊D7 | 2 |
| #10587963 | 2 |
| D5×C22⋊D7 | 2 |
| D7×C22⋊D5 | 2 |
| #10587955 | 2 |
| #10587953 | 2 |
| #10587952 | 2 |
| #6548 | 5 |
| #4144 | 7 |