Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
224 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 41 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 | #16811353 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #16810808 | C2 | 1 | 2 | yes | maximal |
| 3 | #16810805 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #16810421 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C53×#356 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #13133265 | C22 | 1 | 1 | yes | characteristic |
| 7 | #13133250 | — | 2 | 2 | no | — |
| 8 | #13133248 | — | 2 | 1 | no | — |
| 9 | #13133216 | — | 2 | 2 | no | — |
| 10 | #13133215 | — | 2 | 1 | no | — |
| 11 | #13133215 | C22 | 1 | 2 | yes | — |
| 12 | C4×Dic106 | — | 2 | 1 | no | — |
| 13 | #13133182 | — | 2 | 2 | no | — |
| 14 | #13133145 | — | 2 | 1 | no | — |
| 15 | #13133143 | — | 2 | 2 | no | — |
| 16 | #13133136 | C22 | 1 | 1 | yes | characteristic |
| 17 | C4⋊Q8×C53 | C22 | 1 | 1 | yes | characteristic |
| 18 | #101×C53 | C22 | 1 | 2 | yes | — |
| 19 | C2×Q8×C53 | D4 | 1 | 1 | yes | characteristic |
| 20 | C2×Q8×C53 | C23 | 1 | 1 | yes | characteristic |
| 21 | #11719949 | — | 4 | 1 | no | — |
| 22 | Q8×D53 | — | 4 | 1 | no | — |
| 23 | #11719945 | — | 2 | 2 | no | — |
| 24 | #11719945 | — | 2 | 1 | no | — |
| 25 | C2×Dic106 | — | 2 | 1 | no | — |
| 26 | C2×Dic106 | — | 2 | 1 | no | — |
| 27 | C2×Dic106 | D4 | 1 | 2 | yes | — |
| 28 | M16×C53 | — | 2 | 2 | no | — |
| 29 | C4⋊C4×C53 | — | 2 | 2 | no | — |
| 30 | C42×C53 | D4 | 1 | 1 | yes | characteristic |
| 31 | #11719925 | — | 4 | 1 | no | — |
| 32 | #11719924 | — | 4 | 1 | no | — |
| 33 | C4⋊Dic53 | — | 2 | 1 | no | — |
| 34 | #11719919 | — | 2 | 2 | no | — |
| 35 | #11719919 | — | 2 | 2 | no | — |
| 36 | #11719919 | — | 2 | 2 | no | — |
| 37 | C4×Dic53 | — | 2 | 1 | no | — |
| 38 | C4×Dic53 | — | 2 | 2 | no | — |
| 39 | C4×Dic53 | D4 | 1 | 2 | yes | — |
| 40 | #11719917 | — | 2 | 1 | no | — |
| 41 | Dic212 | — | 4 | 2 | no | — |
| 42 | #11719913 | — | 4 | 2 | no | — |
| 43 | Q8×C53 | — | 4 | 1 | no | — |
| 44 | Q8×C53 | — | 4 | 1 | no | — |
| 45 | C2×C4×C53 | — | 2 | 1 | no | — |
| 46 | C2×C4×C53 | — | 2 | 1 | no | — |
| 47 | C2×C4×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 48 | C2×C4×C53 | C2×D4 | 1 | 2 | yes | — |
| 49 | C22⋊D53 | — | 4 | 1 | no | — |
| 50 | C2×Dic53 | — | 2 | 2 | no | — |
| 51 | C2×Dic53 | — | 2 | 2 | no | — |
| 52 | D212 | — | 2 | 2 | no | — |
| 53 | D212 | — | 2 | 1 | no | — |
| 54 | C4×D53 | — | 4 | 1 | no | — |
| 55 | C4×D53 | — | 4 | 2 | no | — |
| 56 | Dic106 | — | 2 | 4 | no | — |
| 57 | Dic106 | — | 2 | 2 | no | — |
| 58 | Dic106 | — | 2 | 1 | no | — |
| 59 | Dic106 | — | 4 | 2 | no | — |
| 60 | Dic106 | — | 8 | 1 | no | — |
| 61 | C8×C53 | — | 4 | 2 | no | — |
| 62 | #88322 | — | 4 | 1 | no | — |
| 63 | C22×C53 | #120 | 1 | 1 | yes | characteristic |
| 64 | C2×D53 | — | 4 | 1 | no | — |
| 65 | C4×C53 | — | 2 | 2 | no | — |
| 66 | C4×C53 | — | 2 | 1 | no | — |
| 67 | C4×C53 | — | 2 | 2 | no | — |
| 68 | C4×C53 | — | 4 | 1 | no | — |
| 69 | Dic53 | — | 2 | 4 | no | — |
| 70 | Dic53 | — | 4 | 1 | no | — |
| 71 | Dic53 | — | 4 | 2 | no | — |
| 72 | #2205 | — | 53 | 1 | no | maximal, Sylow |
| 73 | C2×C53 | — | 2 | 1 | no | — |
| 74 | C2×C53 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 75 | D53 | — | 8 | 1 | no | — |
| 76 | Q82 | — | 53 | 1 | no | — |
| 77 | #456 | — | 53 | 1 | no | — |
| 78 | #456 | — | 53 | 2 | no | — |
| 79 | #356 | — | 53 | 2 | no | — |
| 80 | #356 | D53 | 1 | 1 | yes | characteristic, complemented |
| 81 | C53 | #2205 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 82 | 2−1+4 | — | 53 | 1 | no | — |
| 83 | #137 | — | 106 | 2 | no | — |
| 84 | #137 | — | 106 | 1 | no | — |
| 85 | C4⋊Q8 | — | 53 | 2 | no | — |
| 86 | C4⋊Q8 | — | 106 | 2 | no | — |
| 87 | C4⋊Q8 | — | 106 | 1 | no | — |
| 88 | C4⋊Q8 | C2×D53 | 1 | 1 | yes | characteristic |
| 89 | C4×Q8 | — | 106 | 2 | no | — |
| 90 | C4×Q8 | — | 106 | 1 | no | — |
| 91 | #104 | — | 106 | 2 | no | — |
| 92 | #104 | — | 106 | 1 | no | — |
| 93 | #101 | — | 53 | 1 | no | — |
| 94 | #101 | C2×D53 | 1 | 2 | yes | — |
| 95 | #41 | — | 106 | 1 | no | — |
| 96 | #41 | — | 106 | 2 | no | — |
| 97 | #41 | — | 212 | 1 | no | — |
| 98 | C2×Q8 | — | 53 | 2 | no | — |
| 99 | C2×Q8 | — | 106 | 1 | no | — |
| 100 | C2×Q8 | — | 106 | 1 | no | — |
| 101 | C2×Q8 | — | 212 | 1 | no | — |
| 102 | C2×Q8 | D212 | 1 | 1 | yes | characteristic |
| 103 | C2×Q8 | C22×D53 | 1 | 1 | yes | characteristic |
| 104 | Q16 | — | 212 | 2 | no | — |
| 105 | Q16 | — | 212 | 1 | no | — |
| 106 | SD16 | — | 212 | 2 | no | — |
| 107 | SD16 | — | 212 | 1 | no | — |
| 108 | M16 | — | 2 | 2 | no | — |
| 109 | M16 | — | 106 | 1 | no | — |
| 110 | C4⋊C4 | — | 2 | 2 | no | — |
| 111 | C4⋊C4 | — | 106 | 2 | no | — |
| 112 | C4⋊C4 | — | 106 | 1 | no | — |
| 113 | C4⋊C4 | — | 106 | 2 | no | — |
| 114 | C4⋊C4 | — | 106 | 2 | no | — |
| 115 | C42 | — | 53 | 2 | no | — |
| 116 | C42 | — | 106 | 2 | no | — |
| 117 | C42 | — | 106 | 1 | no | — |
| 118 | C42 | D212 | 1 | 1 | yes | characteristic |
| 119 | Q8 | — | 4 | 1 | no | — |
| 120 | Q8 | — | 4 | 1 | no | — |
| 121 | Q8 | — | 106 | 4 | no | — |
| 122 | Q8 | — | 106 | 2 | no | — |
| 123 | Q8 | — | 106 | 1 | no | — |
| 124 | Q8 | — | 424 | 1 | no | — |
| 125 | Q8 | — | 212 | 2 | no | — |
| 126 | D4 | — | 106 | 2 | no | — |
| 127 | D4 | — | 106 | 1 | no | — |
| 128 | D4 | — | 212 | 1 | no | — |
| 129 | C2×C4 | — | 2 | 1 | no | — |
| 130 | C2×C4 | — | 2 | 1 | no | — |
| 131 | C2×C4 | — | 106 | 2 | no | — |
| 132 | C2×C4 | — | 106 | 2 | no | — |
| 133 | C2×C4 | — | 212 | 2 | no | — |
| 134 | C2×C4 | — | 212 | 1 | no | — |
| 135 | C2×C4 | C2×D212 | 1 | 1 | yes | characteristic |
| 136 | C2×C4 | D4×D53 | 1 | 2 | yes | — |
| 137 | C8 | — | 4 | 2 | no | — |
| 138 | C8 | — | 212 | 1 | no | — |
| 139 | C22 | — | 212 | 1 | no | — |
| 140 | C22 | #13133187 | 1 | 1 | yes | characteristic |
| 141 | C4 | — | 2 | 2 | no | — |
| 142 | C4 | — | 2 | 2 | no | — |
| 143 | C4 | — | 2 | 1 | no | — |
| 144 | C4 | — | 4 | 1 | no | — |
| 145 | C4 | — | 106 | 4 | no | — |
| 146 | C4 | — | 212 | 2 | no | — |
| 147 | C4 | — | 212 | 1 | no | — |
| 148 | C2 | — | 2 | 1 | no | — |
| 149 | C2 | — | 424 | 1 | no | — |
| 150 | C2 | #16810811 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #16811353 | C2 | semidirect |
| 4. #16810421 | C2 | semidirect |
| 5. C53×#356 | C2 | semidirect |
| 80. #356 | D53 | semidirect |
| 81. C53 | #2205 | semidirect |