Finite Group DatabasebetaFinite Group DatabasebetaOrder 1664.
1058 conjugacy classes of subgroups; only the 451 normal subgroups are shown.
This group has 1058 conjugacy classes of subgroups — too many to draw. Only its 451 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D13 |
| commutator, socle, cosocle | C22×C13 | C25 |
| Fitting | C13×H(3,𝔽4) | C2 |
| hypercenter | H(3,𝔽4) | D13 |
| hypocenter | C13 | #3529 |
| LCS | G ▷ C22×C13 ▷ C13 | |
| UCS | C1 ◁ C22 ◁ H(3,𝔽4) | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C13×H(3,𝔽4) | C2 | semidirect |
| 2. #11719389 | C2 | semidirect |
| 3. #11719385 | C2 | semidirect |
| 4. #11719377 | C2 | semidirect |
| 5. #11719373 | C2 | semidirect |
| 6. #11719300 | C2 | semidirect |
| 7. #11719291 | C2 | semidirect |
| 8. C23×Dic13 | C22 | semidirect |
| 9. C2×#6131 | C22 | semidirect |
| 14. #88207 | C22 | semidirect |
| 15. #88200 | C22 | semidirect |
| 16. Q8×Dic13 | C22 | semidirect |
| 17. #88198 | C22 | semidirect |
| 23. #88140 | C22 | semidirect |
| 24. #88138 | C22 | semidirect |
| 25. #88137 | C22 | semidirect |
| 26. #88135 | C22 | semidirect |
| 27. #88133 | C22 | semidirect |
| 28. #88132 | C22 | semidirect |
| 29. #88131 | C22 | semidirect |
| 30. #88129 | C22 | semidirect |
| 31. C42×D13 | C22 | semidirect |
| 32. #88124 | C22 | semidirect |
| 85. H(3,𝔽4) | D13 | semidirect |
| 158. C13 | #3529 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C13×H(3,𝔽4) | 2 |
| #11719389 | 2 |
| #11719385 | 2 |
| #11719377 | 2 |
| #11719373 | 2 |
| #11719300 | 2 |
| #11719291 | 2 |
| #3529 | 13 |