Finite Group DatabasebetaFinite Group DatabasebetaOrder 10688.
154 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 154 conjugacy classes of subgroups — too many to draw. Only its 78 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4×C167 | C2×D4 |
| commutator | C4 | C22×C4×C167 |
| Frattini, cosocle | C2×C4 | C23×C167 |
| socle | C2×C167 | C2×C22⋊C4 |
| prime radical | C2×#41×C167 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C167 ◁ C22×C4×C167 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×M16×C167 | C2 | semidirect |
| 3. #117×C167 | C2 | semidirect |
| 4. #104×C167 | C2 | semidirect |
| 7. #41×C167 | C4 | semidirect |
| 9. C2×Q8×C167 | C4 | semidirect |
| 11. C2×D4×C167 | C4 | semidirect |
| 15. M16×C167 | C22 | semidirect |
| 19. C42×C167 | C22 | semidirect |
| 56. C167 | #421 | direct (N × H) |
| 57. #421 | C167 | direct (N × H) |
| 59. C2×M16 | C2×C167 | semidirect |
| 60. #117 | C2×C167 | semidirect |
| 61. #104 | C2×C167 | semidirect |
| 64. #41 | C4×C167 | semidirect |
| 66. C2×Q8 | C4×C167 | semidirect |
| 68. C2×D4 | C4×C167 | semidirect |
| 72. M16 | C22×C167 | semidirect |
| 76. C42 | C22×C167 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#41×C167 | 2 |
| C2×M16×C167 | 2 |
| #117×C167 | 2 |
| #104×C167 | 2 |
| #421 | 167 |