Finite Group DatabasebetaFinite Group DatabasebetaOrder 20224.
482 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 482 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 | C4⋊Q8 | 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×#2132 | C2 | semidirect |
| 3. C79×#1543 | C2 | semidirect |
| 4. C79×#1405 | C2 | semidirect |
| 16. #356×C79 | C22 | semidirect |
| 18. #333×C79 | C22 | semidirect |
| 166. #13572 | C79 | direct (N × H) |
| 174. #2132 | C2×C79 | semidirect |
| 175. #1543 | C2×C79 | semidirect |
| 176. #1405 | C2×C79 | semidirect |
| 177. C79 | #13572 | direct (N × H) |
| 189. #356 | C22×C79 | semidirect |
| 191. #333 | 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×#2132 | 2 |
| C79×#1543 | 2 |
| C79×#1405 | 2 |
| #13572 | 79 |