Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
404 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 404 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 | C24×C7 | C2×D4 |
| 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 |
|---|---|---|
| 27. #10383 | C7 | direct (N × H) |
| 29. C22×C7×Q8 | Q8 | semidirect |
| 42. C2×C7×C4⋊C4 | Q8 | semidirect |
| 185. C22×Q8 | C7×Q8 | semidirect |
| 198. C2×C4⋊C4 | C7×Q8 | semidirect |
| 347. C7 | #10383 | 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×#1996 | 2 |
| C7×#1511 | 2 |
| C7×#1508 | 2 |
| C7×#1305 | 2 |
| C7×#1297 | 2 |
| C7×#1284 | 2 |
| #10383 | 7 |