Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
396 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 396 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C7 | #120 |
| commutator, Frattini, cosocle | C2×C42 | C23×C7 |
| prime radical | C22×C7×D4 | D4 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C7 ◁ C2×C42×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C7×#2100 | C2 | semidirect |
| 3. C7×#2095 | C2 | semidirect |
| 5. C7×#1272 | C2 | semidirect |
| 6. C7×#1271 | C2 | semidirect |
| 27. #7834 | C7 | direct (N × H) |
| 60. #2100 | C2×C7 | semidirect |
| 61. #2095 | C2×C7 | semidirect |
| 63. #1272 | C2×C7 | semidirect |
| 64. #1271 | C2×C7 | semidirect |
| 338. C7 | #7834 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#2808 | 2 |
| C7×#2100 | 2 |
| C7×#2095 | 2 |
| C7×#2094 | 2 |
| C7×#1272 | 2 |
| C7×#1271 | 2 |
| C7×#1270 | 2 |
| #7834 | 7 |