Finite Group DatabasebetaFinite Group DatabasebetaOrder 17152.
1786 conjugacy classes of subgroups; only the 366 normal subgroups are shown.
This group has 1786 conjugacy classes of subgroups — too many to draw. Only its 366 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C67 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C67 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C67 ◁ C24×C4×C67 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C24×C8×C67 | C2 | semidirect |
| 2. C2×#466×C67 | C2 | semidirect |
| 3. C67×#2847 | C2 | semidirect |
| 4. C67×#1893 | C2 | semidirect |
| 5. C67×#1808 | C2 | semidirect |
| 7. C22×D8×C67 | C22 | semidirect |
| 14. C2×#107×C67 | C22 | semidirect |
| 15. C2×#102×C67 | C22 | semidirect |
| 18. #340×C67 | C22 | semidirect |
| 31. C22×C8×C67 | D4 | semidirect |
| 46. C24×C67 | D8 | semidirect |
| 101. C8×C67 | #120 | semidirect |
| 110. C22×C67 | #466 | semidirect |
| 118. #32930 | C67 | direct (N × H) |
| 124. C24×C8 | C2×C67 | semidirect |
| 125. C2×#466 | C2×C67 | semidirect |
| 126. #2847 | C2×C67 | semidirect |
| 127. #1893 | C2×C67 | semidirect |
| 128. #1808 | C2×C67 | semidirect |
| 129. C67 | #32930 | direct (N × H) |
| 131. C22×D8 | C22×C67 | semidirect |
| 138. C2×#107 | C22×C67 | semidirect |
| 139. C2×#102 | C22×C67 | semidirect |
| 142. #340 | C22×C67 | semidirect |
| 155. C22×C8 | D4×C67 | semidirect |
| 170. C24 | D8×C67 | semidirect |
| 225. C8 | #120×C67 | semidirect |
| 234. C22 | #466×C67 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×C8×C67 | 2 |
| C2×#466×C67 | 2 |
| C67×#2847 | 2 |
| C67×#1893 | 2 |
| C67×#1808 | 2 |
| #32930 | 67 |