Finite Group DatabasebetaFinite Group DatabasebetaOrder 1664.
840 conjugacy classes of subgroups; only the 205 normal subgroups are shown.
This group has 840 conjugacy classes of subgroups — too many to draw. Only its 205 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 | C23×Q8×C13 | C2 |
| prime radical | C22×C22⋊D13 | C22 |
| hypercenter | C23×Q8 | D13 |
| hypocenter | C13 | #2409 |
| LCS | G ▷ C23×C13 ▷ C13 | |
| UCS | C1 ◁ C23 ◁ C23×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×Q8×C13 | C2 | semidirect |
| 2. C2×#88200 | C2 | semidirect |
| 4. #11718892 | C2 | semidirect |
| 5. #11718891 | C2 | semidirect |
| 6. #11718890 | C2 | semidirect |
| 8. #11718873 | C2 | semidirect |
| 9. #11718871 | C2 | semidirect |
| 38. C2×Q8×C13 | D4 | semidirect |
| 48. C2×C22⋊D13 | Q8 | semidirect |
| 74. C22×D13 | C2×Q8 | semidirect |
| 96. C2×Dic13 | C2×Q8 | semidirect |
| 98. C23×Q8 | D13 | semidirect |
| 111. C22×C13 | C22⋊Q8 | semidirect |
| 164. C2×Q8 | C22⋊D13 | semidirect |
| 189. C13 | #2409 | semidirect |
| 233. C22 | #88200 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×Q8×C13 | 2 |
| C2×#88200 | 2 |
| C2×C4×C22⋊D13 | 2 |
| #11718892 | 2 |
| #11718891 | 2 |
| #11718890 | 2 |
| #11718874 | 2 |
| #11718873 | 2 |
| #11718871 | 2 |
| #2409 | 13 |