Finite Group DatabasebetaFinite Group DatabasebetaOrder 1664.
456 conjugacy classes of subgroups; only the 173 normal subgroups are shown.
This group has 456 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×D13 |
| commutator | C22×C13 | C23×C4 |
| Fitting | C13×#384 | C2 |
| socle, cosocle | C23×C13 | C24 |
| prime radical | C23×D13 | C23 |
| hypercenter | #384 | D13 |
| hypocenter | C13 | #2383 |
| LCS | G ▷ C22×C13 ▷ C13 | |
| UCS | C1 ◁ C23 ◁ #384 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C13×#384 | C2 | semidirect |
| 5. #11718639 | C2 | semidirect |
| 6. #11718633 | C2 | semidirect |
| 7. #11718598 | C2 | semidirect |
| 17. #88130 | C4 | semidirect |
| 32. C42×C13 | C2×C4 | semidirect |
| 36. #6101 | C2×C4 | semidirect |
| 62. #384 | D13 | semidirect |
| 109. C42 | C4×D13 | semidirect |
| 110. C13 | #2383 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#88130 | 2 |
| C13×#384 | 2 |
| #11718651 | 2 |
| #11718650 | 2 |
| #11718639 | 2 |
| #11718633 | 2 |
| #11718598 | 2 |
| #2383 | 13 |