Finite Group DatabasebetaFinite Group DatabasebetaOrder 10688.
218 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 218 conjugacy classes of subgroups — too many to draw. Only its 146 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×C167 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C167 |
| prime radical | #121×C167 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C167 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #126×C167 | C2 | semidirect |
| 2. #125×C167 | C2 | semidirect |
| 3. #123×C167 | C2 | semidirect |
| 4. #123×C167 | C2 | semidirect |
| 5. C22⋊Q8×C167 | C2 | semidirect |
| 6. C22⋊Q8×C167 | C2 | semidirect |
| 8. C4×D4×C167 | C2 | semidirect |
| 9. C2×C4⋊C4×C167 | C2 | semidirect |
| 66. C167 | #539 | direct (N × H) |
| 67. #539 | C167 | direct (N × H) |
| 68. #126 | C2×C167 | semidirect |
| 69. #125 | C2×C167 | semidirect |
| 70. #123 | C2×C167 | semidirect |
| 71. #123 | C2×C167 | semidirect |
| 72. C22⋊Q8 | C2×C167 | semidirect |
| 73. C22⋊Q8 | C2×C167 | semidirect |
| 75. C4×D4 | C2×C167 | semidirect |
| 76. C2×C4⋊C4 | C2×C167 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #126×C167 | 2 |
| #125×C167 | 2 |
| #123×C167 | 2 |
| C22⋊Q8×C167 | 2 |
| #121×C167 | 2 |
| C4×D4×C167 | 2 |
| C2×C4⋊C4×C167 | 2 |
| #539 | 167 |