Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
446 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 446 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 | C22×C7 | #393 |
| commutator, Frattini, cosocle | C23×C4 | C23×C7 |
| prime radical | C2×C7×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×#2665 | C2 | semidirect |
| 5. C7×#1305 | C2 | semidirect |
| 6. C7×#1297 | C2 | semidirect |
| 7. C7×#1284 | C2 | semidirect |
| 27. #10382 | C7 | direct (N × H) |
| 30. C22×C7×D4 | Q8 | semidirect |
| 42. C2×C7×C4⋊C4 | Q8 | semidirect |
| 68. #2665 | C2×C7 | semidirect |
| 72. #1305 | C2×C7 | semidirect |
| 73. #1297 | C2×C7 | semidirect |
| 74. #1284 | C2×C7 | semidirect |
| 200. C22×D4 | C7×Q8 | semidirect |
| 212. C2×C4⋊C4 | C7×Q8 | semidirect |
| 380. C7 | #10382 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#2665 | 2 |
| C7×#1964 | 2 |
| C7×#1510 | 2 |
| C7×#1507 | 2 |
| C7×#1305 | 2 |
| C7×#1297 | 2 |
| C7×#1284 | 2 |
| #10382 | 7 |