Finite Group DatabasebetaFinite Group DatabasebetaOrder 28544.
400 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 400 conjugacy classes of subgroups — too many to draw. Only its 184 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×C223 | C2×D8 |
| commutator, Frattini, cosocle | C8 | C24×C223 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C223 ◁ C22×C4×C223 ◁ C22×C8×C223 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#135×C223 | C2 | semidirect |
| 2. C22×D8×C223 | C2 | semidirect |
| 3. #509×C223 | C2 | direct (N × H) |
| 4. C2×SD32×C223 | C2 | semidirect |
| 5. C2×D16×C223 | C2 | semidirect |
| 6. C2×M32×C223 | C2 | semidirect |
| 10. #135×C223 | C22 | semidirect |
| 11. C2×Q16×C223 | C22 | semidirect |
| 16. C2×D8×C223 | C22 | semidirect |
| 17. C2×D8×C223 | C22 | semidirect |
| 19. SD32×C223 | C22 | semidirect |
| 20. D16×C223 | C22 | semidirect |
| 21. M32×C223 | C22 | semidirect |
| 22. C2×C16×C223 | C22 | semidirect |
| 36. Q16×C223 | C23 | semidirect |
| 42. D8×C223 | C23 | semidirect |
| 47. C16×C223 | C23 | semidirect |
| 85. C2×C223 | #509 | direct (N × H) |
| 87. C223 | C2×#509 | direct (N × H) |
| 88. C2×#509 | C223 | direct (N × H) |
| 89. C2×#135 | C2×C223 | semidirect |
| 90. C22×D8 | C2×C223 | semidirect |
| 91. #509 | C2×C223 | direct (N × H) |
| 92. C2×SD32 | C2×C223 | semidirect |
| 93. C2×D16 | C2×C223 | semidirect |
| 94. C2×M32 | C2×C223 | semidirect |
| 98. #135 | C22×C223 | semidirect |
| 99. C2×Q16 | C22×C223 | semidirect |
| 104. C2×D8 | C22×C223 | semidirect |
| 105. C2×D8 | C22×C223 | semidirect |
| 107. SD32 | C22×C223 | semidirect |
| 108. D16 | C22×C223 | semidirect |
| 109. M32 | C22×C223 | semidirect |
| 110. C2×C16 | C22×C223 | semidirect |
| 124. Q16 | C23×C223 | semidirect |
| 130. D8 | C23×C223 | semidirect |
| 135. C16 | C23×C223 | semidirect |
| 173. C2 | #509×C223 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#135×C223 | 2 |
| C22×D8×C223 | 2 |
| #509×C223 | 2 |
| C2×SD32×C223 | 2 |
| C2×D16×C223 | 2 |
| C2×M32×C223 | 2 |
| C2×#509 | 223 |