Finite Group DatabasebetaFinite Group DatabasebetaOrder 22336.
898 conjugacy classes of subgroups; only the 447 normal subgroups are shown.
This group has 898 conjugacy classes of subgroups — too many to draw. Only its 447 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C23×D349 |
| commutator, cosocle | C2×C349 | C25 |
| Frattini | C2 | C24×D349 |
| Fitting | C22×D4×C349 | C2 |
| socle | C22×C349 | C24 |
| hypercenter | C22×D4 | D349 |
| hypocenter | C349 | C2×H(5,𝔽2) |
| LCS | G ▷ C2×C349 ▷ C349 | |
| UCS | C1 ◁ C22 ◁ C22×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #26013684 | C2 | direct (N × H) |
| 2. C2×#19150233 | C2 | semidirect |
| 3. C2×#19150232 | C2 | semidirect |
| 4. C22×C22⋊D349 | C2 | semidirect |
| 5. C2×D4×D349 | C2 | semidirect |
| 6. C22×D4×C349 | C2 | semidirect |
| 7. C23×D349 | C22 | semidirect |
| 8. #19150233 | C22 | semidirect |
| 9. #19150232 | C22 | semidirect |
| 10. C2×Dic698 | C22 | semidirect |
| 11. C2×D1396 | C22 | semidirect |
| 12. C2×C22⋊D349 | C22 | semidirect |
| 13. C2×C22⋊D349 | C22 | semidirect |
| 14. C2×C22⋊D349 | C22 | semidirect |
| 15. D4×D349 | C22 | semidirect |
| 16. C22×Dic349 | C22 | semidirect |
| 17. C2×C4×D349 | C22 | semidirect |
| 18. C24×C349 | C22 | semidirect |
| 19. C2×D4×C349 | C22 | semidirect |
| 20. C2×D4×C349 | C22 | semidirect |
| 21. C22×C4×C349 | C22 | semidirect |
| 24. C22×D349 | C23 | semidirect |
| 25. Dic698 | C23 | semidirect |
| 26. D1396 | C23 | semidirect |
| 27. C22⋊D349 | C23 | semidirect |
| 28. C22⋊D349 | C23 | semidirect |
| 30. C2×Dic349 | C23 | semidirect |
| 31. C4×D349 | C23 | semidirect |
| 35. C23×C349 | C23 | semidirect |
| 36. D4×C349 | C23 | semidirect |
| 38. C2×C4×C349 | C23 | semidirect |
| 52. C2×C349 | H(5,𝔽2) | semidirect |
| 55. C349 | C2×H(5,𝔽2) | semidirect |
| 62. C22×D4 | D349 | semidirect |
| 64. C24 | C2×D349 | semidirect |
| 73. C2×D4 | C2×D349 | semidirect |
| 74. C2×D4 | C2×D349 | semidirect |
| 77. C22×C4 | C2×D349 | semidirect |
| 84. C23 | C22×D349 | semidirect |
| 89. D4 | C22×D349 | semidirect |
| 94. C2×C4 | C22×D349 | semidirect |
| 109. C2 | #26013684 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #26013684 | 2 |
| C2×#19150233 | 2 |
| C2×#19150232 | 2 |
| C22×C22⋊D349 | 2 |
| C2×D4×D349 | 2 |
| C22×D4×C349 | 2 |
| C2×H(5,𝔽2) | 349 |