Finite Group DatabasebetaFinite Group DatabasebetaOrder 17152.
528 conjugacy classes of subgroups; only the 234 normal subgroups are shown.
This group has 528 conjugacy classes of subgroups — too many to draw. Only its 234 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 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C67 |
| prime radical | C2×D16×C67 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C67 ◁ C4×Q8×C67 ◁ C8×Q8×C67 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. Q8×D8×C67 | C2 | semidirect |
| 2. C67×#2252 | C2 | semidirect |
| 3. C67×#2224 | C2 | semidirect |
| 4. Q8×C16×C67 | C2 | semidirect |
| 23. D16×C67 | Q8 | direct (N × H) |
| 37. D8×C67 | C2×Q8 | semidirect |
| 46. C16×C67 | C2×Q8 | semidirect |
| 49. Q8×C67 | D16 | direct (N × H) |
| 69. Q8×D16 | C67 | direct (N × H) |
| 74. Q8×D8 | C2×C67 | semidirect |
| 75. #2252 | C2×C67 | semidirect |
| 76. #2224 | C2×C67 | semidirect |
| 77. Q8×C16 | C2×C67 | semidirect |
| 79. C67 | Q8×D16 | direct (N × H) |
| 97. D16 | Q8×C67 | direct (N × H) |
| 111. D8 | C2×Q8×C67 | semidirect |
| 120. C16 | C2×Q8×C67 | semidirect |
| 123. Q8 | D16×C67 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×D8×C67 | 2 |
| C67×#2252 | 2 |
| C67×#2224 | 2 |
| Q8×C16×C67 | 2 |
| C4×D16×C67 | 2 |
| Q8×D16 | 67 |