Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
316 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 316 conjugacy classes of subgroups — too many to draw. Only its 78 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 | C22×C8 | C23×C7 |
| prime radical | C7×#466 | C22 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C2×C7×C4⋊C8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C7×#2235 | C2 | semidirect |
| 4. C7×#2086 | C2 | semidirect |
| 5. C7×#1348 | C2 | semidirect |
| 7. C7×#1322 | C2 | semidirect |
| 27. #12290 | C7 | direct (N × H) |
| 62. #2235 | C2×C7 | semidirect |
| 64. #2086 | C2×C7 | semidirect |
| 65. #1348 | C2×C7 | semidirect |
| 67. #1322 | C2×C7 | semidirect |
| 280. C7 | #12290 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#3250 | 2 |
| C7×#2235 | 2 |
| C7×#2232 | 2 |
| C7×#2086 | 2 |
| C7×#1348 | 2 |
| C7×#1347 | 2 |
| C7×#1322 | 2 |
| #12290 | 7 |