Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
252 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 252 conjugacy classes of subgroups — too many to draw. Only its 130 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C7×C8 | C2×D4 |
| commutator | C4 | C22×C7×C16 |
| Frattini, cosocle | C4×C8 | C23×C7 |
| socle | C22×C7 | C8×D4 |
| prime radical | C2×C7×D8 | C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7×C8 ◁ C4×C7×C16 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×D4×C16 | C2 | semidirect |
| 3. C7×#1367 | C2 | semidirect |
| 4. C7×D4⋊C16 | C2 | semidirect |
| 5. C7×#1312 | C2 | semidirect |
| 16. #12062 | C7 | direct (N × H) |
| 31. D4×C16 | C2×C7 | semidirect |
| 33. #1367 | C2×C7 | semidirect |
| 34. D4⋊C16 | C2×C7 | semidirect |
| 35. #1312 | C2×C7 | semidirect |
| 38. C7×D8 | C16 | semidirect |
| 65. C7×D4 | C2×C16 | semidirect |
| 76. C7×C8 | C2×C16 | semidirect |
| 104. D8 | C7×C16 | semidirect |
| 125. D4 | C2×C7×C16 | semidirect |
| 136. C8 | C2×C7×C16 | semidirect |
| 138. C7 | #12062 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×D4×C16 | 2 |
| C7×C8×D8 | 2 |
| C7×#1367 | 2 |
| C7×D4⋊C16 | 2 |
| C7×#1312 | 2 |
| #12062 | 7 |