Finite Group DatabasebetaFinite Group DatabasebetaOrder 21376.
528 conjugacy classes of subgroups; only the 159 normal subgroups are shown.
This group has 528 conjugacy classes of subgroups — too many to draw. Only its 159 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D167 |
| commutator, socle, cosocle | C23×C167 | C24 |
| Fitting | #395×C167 | C2 |
| prime radical | C23×D167 | C23 |
| hypercenter | #395 | D167 |
| hypocenter | C167 | #2396 |
| LCS | G ▷ C23×C167 ▷ C167 | |
| UCS | C1 ◁ C23 ◁ #395 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×Q8×Dic167 | C2 | semidirect |
| 3. #24923208 | C2 | semidirect |
| 5. #24923201 | C2 | semidirect |
| 8. C4⋊C4×Dic167 | C2 | semidirect |
| 9. #24923120 | C2 | semidirect |
| 11. #395×C167 | C2 | semidirect |
| 45. C2×C4×D167 | Q8 | semidirect |
| 128. C167 | #2396 | semidirect |
| 136. #395 | D167 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×Q8×Dic167 | 2 |
| C2×#19128790 | 2 |
| #24923208 | 2 |
| #24923206 | 2 |
| #24923201 | 2 |
| #24923191 | 2 |
| #24923180 | 2 |
| C4⋊C4×Dic167 | 2 |
| #24923120 | 2 |
| #24923116 | 2 |
| #395×C167 | 2 |
| #2396 | 167 |