Finite Group DatabasebetaFinite Group DatabasebetaOrder 12736.
124 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 124 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C199 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C23×C199 |
| prime radical | C2×D8×C199 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C199 ◁ C42×C199 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×SD16×C199 | C2 | semidirect |
| 3. #124×C199 | C2 | semidirect |
| 4. C4×D4×C199 | C2 | semidirect |
| 5. C4⋊C8×C199 | C2 | semidirect |
| 6. #103×C199 | C2 | semidirect |
| 7. #102×C199 | C2 | semidirect |
| 55. C199 | #463 | direct (N × H) |
| 56. #463 | C199 | direct (N × H) |
| 57. C2×SD16 | C2×C199 | semidirect |
| 59. #124 | C2×C199 | semidirect |
| 60. C4×D4 | C2×C199 | semidirect |
| 61. C4⋊C8 | C2×C199 | semidirect |
| 62. #103 | C2×C199 | semidirect |
| 63. #102 | C2×C199 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×SD16×C199 | 2 |
| C2×D8×C199 | 2 |
| #124×C199 | 2 |
| C4×D4×C199 | 2 |
| C4⋊C8×C199 | 2 |
| #103×C199 | 2 |
| #102×C199 | 2 |
| #463 | 199 |