Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
432 conjugacy classes of subgroups; only the 200 normal subgroups are shown.
This group has 432 conjugacy classes of subgroups — too many to draw. Only its 200 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C97 | C23 |
| commutator, Frattini, cosocle | C23 | C24×C97 |
| socle | C23×C97 | C24 |
| prime radical | C24×C97 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C97 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #401×C97 | C2 | semidirect |
| 2. #400×C97 | C2 | semidirect |
| 5. C4×C4⋊C4×C97 | C2 | semidirect |
| 7. #376×C97 | C2 | semidirect |
| 70. #2401 | C97 | direct (N × H) |
| 71. C97 | #2401 | direct (N × H) |
| 72. #401 | C2×C97 | semidirect |
| 73. #400 | C2×C97 | semidirect |
| 76. C4×C4⋊C4 | C2×C97 | semidirect |
| 78. #376 | C2×C97 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #401×C97 | 2 |
| #400×C97 | 2 |
| #397×C97 | 2 |
| #396×C97 | 2 |
| C4×C4⋊C4×C97 | 2 |
| C4×C22⋊C4×C97 | 2 |
| #376×C97 | 2 |
| #2401 | 97 |