Finite Group DatabasebetaFinite Group DatabasebetaOrder 20864.
952 conjugacy classes of subgroups; only the 173 normal subgroups are shown.
This group has 952 conjugacy classes of subgroups — too many to draw. Only its 173 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×D163 |
| commutator, socle, cosocle | C23×C163 | C24 |
| Fitting | #391×C163 | C2 |
| hypercenter | #391 | D163 |
| hypocenter | C163 | #2431 |
| LCS | G ▷ C23×C163 ▷ C163 | |
| UCS | C1 ◁ C23 ◁ #391 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#14579700 | C2 | semidirect |
| 2. C2×#19104732 | C2 | semidirect |
| 3. C2×#19104724 | C2 | semidirect |
| 4. C2×#19104710 | C2 | semidirect |
| 5. #23781575 | C2 | semidirect |
| 6. #23781566 | C2 | semidirect |
| 7. #23781543 | C2 | semidirect |
| 8. C4×(C2×D163)⋊C4 | C2 | semidirect |
| 9. #391×C163 | C2 | semidirect |
| 50. C2×D652 | D4 | semidirect |
| 58. C2×C4×D163 | D4 | semidirect |
| 66. C2×Q8×C163 | D4 | semidirect |
| 113. C2×C4×C163 | #41 | semidirect |
| 148. C163 | #2431 | semidirect |
| 155. #391 | D163 | semidirect |
| 188. C2×Q8 | D652 | semidirect |
| 262. C2×C4 | #14579700 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#14579700 | 2 |
| C2×#19104732 | 2 |
| C2×#19104724 | 2 |
| C2×#19104710 | 2 |
| #23781575 | 2 |
| #23781566 | 2 |
| #23781543 | 2 |
| C4×(C2×D163)⋊C4 | 2 |
| #391×C163 | 2 |
| #2431 | 163 |