Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
512 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 512 conjugacy classes of subgroups — too many to draw. Only its 102 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 | C2×C42 | C23×C7 |
| prime radical | C22×C7×D4 | D4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C7×#3139 | C2 | semidirect |
| 4. C7×#1558 | C2 | semidirect |
| 6. C7×#1506 | C2 | semidirect |
| 7. C7×#1300 | C2 | semidirect |
| 31. #10098 | C7 | direct (N × H) |
| 77. #3139 | C2×C7 | semidirect |
| 79. #1558 | C2×C7 | semidirect |
| 81. #1506 | C2×C7 | semidirect |
| 82. #1300 | C2×C7 | semidirect |
| 334. C7 | #10098 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#3215 | 2 |
| C7×#3139 | 2 |
| C7×#2640 | 2 |
| C7×#1558 | 2 |
| C7×#1527 | 2 |
| C7×#1506 | 2 |
| C7×#1300 | 2 |
| #10098 | 7 |