Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
1044 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 1044 conjugacy classes of subgroups — too many to draw. Only its 106 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 | #392 |
| commutator, Frattini, cosocle | C23×C4 | C23×C7 |
| prime radical | C7×#2433 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×#3193 | C2 | semidirect |
| 2. C7×#2435 | C2 | semidirect |
| 4. C7×#1783 | C2 | semidirect |
| 5. C7×#1508 | C2 | semidirect |
| 6. C7×#1507 | C2 | semidirect |
| 7. C7×#1297 | C2 | semidirect |
| 31. #10329 | C7 | direct (N × H) |
| 33. C22×C7×Q8 | D4 | semidirect |
| 47. C2×C7×M16 | D4 | semidirect |
| 61. C2×C7×C4⋊C4 | D4 | semidirect |
| 80. C7×#95 | D4 | semidirect |
| 81. #3193 | C2×C7 | semidirect |
| 82. #2435 | C2×C7 | semidirect |
| 84. #1783 | C2×C7 | semidirect |
| 85. #1508 | C2×C7 | semidirect |
| 86. #1507 | C2×C7 | semidirect |
| 87. #1297 | C2×C7 | semidirect |
| 238. C22×Q8 | C7×D4 | semidirect |
| 252. C2×M16 | C7×D4 | semidirect |
| 266. C2×C4⋊C4 | C7×D4 | semidirect |
| 285. #95 | C7×D4 | semidirect |
| 453. C7 | #10329 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#3193 | 2 |
| C7×#2435 | 2 |
| C7×#2433 | 2 |
| C7×#1783 | 2 |
| C7×#1508 | 2 |
| C7×#1507 | 2 |
| C7×#1297 | 2 |
| #10329 | 7 |