Finite Group DatabasebetaFinite Group DatabasebetaOrder 17152.
528 conjugacy classes of subgroups; only the 218 normal subgroups are shown.
This group has 528 conjugacy classes of subgroups — too many to draw. Only its 218 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 | C67×#3391 | C2 |
| 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 |
|---|---|---|
| 2. Q8×Q16×C67 | C2 | semidirect |
| 3. C67×#2240 | C2 | semidirect |
| 4. C67×#2226 | C2 | semidirect |
| 5. C67×#2207 | C2 | semidirect |
| 6. Q8×C16×C67 | C2 | semidirect |
| 7. C4×SD32×C67 | C2 | semidirect |
| 55. Q16×C67 | #41 | semidirect |
| 75. Q8×C67 | SD32 | semidirect |
| 101. #33929 | C67 | direct (N × H) |
| 107. Q8×Q16 | C2×C67 | semidirect |
| 108. #2240 | C2×C67 | semidirect |
| 109. #2226 | C2×C67 | semidirect |
| 110. #2207 | C2×C67 | semidirect |
| 111. Q8×C16 | C2×C67 | semidirect |
| 112. C4×SD32 | C2×C67 | semidirect |
| 113. C67 | #33929 | direct (N × H) |
| 161. Q16 | #41×C67 | semidirect |
| 181. Q8 | SD32×C67 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C67×#3391 | 2 |
| Q8×Q16×C67 | 2 |
| C67×#2240 | 2 |
| C67×#2226 | 2 |
| C67×#2207 | 2 |
| Q8×C16×C67 | 2 |
| C4×SD32×C67 | 2 |
| #33929 | 67 |