Finite Group DatabasebetaFinite Group DatabasebetaOrder 21376.
278 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 278 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C167 | #393 |
| commutator, Frattini, cosocle | C24 | C23×C167 |
| prime radical | C22×D4×C167 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C167 ◁ C24×C167 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #543×C167 | C2 | semidirect |
| 3. #342×C167 | C2 | semidirect |
| 9. C24×C167 | Q8 | semidirect |
| 16. C22×C4×C167 | Q8 | semidirect |
| 52. C167 | #2032 | direct (N × H) |
| 53. #2032 | C167 | direct (N × H) |
| 54. #543 | C2×C167 | semidirect |
| 56. #342 | C2×C167 | semidirect |
| 62. C24 | Q8×C167 | semidirect |
| 69. C22×C4 | Q8×C167 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #543×C167 | 2 |
| C2×#99×C167 | 2 |
| #342×C167 | 2 |
| #2032 | 167 |