Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
672 conjugacy classes of subgroups; only the 189 normal subgroups are shown.
This group has 672 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 | #380×C97 | C2 |
| socle, cosocle | C23×C97 | C24 |
| prime radical | C24×D97 | C22 |
| hypercenter | #380 | D97 |
| hypocenter | C97 | #2312 |
| LCS | G ▷ C22×C97 ▷ C97 | |
| UCS | C1 ◁ C23 ◁ #380 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #20260627 | C2 | semidirect |
| 3. #20260625 | C2 | semidirect |
| 4. #20260595 | C2 | semidirect |
| 6. C22⋊C4×Dic97 | C2 | semidirect |
| 7. #380×C97 | C2 | semidirect |
| 10. C2×C4⋊Dic97 | C22 | semidirect |
| 12. C2×#13095488 | C4 | semidirect |
| 17. C2×C4×Dic97 | C22 | semidirect |
| 18. #15690717 | C22 | semidirect |
| 19. #15690717 | C22 | semidirect |
| 21. C2×C4⋊C4×C97 | C22 | semidirect |
| 23. #95×C97 | C22 | semidirect |
| 108. C97 | #2312 | semidirect |
| 113. #380 | D97 | semidirect |
| 120. C2×C4⋊C4 | C2×D97 | semidirect |
| 131. #95 | C2×D97 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#13095488 | 2 |
| #20260627 | 2 |
| #20260625 | 2 |
| #20260595 | 2 |
| #20260590 | 2 |
| C22⋊C4×Dic97 | 2 |
| #380×C97 | 2 |
| #2312 | 97 |