Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
254 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 254 conjugacy classes of subgroups — too many to draw. Only its 114 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C2×D8 |
| commutator | C8 | C22×C7×C8 |
| Frattini, cosocle | C4×C8 | C23×C7 |
| socle | C22×C7 | C4×D8 |
| prime radical | C2×C7×D16 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C4×C7×C8 ◁ C7×C82 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C7×C8×D8 | C2 | semidirect |
| 3. C7×#1372 | C2 | semidirect |
| 4. C7×#1331 | C2 | semidirect |
| 5. C7×#1311 | C2 | semidirect |
| 16. #12045 | C7 | direct (N × H) |
| 20. C7×D16 | C8 | semidirect |
| 32. C8×D8 | C2×C7 | semidirect |
| 33. #1372 | C2×C7 | semidirect |
| 34. #1331 | C2×C7 | semidirect |
| 35. #1311 | C2×C7 | semidirect |
| 39. C7×D8 | C2×C8 | semidirect |
| 51. C7×C16 | C2×C8 | semidirect |
| 80. D16 | C7×C8 | semidirect |
| 103. D8 | C2×C7×C8 | semidirect |
| 115. C16 | C2×C7×C8 | semidirect |
| 135. C7 | #12045 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×C7×D16 | 2 |
| C7×C8×D8 | 2 |
| C7×#1372 | 2 |
| C7×#1331 | 2 |
| C7×#1311 | 2 |
| #12045 | 7 |