Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
712 conjugacy classes of subgroups; only the 440 normal subgroups are shown.
This group has 712 conjugacy classes of subgroups — too many to draw. Only its 440 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C97 | C23 |
| commutator | C22 | C23×C4×C97 |
| Frattini, cosocle | C23 | C24×C97 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24×C97 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #382×C97 | C2 | direct (N × H) |
| 10. C2×C42×C97 | C4 | semidirect |
| 25. C42×C97 | C2×C4 | semidirect |
| 52. C2×C97 | #382 | direct (N × H) |
| 57. C2×#382 | C97 | direct (N × H) |
| 58. C97 | C2×#382 | direct (N × H) |
| 61. #382 | C2×C97 | direct (N × H) |
| 68. C2×C42 | C4×C97 | semidirect |
| 83. C42 | C2×C4×C97 | semidirect |
| 110. C2 | #382×C97 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C4⋊C4×C97 | 2 |
| C22×C42×C97 | 2 |
| #382×C97 | 2 |
| C2×#95×C97 | 2 |
| C2×#382 | 97 |