Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
752 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 752 conjugacy classes of subgroups — too many to draw. Only its 118 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 | C2×D4 | C22×C4×C7 |
| Frattini, cosocle | C22×D4 | C23×C7 |
| prime radical | C2×C7×H(5,𝔽2) | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C23×C7 ◁ C22×C7×D4 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C7×#2372 | C2 | semidirect |
| 3. C2×C7×#354 | C2 | semidirect |
| 5. C7×#1390 | C2 | semidirect |
| 10. C7×#455 | C4 | semidirect |
| 22. #13048 | C7 | direct (N × H) |
| 63. #2372 | C2×C7 | semidirect |
| 64. C2×#354 | C2×C7 | semidirect |
| 66. #1390 | C2×C7 | semidirect |
| 80. C2×C7×Q8 | C22⋊C4 | semidirect |
| 122. C42×C7 | C22⋊C4 | semidirect |
| 127. #455 | C4×C7 | semidirect |
| 265. C2×Q8 | C7×C22⋊C4 | semidirect |
| 307. C42 | C7×C22⋊C4 | semidirect |
| 366. C7 | #13048 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C7×#455 | 2 |
| C7×#2372 | 2 |
| C2×C7×#354 | 2 |
| C7×#1790 | 2 |
| C7×#1390 | 2 |
| #13048 | 7 |