Finite Group DatabasebetaFinite Group DatabasebetaOrder 1664.
752 conjugacy classes of subgroups; only the 177 normal subgroups are shown.
This group has 752 conjugacy classes of subgroups — too many to draw. Only its 177 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 | C13×#390 | C2 |
| prime radical | C2×#88194 | C2 |
| hypercenter | #390 | D13 |
| hypocenter | C13 | #2858 |
| LCS | G ▷ C23×C13 ▷ C13 | |
| UCS | C1 ◁ C23 ◁ #390 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#88142 | C2 | semidirect |
| 3. C2×#88139 | C2 | semidirect |
| 4. C13×#390 | C2 | semidirect |
| 5. #11718882 | C2 | semidirect |
| 6. #11718649 | C2 | semidirect |
| 7. #11718618 | C2 | semidirect |
| 8. #11718600 | C2 | semidirect |
| 9. C42×Dic13 | C2 | semidirect |
| 62. C4×Dic13 | D4 | semidirect |
| 104. #390 | D13 | semidirect |
| 123. C4×C13 | #124 | semidirect |
| 126. Dic13 | #124 | semidirect |
| 188. C13 | #2858 | semidirect |
| 243. C4 | #88139 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#88194 | 2 |
| C2×#88142 | 2 |
| C2×#88139 | 2 |
| C13×#390 | 2 |
| #11718882 | 2 |
| #11718649 | 2 |
| #11718618 | 2 |
| #11718600 | 2 |
| C42×Dic13 | 2 |
| #2858 | 13 |