Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
884 conjugacy classes of subgroups; only the 230 normal subgroups are shown.
This group has 884 conjugacy classes of subgroups — too many to draw. Only its 230 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×C7 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C7 |
| prime radical | C2×C7×#127 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×#3556 | C2 | semidirect |
| 4. C7×#1626 | C2 | semidirect |
| 5. C7×#1544 | C2 | semidirect |
| 6. C7×#1485 | C2 | semidirect |
| 10. C7×#557 | C22 | semidirect |
| 25. C7×#332 | C22 | semidirect |
| 26. C7×Q8⋊C8 | C22 | semidirect |
| 27. #18731 | C7 | direct (N × H) |
| 67. #3556 | C2×C7 | semidirect |
| 70. #1626 | C2×C7 | semidirect |
| 71. #1544 | C2×C7 | semidirect |
| 72. #1485 | C2×C7 | semidirect |
| 112. #557 | C22×C7 | semidirect |
| 127. #332 | C22×C7 | semidirect |
| 128. Q8⋊C8 | C22×C7 | semidirect |
| 265. C7 | #18731 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#3556 | 2 |
| C7×#3208 | 2 |
| C7×#3138 | 2 |
| C7×#1626 | 2 |
| C7×#1544 | 2 |
| C7×#1485 | 2 |
| #18731 | 7 |