Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
156 conjugacy classes of subgroups; only the 22 normal subgroups are shown.
This group has 156 conjugacy classes of subgroups — too many to draw. Only its 22 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #20289847 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #20289847 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #20274344 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #16811528 | — | 2 | 1 | no | — |
| 5 | #16811528 | — | 2 | 1 | no | — |
| 6 | #16811000 | C22 | 1 | 1 | yes | characteristic |
| 7 | #16810997 | — | 2 | 1 | no | — |
| 8 | #16810971 | — | 2 | 1 | no | — |
| 9 | #16810344 | C4 | 1 | 1 | yes | characteristic, complemented |
| 10 | C53×#498 | C4 | 1 | 1 | yes | characteristic, complemented |
| 11 | #13133276 | — | 4 | 1 | no | — |
| 12 | #13133276 | — | 4 | 1 | no | — |
| 13 | C2×#11719924 | — | 4 | 2 | no | — |
| 14 | C2×D424 | — | 4 | 1 | no | — |
| 15 | #13133214 | D4 | 1 | 1 | yes | characteristic |
| 16 | #13133214 | D4 | 1 | 1 | yes | characteristic |
| 17 | #13133137 | — | 4 | 2 | no | — |
| 18 | C4×#88322 | — | 2 | 1 | no | — |
| 19 | C4⋊Q8×C53 | — | 2 | 1 | no | — |
| 20 | #106×C53 | — | 4 | 1 | no | — |
| 21 | C4×C8×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 22 | C2×Q8×C53 | — | 4 | 1 | no | — |
| 23 | C2×D212 | — | 2 | 1 | no | — |
| 24 | C2×D212 | — | 2 | 1 | no | — |
| 25 | C2×D212 | — | 4 | 1 | no | — |
| 26 | C2×D212 | — | 4 | 1 | no | — |
| 27 | C22⋊#6157 | — | 8 | 1 | no | — |
| 28 | C22⋊#6157 | — | 8 | 1 | no | — |
| 29 | C2×C8×C53 | — | 2 | 1 | no | — |
| 30 | C4⋊C4×C53 | — | 4 | 1 | no | — |
| 31 | C4⋊C4×C53 | — | 4 | 1 | no | — |
| 32 | C42×C53 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 33 | #11719924 | — | 8 | 4 | no | — |
| 34 | C2×#88322 | — | 2 | 2 | no | — |
| 35 | D424 | — | 8 | 2 | no | — |
| 36 | C22×D53 | — | 4 | 1 | no | — |
| 37 | C22×D53 | — | 4 | 1 | no | — |
| 38 | C2×#6157 | — | 16 | 1 | no | — |
| 39 | C2×#6157 | — | 16 | 1 | no | — |
| 40 | Q8×C53 | — | 8 | 2 | no | — |
| 41 | C2×C4×C53 | — | 2 | 1 | no | — |
| 42 | C2×C4×C53 | — | 8 | 1 | no | — |
| 43 | C2×C4×C53 | #99 | 1 | 1 | yes | characteristic |
| 44 | D212 | — | 8 | 1 | no | — |
| 45 | D212 | — | 8 | 1 | no | — |
| 46 | D212 | — | 8 | 2 | no | — |
| 47 | D212 | — | 8 | 2 | no | — |
| 48 | C8×C53 | — | 2 | 2 | no | — |
| 49 | #88322 | — | 4 | 2 | no | — |
| 50 | #7533 | — | 53 | 1 | no | maximal, Sylow |
| 51 | C22×C53 | #353 | 1 | 1 | yes | characteristic |
| 52 | C2×D53 | — | 8 | 1 | no | — |
| 53 | C2×D53 | — | 8 | 1 | no | — |
| 54 | C2×D53 | — | 16 | 1 | no | — |
| 55 | C2×D53 | — | 16 | 1 | no | — |
| 56 | #6157 | — | 32 | 1 | no | — |
| 57 | #6157 | — | 32 | 1 | no | — |
| 58 | C4×C53 | — | 2 | 2 | no | — |
| 59 | C4×C53 | — | 2 | 1 | no | — |
| 60 | C4×C53 | — | 16 | 1 | no | — |
| 61 | #1715 | — | 53 | 1 | no | — |
| 62 | #1408 | — | 53 | 1 | no | — |
| 63 | #1408 | — | 53 | 1 | no | — |
| 64 | C2×C53 | — | 2 | 1 | no | — |
| 65 | C2×C53 | #1409 | 1 | 1 | yes | characteristic |
| 66 | D53 | — | 16 | 1 | no | — |
| 67 | D53 | — | 16 | 1 | no | — |
| 68 | #498 | #6157 | 1 | 1 | yes | characteristic, complemented |
| 69 | #493 | — | 53 | 1 | no | — |
| 70 | #492 | — | 106 | 1 | no | — |
| 71 | #486 | — | 106 | 1 | no | — |
| 72 | #353 | — | 106 | 1 | no | — |
| 73 | #353 | — | 106 | 1 | no | — |
| 74 | #322 | — | 53 | 1 | no | — |
| 75 | C53 | #7533 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 76 | C2×SD16 | — | 212 | 2 | no | — |
| 77 | C2×D8 | — | 212 | 1 | no | — |
| 78 | C4⋊Q8 | — | 2 | 1 | no | — |
| 79 | #127 | — | 53 | 1 | no | — |
| 80 | #127 | — | 53 | 1 | no | — |
| 81 | #106 | — | 4 | 1 | no | — |
| 82 | #102 | — | 212 | 2 | no | — |
| 83 | #99 | — | 212 | 1 | no | — |
| 84 | #99 | — | 212 | 1 | no | — |
| 85 | C4×C8 | — | 106 | 1 | no | — |
| 86 | C4×C8 | C2×#6157 | 1 | 1 | yes | characteristic |
| 87 | C2×Q8 | — | 4 | 1 | no | — |
| 88 | C2×D4 | — | 106 | 1 | no | — |
| 89 | C2×D4 | — | 106 | 1 | no | — |
| 90 | C2×D4 | — | 212 | 1 | no | — |
| 91 | C2×D4 | — | 212 | 1 | no | — |
| 92 | SD16 | — | 424 | 4 | no | — |
| 93 | D8 | — | 424 | 2 | no | — |
| 94 | C2×C8 | — | 2 | 1 | no | — |
| 95 | C2×C8 | — | 106 | 2 | no | — |
| 96 | C4⋊C4 | — | 4 | 1 | no | — |
| 97 | C4⋊C4 | — | 4 | 1 | no | — |
| 98 | C22⋊C4 | — | 424 | 1 | no | — |
| 99 | C22⋊C4 | — | 424 | 1 | no | — |
| 100 | C42 | C22⋊#6157 | 1 | 1 | yes | characteristic |
| 101 | C23 | — | 212 | 1 | no | — |
| 102 | C23 | — | 212 | 1 | no | — |
| 103 | Q8 | — | 8 | 2 | no | — |
| 104 | D4 | — | 424 | 2 | no | — |
| 105 | D4 | — | 424 | 2 | no | — |
| 106 | D4 | — | 424 | 1 | no | — |
| 107 | D4 | — | 424 | 1 | no | — |
| 108 | C2×C4 | — | 2 | 1 | no | — |
| 109 | C2×C4 | — | 8 | 1 | no | — |
| 110 | C2×C4 | — | 848 | 1 | no | — |
| 111 | C2×C4 | — | 848 | 1 | no | — |
| 112 | C2×C4 | #13133275 | 1 | 1 | yes | characteristic |
| 113 | C8 | — | 2 | 2 | no | — |
| 114 | C8 | — | 212 | 2 | no | — |
| 115 | C22 | — | 848 | 1 | no | — |
| 116 | C22 | — | 848 | 1 | no | — |
| 117 | C22 | — | 424 | 1 | no | — |
| 118 | C22 | — | 424 | 1 | no | — |
| 119 | C22 | #16811527 | 1 | 1 | yes | characteristic |
| 120 | C4 | — | 2 | 2 | no | — |
| 121 | C4 | — | 2 | 1 | no | — |
| 122 | C4 | — | 16 | 1 | no | — |
| 123 | C4 | — | 1696 | 1 | no | — |
| 124 | C4 | — | 1696 | 1 | no | — |
| 125 | C2 | — | 2 | 1 | no | — |
| 126 | C2 | — | 848 | 1 | no | — |
| 127 | C2 | — | 848 | 1 | no | — |
| 128 | C2 | #20289848 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 9. #16810344 | C4 | semidirect |
| 10. C53×#498 | C4 | semidirect |
| 68. #498 | #6157 | semidirect |
| 75. C53 | #7533 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20289847 | 2 |
| #20274344 | 2 |
| #7533 | 53 |