Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
540 conjugacy classes of subgroups; only the 189 normal subgroups are shown.
This group has 540 conjugacy classes of subgroups — too many to draw. Only its 189 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 | C22×C97 | C23×C4 |
| Fitting | #376×C97 | C2 |
| socle, cosocle | C23×C97 | C24 |
| prime radical | C22×D388 | C22 |
| hypercenter | #376 | D97 |
| hypocenter | C97 | #2319 |
| LCS | G ▷ C22×C97 ▷ C97 | |
| UCS | C1 ◁ C23 ◁ #376 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #20260633 | C2 | semidirect |
| 4. #20260624 | C2 | semidirect |
| 5. #20260597 | C2 | semidirect |
| 6. #20260594 | C2 | semidirect |
| 7. C4×C4⋊Dic97 | C2 | semidirect |
| 8. #376×C97 | C2 | semidirect |
| 11. C4×D388 | C4 | semidirect |
| 97. C97 | #2319 | semidirect |
| 105. #376 | D97 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×D388 | 2 |
| #20260637 | 2 |
| #20260633 | 2 |
| #20260624 | 2 |
| #20260597 | 2 |
| #20260594 | 2 |
| C4×C4⋊Dic97 | 2 |
| #376×C97 | 2 |
| #2319 | 97 |