Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
288 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 288 conjugacy classes of subgroups — too many to draw. Only its 82 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 | #447 |
| commutator, Frattini, cosocle | C4×C8 | C23×C7 |
| prime radical | C2×C7×D8 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C42×C7 ◁ C7×C8×Q8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C7×#2241 | C2 | semidirect |
| 4. C7×#1676 | C2 | semidirect |
| 5. C7×#1362 | C2 | semidirect |
| 6. C7×Q8⋊C16 | C2 | semidirect |
| 25. #12387 | C7 | direct (N × H) |
| 54. #2241 | C2×C7 | semidirect |
| 56. #1676 | C2×C7 | semidirect |
| 57. #1362 | C2×C7 | semidirect |
| 58. Q8⋊C16 | C2×C7 | semidirect |
| 230. C7 | #12387 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#3393 | 2 |
| C7×#2241 | 2 |
| C7×#2226 | 2 |
| C7×#1676 | 2 |
| C7×#1362 | 2 |
| C7×Q8⋊C16 | 2 |
| C7×#1331 | 2 |
| #12387 | 7 |