Finite Group DatabasebetaFinite Group DatabasebetaOrder 12928.
1056 conjugacy classes of subgroups; only the 173 normal subgroups are shown.
This group has 1056 conjugacy classes of subgroups — too many to draw. Only its 173 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D101 |
| commutator, socle, cosocle | C23×C101 | C24 |
| Fitting | C22×C22⋊C4×C101 | C2 |
| hypercenter | C22×C22⋊C4 | D101 |
| hypocenter | C101 | #2404 |
| LCS | G ▷ C23×C101 ▷ C101 | |
| UCS | C1 ◁ C23 ◁ C22×C22⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#13108074 | C2 | semidirect |
| 2. C2×#15692224 | C2 | semidirect |
| 3. #20265089 | C2 | semidirect |
| 4. #20265086 | C2 | semidirect |
| 5. #20265056 | C2 | semidirect |
| 6. #20265053 | C2 | semidirect |
| 7. #20265040 | C2 | semidirect |
| 8. C22×C22⋊C4×C101 | C2 | semidirect |
| 11. C22×Dic202 | C22 | semidirect |
| 12. C22×D404 | C22 | semidirect |
| 15. C22×C4×D101 | C22 | semidirect |
| 19. C2×#13108051 | C22 | semidirect |
| 20. #15692162 | C22 | semidirect |
| 21. #15692162 | C22 | semidirect |
| 22. #15692162 | C22 | semidirect |
| 40. C2×C22⋊D101 | D4 | semidirect |
| 58. C22×C4×C101 | D4 | semidirect |
| 76. C23×C101 | #41 | semidirect |
| 134. C101 | #2404 | semidirect |
| 137. C22×C22⋊C4 | D101 | semidirect |
| 189. C22×C4 | C22⋊D101 | semidirect |
| 215. C23 | #13108074 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#13108074 | 2 |
| C2×#15692224 | 2 |
| #20265089 | 2 |
| #20265086 | 2 |
| #20265056 | 2 |
| #20265053 | 2 |
| #20265040 | 2 |
| C22×C22⋊C4×C101 | 2 |
| #2404 | 101 |