Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
394 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 394 conjugacy classes of subgroups — too many to draw. Only its 74 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×C4⋊C4 | C23×C7 |
| prime radical | C22×C7×D4 | D4 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C23×C7 ◁ C2×C7×C4⋊C4 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C7×#1617 | C2 | semidirect |
| 4. C7×#1605 | C2 | semidirect |
| 5. C7×#1354 | C2 | semidirect |
| 6. C7×#1353 | C2 | semidirect |
| 27. #12223 | C7 | direct (N × H) |
| 65. #1617 | C2×C7 | semidirect |
| 66. #1605 | C2×C7 | semidirect |
| 67. #1354 | C2×C7 | semidirect |
| 68. #1353 | C2×C7 | semidirect |
| 78. C2×C7×D4 | Q16 | semidirect |
| 243. C2×D4 | C7×Q16 | semidirect |
| 329. C7 | #12223 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#2784 | 2 |
| C7×#2027 | 2 |
| C7×#1617 | 2 |
| C7×#1605 | 2 |
| C7×#1354 | 2 |
| C7×#1353 | 2 |
| C7×#1318 | 2 |
| #12223 | 7 |