Finite Group DatabasebetaFinite Group DatabasebetaOrder 1568.
470 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 470 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×C7⋊D7 |
| commutator, socle, cosocle | C22×C72 | C23 |
| Fitting | C2×C72×D4 | C2 |
| hypercenter | C2×D4 | C7⋊D7 |
| hypocenter | C72 | #121 |
| LCS | G ▷ C22×C72 ▷ C72 | |
| UCS | C1 ◁ C22 ◁ C2×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C72×D4 | C2 | semidirect |
| 2. C2×C22⋊(C7⋊D7) | C2 | semidirect |
| 3. C2×C22⋊(C7⋊D7) | C2 | semidirect |
| 4. C22×C7⋊Dic7 | C2 | semidirect |
| 5. C22⋊(C7⋊Dic7) | C2 | semidirect |
| 6. #11716070 | C2 | semidirect |
| 7. #11716068 | C2 | semidirect |
| 25. C22×C72 | D4 | semidirect |
| 33. C7⋊Dic7 | D4 | semidirect |
| 34. C2×C7×D4 | D7 | semidirect |
| 58. C72 | #121 | semidirect |
| 65. C22×C7 | C22⋊D7 | semidirect |
| 76. C2×D4 | C7⋊D7 | semidirect |
| 98. C7 | #6502 | semidirect |
| 107. C22 | C22⋊(C7⋊D7) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C72×D4 | 2 |
| C2×C22⋊(C7⋊D7) | 2 |
| C22×C7⋊Dic7 | 2 |
| C22⋊(C7⋊Dic7) | 2 |
| #11716070 | 2 |
| #11716068 | 2 |
| #6502 | 7 |