Finite Group DatabasebetaFinite Group DatabasebetaOrder 20224.
694 conjugacy classes of subgroups; only the 214 normal subgroups are shown.
This group has 694 conjugacy classes of subgroups — too many to draw. Only its 214 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C79 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C23×C4×C79 ◁ C23×C8×C79 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#466×C79 | C2 | semidirect |
| 2. C79×#2232 | C2 | semidirect |
| 3. C79×#2215 | C2 | semidirect |
| 4. C79×#2184 | C2 | semidirect |
| 5. C2×C22⋊C16×C79 | C2 | semidirect |
| 6. C22×D8×C79 | C22 | semidirect |
| 9. C2×D16×C79 | C22 | semidirect |
| 10. C22×C16×C79 | C22 | semidirect |
| 14. #466×C79 | C22 | semidirect |
| 15. C2×#107×C79 | C22 | semidirect |
| 17. #366×C79 | C22 | semidirect |
| 18. #357×C79 | C22 | semidirect |
| 19. C22⋊C16×C79 | C22 | semidirect |
| 80. C23×C79 | D16 | semidirect |
| 119. #33752 | C79 | direct (N × H) |
| 126. C2×#466 | C2×C79 | semidirect |
| 127. #2232 | C2×C79 | semidirect |
| 128. #2215 | C2×C79 | semidirect |
| 129. #2184 | C2×C79 | semidirect |
| 130. C2×C22⋊C16 | C2×C79 | semidirect |
| 131. C79 | #33752 | direct (N × H) |
| 132. C22×D8 | C22×C79 | semidirect |
| 135. C2×D16 | C22×C79 | semidirect |
| 136. C22×C16 | C22×C79 | semidirect |
| 140. #466 | C22×C79 | semidirect |
| 141. C2×#107 | C22×C79 | semidirect |
| 143. #366 | C22×C79 | semidirect |
| 144. #357 | C22×C79 | semidirect |
| 145. C22⋊C16 | C22×C79 | semidirect |
| 206. C23 | D16×C79 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#466×C79 | 2 |
| C79×#2232 | 2 |
| C79×#2215 | 2 |
| C79×#2184 | 2 |
| C2×C22⋊C16×C79 | 2 |
| #33752 | 79 |