Finite Group DatabasebetaFinite Group DatabasebetaOrder 21376.
1492 conjugacy classes of subgroups; only the 557 normal subgroups are shown.
This group has 1492 conjugacy classes of subgroups — too many to draw. Only its 557 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D167 |
| commutator, socle, cosocle | C22×C167 | C25 |
| Fitting | C23×Q8×C167 | C2 |
| hypercenter | C23×Q8 | D167 |
| hypocenter | C167 | #3486 |
| LCS | G ▷ C22×C167 ▷ C167 | |
| UCS | C1 ◁ C22 ◁ C23×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#19128880 | C2 | semidirect |
| 2. Q8×C22⋊D167 | C2 | semidirect |
| 3. #24923930 | C2 | semidirect |
| 4. #24923843 | C2 | semidirect |
| 5. #24923780 | C2 | semidirect |
| 6. #24923747 | C2 | semidirect |
| 7. C23×Q8×C167 | C2 | semidirect |
| 9. C2×#14581951 | C22 | semidirect |
| 11. C2×Q8×D167 | C22 | semidirect |
| 12. #19128830 | C22 | semidirect |
| 13. #19128821 | C22 | semidirect |
| 16. #19128808 | C22 | semidirect |
| 19. Q8×Dic167 | C22 | semidirect |
| 44. C2×Q8×C167 | D4 | semidirect |
| 75. C22×C167 | 2−1+4 | semidirect |
| 88. C167 | #3486 | semidirect |
| 91. C23×Q8 | D167 | semidirect |
| 123. C2×Q8 | C22⋊D167 | semidirect |
| 166. C22 | #19128880 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#19128880 | 2 |
| Q8×C22⋊D167 | 2 |
| #24923930 | 2 |
| #24923843 | 2 |
| #24923780 | 2 |
| #24923747 | 2 |
| C23×Q8×C167 | 2 |
| #3486 | 167 |