Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
456 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 456 conjugacy classes of subgroups — too many to draw. Only its 70 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 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C22⋊C4 | C23×C7 |
| prime radical | C7×#1595 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C23×C7 ◁ C2×C7×C22⋊C4 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×#2824 | C2 | semidirect |
| 2. C7×#2044 | C2 | semidirect |
| 3. C7×#1601 | C2 | semidirect |
| 5. C7×#1351 | C2 | semidirect |
| 6. C7×#1343 | C2 | semidirect |
| 7. C7×#1326 | C2 | semidirect |
| 27. #12136 | C7 | direct (N × H) |
| 64. #2824 | C2×C7 | semidirect |
| 65. #2044 | C2×C7 | semidirect |
| 66. #1601 | C2×C7 | semidirect |
| 68. #1351 | C2×C7 | semidirect |
| 69. #1343 | C2×C7 | semidirect |
| 70. #1326 | C2×C7 | semidirect |
| 392. C7 | #12136 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#2824 | 2 |
| C7×#2044 | 2 |
| C7×#1601 | 2 |
| C7×#1595 | 2 |
| C7×#1351 | 2 |
| C7×#1343 | 2 |
| C7×#1326 | 2 |
| #12136 | 7 |