Finite Group DatabasebetaFinite Group DatabasebetaOrder 18176.
814 conjugacy classes of subgroups; only the 230 normal subgroups are shown.
This group has 814 conjugacy classes of subgroups — too many to draw. Only its 230 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×C71 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C24×C71 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C71 ◁ C2×C42×C71 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C71×#3207 | C2 | semidirect |
| 2. C71×#3205 | C2 | semidirect |
| 3. C71×#1721 | C2 | semidirect |
| 4. C71×#1717 | C2 | semidirect |
| 5. C71×#1712 | C2 | semidirect |
| 6. C71×#1711 | C2 | semidirect |
| 7. C71×#1454 | C2 | semidirect |
| 12. #493×C71 | C22 | semidirect |
| 13. #492×C71 | C22 | semidirect |
| 14. #487×C71 | C22 | semidirect |
| 17. #486×C71 | C22 | semidirect |
| 23. #322×C71 | C22 | semidirect |
| 24. C82×C71 | C22 | semidirect |
| 106. #19743 | C71 | direct (N × H) |
| 112. #3207 | C2×C71 | semidirect |
| 113. #3205 | C2×C71 | semidirect |
| 114. #1721 | C2×C71 | semidirect |
| 115. #1717 | C2×C71 | semidirect |
| 116. #1712 | C2×C71 | semidirect |
| 117. #1711 | C2×C71 | semidirect |
| 118. #1454 | C2×C71 | semidirect |
| 119. C71 | #19743 | direct (N × H) |
| 124. #493 | C22×C71 | semidirect |
| 125. #492 | C22×C71 | semidirect |
| 126. #487 | C22×C71 | semidirect |
| 129. #486 | C22×C71 | semidirect |
| 135. #322 | C22×C71 | semidirect |
| 136. C82 | C22×C71 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C71×#3207 | 2 |
| C71×#3205 | 2 |
| C71×#1721 | 2 |
| C71×#1717 | 2 |
| C71×#1712 | 2 |
| C71×#1711 | 2 |
| C71×#1454 | 2 |
| #19743 | 71 |