Finite Group DatabasebetaFinite Group DatabasebetaOrder 20224.
498 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 498 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4×C79 | AΣL(1,𝔽16) |
| commutator | C2×Q8 | C22×C4×C79 |
| Frattini, cosocle | #124 | C23×C79 |
| socle | C2×C79 | C2×AΣL(1,𝔽16) |
| prime radical | C79×#3018 | C2 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C79 ◁ C2×C4×C79 ◁ C22×C4×C79 ◁ #532×C79 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C79×#2126 | C2 | semidirect |
| 3. C79×#1525 | C2 | semidirect |
| 4. C79×#1404 | C2 | semidirect |
| 5. #585×C79 | C4 | semidirect |
| 8. #456×C79 | C4 | semidirect |
| 13. #413×C79 | C4 | semidirect |
| 16. #354×C79 | C22 | semidirect |
| 19. #329×C79 | C22 | semidirect |
| 168. #13568 | C79 | direct (N × H) |
| 176. #2126 | C2×C79 | semidirect |
| 177. #1525 | C2×C79 | semidirect |
| 178. #1404 | C2×C79 | semidirect |
| 179. C79 | #13568 | direct (N × H) |
| 180. #585 | C4×C79 | semidirect |
| 183. #456 | C4×C79 | semidirect |
| 188. #413 | C4×C79 | semidirect |
| 191. #354 | C22×C79 | semidirect |
| 194. #329 | C22×C79 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C79×#3018 | 2 |
| C79×#2126 | 2 |
| C79×#1525 | 2 |
| C79×#1404 | 2 |
| #13568 | 79 |