Finite Group DatabasebetaFinite Group DatabasebetaOrder 1920.
224 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 34 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 | #11749388 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #11746799 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C3×#64927 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #10598575 | — | 3 | 1 | no | maximal, Hall |
| 5 | #92098 | C4 | 1 | 1 | yes | characteristic, complemented |
| 6 | C3×#4093 | C22 | 1 | 1 | yes | characteristic |
| 7 | #91312 | — | 2 | 1 | no | — |
| 8 | #91228 | — | 2 | 1 | no | — |
| 9 | C3×#4002 | — | 2 | 1 | no | — |
| 10 | C5×#836 | C4 | 1 | 1 | yes | characteristic, complemented |
| 11 | #90981 | — | 2 | 1 | no | — |
| 12 | #67460 | — | 5 | 1 | no | maximal, Hall |
| 13 | #65042 | — | 3 | 1 | no | — |
| 14 | #64928 | — | 3 | 1 | no | — |
| 15 | #64927 | D3 | 1 | 1 | yes | characteristic, complemented |
| 16 | C2×C3×C5×Q8 | C2×C4 | 1 | 1 | yes | characteristic |
| 17 | #6821 | — | 4 | 1 | no | — |
| 18 | Q8×D15 | — | 4 | 1 | no | — |
| 19 | #6817 | — | 2 | 1 | no | — |
| 20 | #6817 | — | 4 | 1 | no | — |
| 21 | C2×C3×D20 | D4 | 1 | 1 | yes | characteristic |
| 22 | #6763 | — | 4 | 1 | no | — |
| 23 | C2×C3×#170 | — | 4 | 1 | no | — |
| 24 | C5×#208 | — | 2 | 1 | no | — |
| 25 | C3×#685 | — | 4 | 1 | no | — |
| 26 | C3×C4×Dic5 | D4 | 1 | 1 | yes | characteristic |
| 27 | #6657 | — | 4 | 1 | no | — |
| 28 | #6653 | — | 4 | 1 | no | — |
| 29 | #5062 | — | 5 | 1 | no | — |
| 30 | C3×#329 | — | 5 | 1 | no | — |
| 31 | #4541 | — | 5 | 1 | no | — |
| 32 | #4147 | — | 3 | 1 | no | — |
| 33 | #4093 | C2×D3 | 1 | 1 | yes | characteristic |
| 34 | #4054 | — | 6 | 1 | no | — |
| 35 | #4002 | — | 2 | 1 | no | — |
| 36 | #3999 | — | 6 | 1 | no | — |
| 37 | C5×#101 | — | 3 | 1 | no | — |
| 38 | #3957 | — | 6 | 1 | no | — |
| 39 | #1404 | — | 15 | 1 | no | Sylow |
| 40 | C22×C3×D5 | — | 2 | 1 | no | — |
| 41 | C2×ΓL(1,𝔽16) | — | 8 | 1 | no | — |
| 42 | C3×C5×Q8 | — | 4 | 1 | no | — |
| 43 | C2×C3×C4×C5 | — | 2 | 1 | no | — |
| 44 | C2×C3×C4×C5 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 45 | #1220 | — | 4 | 1 | no | — |
| 46 | D60 | — | 2 | 1 | no | — |
| 47 | D60 | — | 4 | 1 | no | — |
| 48 | C4×D15 | — | 4 | 1 | no | — |
| 49 | C4×D15 | — | 4 | 2 | no | — |
| 50 | Dic30 | — | 2 | 1 | no | — |
| 51 | Dic30 | — | 4 | 1 | no | — |
| 52 | C2×C3×Dic5 | — | 2 | 1 | no | — |
| 53 | C3×D20 | — | 4 | 1 | no | — |
| 54 | C3×#170 | — | 4 | 2 | no | — |
| 55 | C5×C3⋊C8 | — | 4 | 1 | no | — |
| 56 | #1007 | — | 5 | 1 | no | — |
| 57 | C3×#124 | — | 5 | 1 | no | — |
| 58 | #932 | — | 10 | 1 | no | — |
| 59 | C3×#97 | — | 10 | 1 | no | — |
| 60 | #836 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic, complemented |
| 61 | #834 | — | 10 | 1 | no | — |
| 62 | #805 | — | 10 | 1 | no | — |
| 63 | C2×C5×Q8 | C4×D3 | 1 | 1 | yes | characteristic |
| 64 | #713 | — | 12 | 1 | no | — |
| 65 | Q8×D5 | — | 12 | 1 | no | — |
| 66 | #709 | — | 6 | 1 | no | — |
| 67 | #709 | — | 12 | 1 | no | — |
| 68 | C2×D20 | D12 | 1 | 1 | yes | characteristic |
| 69 | C22⋊AGL(1,𝔽5) | — | 12 | 1 | no | — |
| 70 | C2×#170 | — | 4 | 1 | no | — |
| 71 | C5×M16 | — | 6 | 1 | no | — |
| 72 | #685 | — | 4 | 1 | no | — |
| 73 | C4×Dic5 | #67 | 1 | 1 | yes | characteristic |
| 74 | D40 | — | 12 | 1 | no | — |
| 75 | #677 | — | 12 | 1 | no | — |
| 76 | #454 | — | 15 | 1 | no | — |
| 77 | #354 | — | 15 | 1 | no | — |
| 78 | #329 | — | 5 | 1 | no | — |
| 79 | C22×C3×C5 | #99 | 1 | 1 | yes | characteristic |
| 80 | C2×D15 | — | 4 | 1 | no | — |
| 81 | C2×C3×D5 | — | 4 | 1 | no | — |
| 82 | C2×C3×D5 | — | 8 | 1 | no | — |
| 83 | ΓL(1,𝔽16) | — | 16 | 1 | no | — |
| 84 | C3×C4×C5 | — | 2 | 1 | no | — |
| 85 | C3×C4×C5 | — | 4 | 1 | no | — |
| 86 | Dic15 | — | 4 | 1 | no | — |
| 87 | C3×Dic5 | — | 4 | 1 | no | — |
| 88 | C2×C3×Q8 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 89 | C2×C3×D4 | — | 5 | 1 | no | — |
| 90 | #239 | — | 20 | 1 | no | — |
| 91 | D3×Q8 | — | 20 | 1 | no | — |
| 92 | #235 | — | 10 | 1 | no | — |
| 93 | #235 | — | 20 | 1 | no | — |
| 94 | C2×C3×C8 | — | 20 | 1 | no | — |
| 95 | C3×C22⋊C4 | — | 20 | 1 | no | — |
| 96 | C3×C42 | — | 5 | 1 | no | — |
| 97 | #217 | — | 20 | 1 | no | — |
| 98 | #214 | — | 20 | 1 | no | — |
| 99 | #213 | — | 20 | 1 | no | — |
| 100 | #208 | — | 2 | 1 | no | — |
| 101 | C22×D5 | — | 2 | 1 | no | — |
| 102 | C2×AGL(1,𝔽5) | — | 24 | 1 | no | — |
| 103 | C5×Q8 | — | 4 | 1 | no | — |
| 104 | C2×C4×C5 | — | 2 | 1 | no | — |
| 105 | C2×C4×C5 | (C2×D3)⋊C4 | 1 | 1 | yes | characteristic |
| 106 | C22⋊D5 | — | 12 | 1 | no | — |
| 107 | C2×Dic5 | — | 2 | 1 | no | — |
| 108 | D20 | — | 4 | 1 | no | — |
| 109 | D20 | — | 6 | 1 | no | — |
| 110 | D20 | — | 12 | 1 | no | — |
| 111 | C4×D5 | — | 12 | 1 | no | — |
| 112 | C4×D5 | — | 12 | 2 | no | — |
| 113 | Dic10 | — | 6 | 1 | no | — |
| 114 | Dic10 | — | 12 | 1 | no | — |
| 115 | #170 | — | 4 | 2 | no | — |
| 116 | C5×C8 | — | 12 | 1 | no | — |
| 117 | 2−1+4 | — | 15 | 1 | no | — |
| 118 | AGL(1,ℤ/8) | — | 30 | 1 | no | — |
| 119 | #124 | — | 5 | 1 | no | — |
| 120 | #104 | — | 30 | 1 | no | — |
| 121 | #101 | — | 3 | 1 | no | — |
| 122 | #99 | — | 30 | 1 | no | — |
| 123 | #97 | — | 10 | 1 | no | — |
| 124 | C2×C3×C5 | — | 2 | 1 | no | — |
| 125 | C2×C3×C5 | AΣL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 126 | D15 | — | 8 | 1 | no | — |
| 127 | C3×D5 | — | 8 | 1 | no | — |
| 128 | C23×C3 | — | 10 | 1 | no | — |
| 129 | C3×Q8 | — | 4 | 1 | no | — |
| 130 | C3×D4 | — | 20 | 1 | no | — |
| 131 | C2×C3×C4 | — | 2 | 1 | no | — |
| 132 | C2×C3×C4 | — | 10 | 1 | no | — |
| 133 | C2×C3×C4 | C22⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 134 | #67 | — | 20 | 1 | no | — |
| 135 | C2×Dic3 | — | 40 | 1 | no | — |
| 136 | D12 | — | 10 | 1 | no | — |
| 137 | D12 | — | 20 | 1 | no | — |
| 138 | C4×D3 | — | 20 | 1 | no | — |
| 139 | C4×D3 | — | 20 | 2 | no | — |
| 140 | Dic6 | — | 10 | 1 | no | — |
| 141 | Dic6 | — | 20 | 1 | no | — |
| 142 | C3×C8 | — | 20 | 2 | no | — |
| 143 | C3⋊C8 | — | 4 | 1 | no | — |
| 144 | C22×C5 | #806 | 1 | 1 | yes | characteristic |
| 145 | C2×D5 | — | 4 | 1 | no | — |
| 146 | C2×D5 | — | 8 | 1 | no | — |
| 147 | C2×D5 | — | 12 | 1 | no | — |
| 148 | AGL(1,𝔽5) | — | 48 | 1 | no | — |
| 149 | C4×C5 | — | 2 | 1 | no | — |
| 150 | C4×C5 | — | 4 | 1 | no | — |
| 151 | Dic5 | — | 4 | 1 | no | — |
| 152 | Dic5 | — | 12 | 1 | no | — |
| 153 | #41 | — | 30 | 1 | no | — |
| 154 | #41 | — | 60 | 1 | no | — |
| 155 | #41 | — | 60 | 1 | no | — |
| 156 | C2×Q8 | — | 60 | 1 | no | — |
| 157 | C2×Q8 | D3×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 158 | C2×D4 | — | 5 | 1 | no | — |
| 159 | SD16 | — | 60 | 1 | no | — |
| 160 | D8 | — | 60 | 1 | no | — |
| 161 | M16 | — | 6 | 1 | no | — |
| 162 | C2×C8 | — | 20 | 1 | no | — |
| 163 | C22⋊C4 | — | 20 | 1 | no | — |
| 164 | C22⋊C4 | — | 60 | 1 | no | — |
| 165 | C42 | — | 5 | 1 | no | — |
| 166 | C3×C5 | #1404 | 1 | 1 | yes | characteristic, complemented, Hall |
| 167 | C22×C3 | — | 20 | 1 | no | — |
| 168 | C22×C3 | — | 40 | 1 | no | — |
| 169 | C22×C3 | #3999 | 1 | 1 | yes | characteristic |
| 170 | C2×D3 | — | 20 | 1 | no | — |
| 171 | C3×C4 | — | 2 | 1 | no | — |
| 172 | C3×C4 | — | 4 | 1 | no | — |
| 173 | C3×C4 | — | 20 | 1 | no | — |
| 174 | Dic3 | — | 20 | 1 | no | — |
| 175 | Dic3 | — | 80 | 1 | no | — |
| 176 | C2×C5 | — | 2 | 1 | no | — |
| 177 | C2×C5 | #4475 | 1 | 1 | yes | characteristic |
| 178 | D5 | — | 8 | 1 | no | — |
| 179 | D5 | — | 24 | 1 | no | — |
| 180 | C23 | — | 10 | 1 | no | — |
| 181 | Q8 | — | 4 | 1 | no | — |
| 182 | Q8 | — | 30 | 1 | no | — |
| 183 | Q8 | — | 60 | 1 | no | — |
| 184 | D4 | — | 20 | 1 | no | — |
| 185 | D4 | — | 30 | 1 | no | — |
| 186 | D4 | — | 60 | 1 | no | — |
| 187 | D4 | — | 60 | 1 | no | — |
| 188 | C2×C4 | — | 2 | 1 | no | — |
| 189 | C2×C4 | — | 10 | 1 | no | — |
| 190 | C2×C4 | — | 60 | 1 | no | — |
| 191 | C2×C4 | — | 60 | 2 | no | — |
| 192 | C2×C4 | — | 120 | 1 | no | — |
| 193 | C2×C4 | (C2×D3)⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 194 | C8 | — | 12 | 1 | no | — |
| 195 | C8 | — | 20 | 2 | no | — |
| 196 | C2×C3 | — | 2 | 1 | no | — |
| 197 | C2×C3 | — | 40 | 1 | no | — |
| 198 | C2×C3 | #64864 | 1 | 1 | yes | characteristic |
| 199 | D3 | — | 40 | 1 | no | — |
| 200 | C5 | #67460 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 201 | C22 | — | 20 | 1 | no | — |
| 202 | C22 | — | 40 | 1 | no | — |
| 203 | C22 | — | 60 | 1 | no | — |
| 204 | C22 | #91173 | 1 | 1 | yes | characteristic |
| 205 | C4 | — | 2 | 1 | no | — |
| 206 | C4 | — | 4 | 1 | no | — |
| 207 | C4 | — | 20 | 1 | no | — |
| 208 | C4 | — | 60 | 1 | no | — |
| 209 | C4 | — | 240 | 1 | no | — |
| 210 | C3 | #10598575 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 211 | C2 | — | 2 | 1 | no | — |
| 212 | C2 | — | 40 | 1 | no | — |
| 213 | C2 | — | 120 | 1 | no | — |
| 214 | C2 | #11746580 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #11746799 | C2 | semidirect |
| 3. C3×#64927 | C2 | semidirect |
| 5. #92098 | C4 | semidirect |
| 10. C5×#836 | C4 | semidirect |
| 15. #64927 | D3 | semidirect |
| 60. #836 | AGL(1,𝔽5) | semidirect |
| 166. C3×C5 | #1404 | semidirect |
| 200. C5 | #67460 | semidirect |
| 210. C3 | #10598575 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11749388 | 2 |
| #11746799 | 2 |
| C3×#64927 | 2 |
| #10598575 | 3 |
| #67460 | 5 |