Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
780 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 780 conjugacy classes of subgroups — too many to draw. Only its 106 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 | #392 |
| 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 |
|---|---|---|
| 3. C7×#2434 | C2 | semidirect |
| 4. C7×#1784 | C2 | semidirect |
| 5. C7×#1508 | C2 | semidirect |
| 7. C7×#1297 | C2 | semidirect |
| 31. #10330 | C7 | direct (N × H) |
| 82. #2434 | C2×C7 | semidirect |
| 83. #1784 | C2×C7 | semidirect |
| 84. #1508 | C2×C7 | semidirect |
| 86. #1297 | C2×C7 | semidirect |
| 414. C7 | #10330 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#3193 | 2 |
| C7×#2436 | 2 |
| C7×#2434 | 2 |
| C7×#1784 | 2 |
| C7×#1508 | 2 |
| C7×#1507 | 2 |
| C7×#1297 | 2 |
| #10330 | 7 |