Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
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×D97 |
| commutator, socle, cosocle | C23×C97 | C24 |
| Fitting | #390×C97 | C2 |
| prime radical | C2×#15690806 | C2 |
| hypercenter | #390 | D97 |
| hypocenter | C97 | #2858 |
| LCS | G ▷ C23×C97 ▷ C97 | |
| UCS | C1 ◁ C23 ◁ #390 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#15690809 | C2 | semidirect |
| 3. C2×#15690796 | C2 | semidirect |
| 4. #20260638 | C2 | semidirect |
| 5. #20260637 | C2 | semidirect |
| 6. #20260636 | C2 | semidirect |
| 7. #20260635 | C2 | semidirect |
| 8. C42×Dic97 | C2 | semidirect |
| 9. #390×C97 | C2 | semidirect |
| 52. C4×Dic97 | D4 | semidirect |
| 113. C4×C97 | #124 | semidirect |
| 116. Dic97 | #124 | semidirect |
| 126. C97 | #2858 | semidirect |
| 134. #390 | D97 | semidirect |
| 243. C4 | #15690796 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#15690809 | 2 |
| C2×#15690806 | 2 |
| C2×#15690796 | 2 |
| #20260638 | 2 |
| #20260637 | 2 |
| #20260636 | 2 |
| #20260635 | 2 |
| C42×Dic97 | 2 |
| #390×C97 | 2 |
| #2858 | 97 |