Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
306 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 306 conjugacy classes of subgroups — too many to draw. Only its 82 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 | #447 |
| commutator, Frattini, cosocle | C4×C8 | C23×C7 |
| prime radical | C7×#493 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C42×C7 ◁ C7×C8×Q8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C7×#2225 | C2 | semidirect |
| 4. C7×#1674 | C2 | semidirect |
| 6. C7×Q8⋊C16 | C2 | semidirect |
| 7. C7×#1332 | C2 | semidirect |
| 25. #12381 | C7 | direct (N × H) |
| 55. #2225 | C2×C7 | semidirect |
| 56. #1674 | C2×C7 | semidirect |
| 58. Q8⋊C16 | C2×C7 | semidirect |
| 59. #1332 | C2×C7 | semidirect |
| 238. C7 | #12381 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#3396 | 2 |
| C7×#2240 | 2 |
| C7×#2225 | 2 |
| C7×#1674 | 2 |
| C7×#1361 | 2 |
| C7×Q8⋊C16 | 2 |
| C7×#1332 | 2 |
| #12381 | 7 |