Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
142 conjugacy classes of subgroups; only the 22 normal subgroups are shown.
This group has 142 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 | #20289852 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #20289724 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #20273856 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #16811546 | — | 2 | 1 | no | — |
| 5 | #16811534 | — | 2 | 1 | no | — |
| 6 | #16811353 | C4 | 1 | 1 | yes | characteristic |
| 7 | #16811002 | C22 | 1 | 1 | yes | characteristic |
| 8 | #16810919 | — | 2 | 1 | no | — |
| 9 | #16810353 | — | 2 | 1 | no | — |
| 10 | C53×#333 | C4 | 1 | 1 | yes | characteristic |
| 11 | #13133291 | — | 4 | 1 | no | — |
| 12 | #13133278 | — | 4 | 1 | no | — |
| 13 | C2×#11719925 | — | 4 | 1 | no | — |
| 14 | #13133217 | D4 | 1 | 1 | yes | characteristic |
| 15 | #13133216 | — | 4 | 1 | no | — |
| 16 | #13133215 | — | 4 | 1 | no | — |
| 17 | C4×Dic106 | — | 2 | 1 | no | — |
| 18 | #13133182 | — | 4 | 1 | no | — |
| 19 | #13133151 | — | 4 | 1 | no | — |
| 20 | #13133142 | — | 4 | 1 | no | — |
| 21 | C4×#88322 | D4 | 1 | 1 | yes | characteristic |
| 22 | C4⋊Q8×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 23 | C4⋊C8×C53 | — | 2 | 1 | no | — |
| 24 | C2×Q8×C53 | — | 2 | 1 | no | — |
| 25 | C2×Dic106 | — | 2 | 1 | no | — |
| 26 | C2×Dic106 | — | 2 | 1 | no | — |
| 27 | C2×Dic106 | — | 4 | 1 | no | — |
| 28 | #11719940 | — | 8 | 1 | no | — |
| 29 | C2×C8×C53 | — | 4 | 1 | no | — |
| 30 | C4⋊C4×C53 | — | 4 | 1 | no | — |
| 31 | C42×C53 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 32 | #11719925 | — | 8 | 1 | no | — |
| 33 | #11719925 | — | 8 | 1 | no | — |
| 34 | C4⋊Dic53 | — | 2 | 1 | no | — |
| 35 | C4⋊Dic53 | — | 4 | 1 | no | — |
| 36 | #11719919 | — | 4 | 2 | no | — |
| 37 | #11719919 | — | 4 | 1 | no | — |
| 38 | C4×Dic53 | — | 4 | 1 | no | — |
| 39 | C4×Dic53 | — | 4 | 1 | no | — |
| 40 | C2×#88322 | — | 2 | 1 | no | — |
| 41 | #11719910 | — | 8 | 1 | no | — |
| 42 | Q8×C53 | — | 8 | 1 | no | — |
| 43 | C2×C4×C53 | — | 2 | 1 | no | — |
| 44 | C2×C4×C53 | — | 4 | 1 | no | — |
| 45 | C2×C4×C53 | #99 | 1 | 1 | yes | characteristic |
| 46 | C2×Dic53 | — | 4 | 1 | no | — |
| 47 | C2×Dic53 | — | 4 | 1 | no | — |
| 48 | C2×Dic53 | — | 4 | 1 | no | — |
| 49 | Dic106 | — | 2 | 2 | no | — |
| 50 | Dic106 | — | 4 | 1 | no | — |
| 51 | Dic106 | — | 8 | 1 | no | — |
| 52 | Dic106 | — | 16 | 1 | no | — |
| 53 | #88324 | — | 16 | 1 | no | — |
| 54 | C8×C53 | — | 8 | 1 | no | — |
| 55 | #88322 | — | 2 | 1 | no | — |
| 56 | #88322 | — | 2 | 1 | no | — |
| 57 | #7522 | — | 53 | 1 | no | maximal, Sylow |
| 58 | C22×C53 | AΣL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 59 | C4×C53 | — | 2 | 1 | no | — |
| 60 | C4×C53 | — | 2 | 1 | no | — |
| 61 | C4×C53 | — | 2 | 1 | no | — |
| 62 | C4×C53 | — | 8 | 1 | no | — |
| 63 | Dic53 | — | 4 | 2 | no | — |
| 64 | Dic53 | — | 8 | 1 | no | — |
| 65 | Dic53 | — | 8 | 1 | no | — |
| 66 | #1633 | — | 53 | 1 | no | — |
| 67 | #1411 | — | 53 | 1 | no | — |
| 68 | #1358 | — | 53 | 1 | no | — |
| 69 | C2×C53 | — | 2 | 1 | no | — |
| 70 | C2×C53 | #1402 | 1 | 1 | yes | characteristic |
| 71 | Q82 | — | 53 | 1 | no | — |
| 72 | #494 | — | 53 | 1 | no | — |
| 73 | #477 | — | 106 | 1 | no | — |
| 74 | #364 | — | 106 | 1 | no | — |
| 75 | #356 | — | 106 | 1 | no | — |
| 76 | #333 | #6157 | 1 | 1 | yes | characteristic |
| 77 | Q8⋊C8 | — | 106 | 1 | no | — |
| 78 | C53 | #7522 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 79 | C2×Q16 | — | 212 | 1 | no | — |
| 80 | C4⋊Q8 | — | 53 | 1 | no | — |
| 81 | C4⋊Q8 | — | 212 | 1 | no | — |
| 82 | C4⋊Q8 | — | 212 | 1 | no | — |
| 83 | C4⋊Q8 | C2×#6157 | 1 | 1 | yes | characteristic |
| 84 | C4×Q8 | — | 106 | 1 | no | — |
| 85 | C4×Q8 | — | 212 | 1 | no | — |
| 86 | M32 | — | 212 | 1 | no | — |
| 87 | #107 | — | 212 | 1 | no | — |
| 88 | C4⋊C8 | — | 2 | 1 | no | — |
| 89 | #103 | — | 212 | 1 | no | — |
| 90 | #101 | — | 212 | 1 | no | — |
| 91 | C4×C8 | — | 53 | 1 | no | — |
| 92 | C2×Q8 | — | 2 | 1 | no | — |
| 93 | C2×Q8 | — | 106 | 1 | no | — |
| 94 | C2×Q8 | — | 106 | 1 | no | — |
| 95 | C2×Q8 | — | 212 | 1 | no | — |
| 96 | Q16 | — | 424 | 1 | no | — |
| 97 | Q16 | — | 424 | 1 | no | — |
| 98 | M16 | — | 424 | 1 | no | — |
| 99 | C2×C8 | — | 4 | 1 | no | — |
| 100 | C2×C8 | — | 106 | 1 | no | — |
| 101 | C4⋊C4 | — | 4 | 1 | no | — |
| 102 | C4⋊C4 | — | 106 | 1 | no | — |
| 103 | C4⋊C4 | — | 212 | 1 | no | — |
| 104 | C4⋊C4 | — | 212 | 2 | no | — |
| 105 | C4⋊C4 | — | 212 | 1 | no | — |
| 106 | C42 | — | 212 | 1 | no | — |
| 107 | C42 | — | 212 | 1 | no | — |
| 108 | C42 | C22⋊#6157 | 1 | 1 | yes | characteristic |
| 109 | C16 | — | 424 | 1 | no | — |
| 110 | Q8 | — | 8 | 1 | no | — |
| 111 | Q8 | — | 848 | 1 | no | — |
| 112 | Q8 | — | 106 | 2 | no | — |
| 113 | Q8 | — | 424 | 1 | no | — |
| 114 | Q8 | — | 212 | 1 | no | — |
| 115 | C2×C4 | — | 2 | 1 | no | — |
| 116 | C2×C4 | — | 4 | 1 | no | — |
| 117 | C2×C4 | — | 212 | 1 | no | — |
| 118 | C2×C4 | — | 212 | 1 | no | — |
| 119 | C2×C4 | — | 212 | 1 | no | — |
| 120 | C2×C4 | #13133276 | 1 | 1 | yes | characteristic |
| 121 | C8 | — | 8 | 1 | no | — |
| 122 | C8 | — | 848 | 1 | no | — |
| 123 | C8 | — | 106 | 1 | no | — |
| 124 | C8 | — | 106 | 1 | no | — |
| 125 | C22 | #16811524 | 1 | 1 | yes | characteristic |
| 126 | C4 | — | 2 | 1 | no | — |
| 127 | C4 | — | 2 | 1 | no | — |
| 128 | C4 | — | 2 | 1 | no | — |
| 129 | C4 | — | 8 | 1 | no | — |
| 130 | C4 | — | 424 | 1 | no | — |
| 131 | C4 | — | 424 | 1 | no | — |
| 132 | C4 | — | 212 | 2 | no | — |
| 133 | C2 | — | 2 | 1 | no | — |
| 134 | C2 | #20289835 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 78. C53 | #7522 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20289852 | 2 |
| #20289724 | 2 |
| #20273856 | 2 |
| #7522 | 53 |