Finite Group DatabasebetaFinite Group DatabasebetaOrder 22336.
1362 conjugacy classes of subgroups; only the 503 normal subgroups are shown.
This group has 1362 conjugacy classes of subgroups — too many to draw. Only its 503 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23 | C22×D349 |
| commutator, cosocle | C2×C349 | C25 |
| Frattini | C2 | C24×D349 |
| Fitting | C22×D4×C349 | C2 |
| socle | C23×C349 | C23 |
| hypercenter | C22×D4 | D349 |
| hypocenter | C349 | C23×D4 |
| LCS | G ▷ C2×C349 ▷ C349 | |
| UCS | C1 ◁ C23 ◁ C22×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C24×D349 | C2 | semidirect |
| 2. C22×D1396 | C2 | semidirect |
| 3. C22×C22⋊D349 | C2 | semidirect |
| 4. C2×D4×D349 | C2 | direct (N × H) |
| 5. C22×C4×D349 | C2 | semidirect |
| 6. C22×D4×C349 | C2 | semidirect |
| 9. C23×D349 | C22 | semidirect |
| 10. C23×D349 | C22 | semidirect |
| 11. C2×D1396 | C22 | semidirect |
| 12. C2×C22⋊D349 | C22 | semidirect |
| 13. D4×D349 | C22 | direct (N × H) |
| 14. C22×Dic349 | C22 | semidirect |
| 15. C2×C4×D349 | C22 | semidirect |
| 16. C24×C349 | C22 | semidirect |
| 17. C2×D4×C349 | C22 | semidirect |
| 18. C22×C4×C349 | C22 | semidirect |
| 21. C22×D349 | D4 | direct (N × H) |
| 23. C22×D349 | C23 | semidirect |
| 24. C22×D349 | C23 | semidirect |
| 25. D1396 | C23 | semidirect |
| 26. C22⋊D349 | C23 | semidirect |
| 27. C2×Dic349 | C23 | semidirect |
| 28. C4×D349 | C23 | semidirect |
| 31. C23×C349 | C23 | semidirect |
| 32. D4×C349 | C23 | semidirect |
| 33. C2×C4×C349 | C23 | semidirect |
| 35. C22×C349 | C2×D4 | semidirect |
| 37. C22×C349 | C24 | semidirect |
| 40. C2×D349 | C2×D4 | direct (N × H) |
| 42. C2×D349 | C24 | semidirect |
| 43. C4×C349 | C24 | semidirect |
| 44. Dic349 | C24 | semidirect |
| 46. C2×C349 | C22×D4 | semidirect |
| 49. D349 | C22×D4 | direct (N × H) |
| 50. C349 | C23×D4 | semidirect |
| 56. C22×D4 | D349 | direct (N × H) |
| 62. C24 | C2×D349 | semidirect |
| 66. C2×D4 | C2×D349 | direct (N × H) |
| 69. C22×C4 | C2×D349 | semidirect |
| 78. C23 | C22×D349 | semidirect |
| 81. D4 | C22×D349 | direct (N × H) |
| 84. C2×C4 | C22×D349 | semidirect |
| 91. C22 | D4×D349 | direct (N × H) |
| 93. C22 | C23×D349 | semidirect |
| 95. C4 | C23×D349 | semidirect |
| 99. C2 | C2×D4×D349 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×D349 | 2 |
| C22×D1396 | 2 |
| C22×C22⋊D349 | 2 |
| C2×D4×D349 | 2 |
| C22×C4×D349 | 2 |
| C22×D4×C349 | 2 |
| C23×D4 | 349 |