Finite Group DatabasebetaFinite Group DatabasebetaOrder 1664.
782 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 782 conjugacy classes of subgroups — too many to draw. Only its 165 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, socle, cosocle | C23×C13 | C24 |
| Fitting | C2×C13×#123 | C2 |
| prime radical | C22×D4×C13 | C22 |
| hypercenter | C2×#123 | D13 |
| hypocenter | C13 | #2614 |
| LCS | G ▷ C23×C13 ▷ C13 | |
| UCS | C1 ◁ C23 ◁ C2×#123 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×Dic26 | C2 | semidirect |
| 4. #11718872 | C2 | semidirect |
| 5. #11718640 | C2 | semidirect |
| 6. #11718608 | C2 | semidirect |
| 7. #11718604 | C2 | semidirect |
| 135. C13 | #2614 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×Dic26 | 2 |
| C2×C13×#123 | 2 |
| C2×#88189 | 2 |
| #11718872 | 2 |
| #11718640 | 2 |
| #11718608 | 2 |
| #11718604 | 2 |
| #2614 | 13 |