Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
178 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 178 conjugacy classes of subgroups — too many to draw. Only its 89 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C42 | C22×D97 |
| commutator, socle | C22×C97 | C2×C42 |
| Fitting | #322×C97 | C2 |
| prime radical | C22×D97 | C42 |
| cosocle | C42×C97 | C23 |
| hypercenter | #322 | D97 |
| hypocenter | C97 | #1454 |
| LCS | G ▷ C22×C97 ▷ C97 | |
| UCS | C1 ◁ C42 ◁ #322 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #20260457 | C2 | semidirect |
| 5. #20260456 | C2 | semidirect |
| 6. C8×#11715895 | C2 | semidirect |
| 7. #322×C97 | C2 | semidirect |
| 65. C97 | #1454 | semidirect |
| 71. #322 | D97 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20260786 | 2 |
| #20260457 | 2 |
| #20260456 | 2 |
| C8×#11715895 | 2 |
| #322×C97 | 2 |
| #1454 | 97 |