Finite Group DatabasebetaFinite Group DatabasebetaOrder 20224.
474 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 474 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, socle | C2×C79 | C2×#353 |
| commutator | C42 | C22×C4×C79 |
| Frattini, cosocle | #127 | C23×C79 |
| prime radical | C79×#3151 | C2 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C79 ◁ C23×C79 ◁ C22×C4×C79 ◁ C2×#127×C79 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#353×C79 | C2 | semidirect |
| 3. C79×#1409 | C2 | semidirect |
| 4. C79×#1408 | C2 | semidirect |
| 6. #532×C79 | C4 | semidirect |
| 9. #493×C79 | C4 | semidirect |
| 11. #492×C79 | C4 | semidirect |
| 14. C4×M16×C79 | C4 | semidirect |
| 27. C4⋊Q8×C79 | C2×C4 | semidirect |
| 30. #127×C79 | C2×C4 | semidirect |
| 41. C4×C8×C79 | C2×C4 | semidirect |
| 42. C4×C8×C79 | C2×C4 | semidirect |
| 124. #13584 | C79 | direct (N × H) |
| 132. C2×#353 | C2×C79 | semidirect |
| 133. #1409 | C2×C79 | semidirect |
| 134. #1408 | C2×C79 | semidirect |
| 135. C79 | #13584 | direct (N × H) |
| 137. #532 | C4×C79 | semidirect |
| 140. #493 | C4×C79 | semidirect |
| 142. #492 | C4×C79 | semidirect |
| 145. C4×M16 | C4×C79 | semidirect |
| 158. C4⋊Q8 | C2×C4×C79 | semidirect |
| 161. #127 | C2×C4×C79 | semidirect |
| 172. C4×C8 | C2×C4×C79 | semidirect |
| 173. C4×C8 | C2×C4×C79 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C79×#3151 | 2 |
| C2×#353×C79 | 2 |
| C79×#1409 | 2 |
| C79×#1408 | 2 |
| #13584 | 79 |