Finite Group DatabasebetaFinite Group DatabasebetaOrder 17152.
638 conjugacy classes of subgroups; only the 206 normal subgroups are shown.
This group has 638 conjugacy classes of subgroups — too many to draw. Only its 206 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×C67 | D42 |
| commutator, Frattini, cosocle | C42 | C24×C67 |
| prime radical | C67×#1712 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C67 ◁ C42×C67 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C67×#3396 | C2 | semidirect |
| 3. C67×#1668 | C2 | semidirect |
| 4. C67×#1620 | C2 | semidirect |
| 5. C8×Q16×C67 | C2 | semidirect |
| 79. #20021 | C67 | direct (N × H) |
| 83. #3396 | C2×C67 | semidirect |
| 85. #1668 | C2×C67 | semidirect |
| 86. #1620 | C2×C67 | semidirect |
| 87. C8×Q16 | C2×C67 | semidirect |
| 88. C67 | #20021 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C67×#3396 | 2 |
| C67×#1712 | 2 |
| C67×#1668 | 2 |
| C67×#1620 | 2 |
| C8×Q16×C67 | 2 |
| #20021 | 67 |