Finite Group DatabasebetaFinite Group DatabasebetaOrder 17152.
274 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 274 conjugacy classes of subgroups — too many to draw. Only its 86 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 | #459 |
| commutator, Frattini, cosocle | C4×C8 | C23×C67 |
| prime radical | C67×#1712 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C67 ◁ C42×C67 ◁ C82×C67 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C67×#2207 | C2 | semidirect |
| 3. C8×Q16×C67 | C2 | semidirect |
| 4. C67×#1370 | C2 | semidirect |
| 5. C67×#1361 | C2 | semidirect |
| 40. Q16×C67 | D8 | semidirect |
| 75. C8×C67 | SD32 | semidirect |
| 89. #12425 | C67 | direct (N × H) |
| 94. #2207 | C2×C67 | semidirect |
| 96. C8×Q16 | C2×C67 | semidirect |
| 97. #1370 | C2×C67 | semidirect |
| 98. #1361 | C2×C67 | semidirect |
| 99. C67 | #12425 | direct (N × H) |
| 134. Q16 | D8×C67 | semidirect |
| 169. C8 | SD32×C67 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C67×#2207 | 2 |
| C67×#1712 | 2 |
| C8×Q16×C67 | 2 |
| C67×#1370 | 2 |
| C67×#1361 | 2 |
| #12425 | 67 |