Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
644 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 644 conjugacy classes of subgroups — too many to draw. Only its 118 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 | C2×#99 |
| commutator | C22×C4 | C22×C4×C7 |
| Frattini, cosocle | C2×C22⋊C4 | C23×C7 |
| prime radical | C7×#2541 | C2 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C2×C4×C7×D4 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C7×#1799 | C2 | semidirect |
| 3. C7×#1459 | C2 | semidirect |
| 4. C7×#1343 | C2 | semidirect |
| 6. C2×C7×#127 | C4 | semidirect |
| 12. C7×#392 | C4 | semidirect |
| 16. C4×C7×C4⋊C4 | C4 | semidirect |
| 20. #11877 | C7 | direct (N × H) |
| 54. #1799 | C2×C7 | semidirect |
| 55. #1459 | C2×C7 | semidirect |
| 56. #1343 | C2×C7 | semidirect |
| 108. C2×#127 | C4×C7 | semidirect |
| 114. #392 | C4×C7 | semidirect |
| 118. C4×C4⋊C4 | C4×C7 | semidirect |
| 347. C7 | #11877 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#2541 | 2 |
| C7×#1799 | 2 |
| C7×#1459 | 2 |
| C7×#1343 | 2 |
| #11877 | 7 |