Finite Group DatabasebetaFinite Group DatabasebetaOrder 1920.
608 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 608 conjugacy classes of subgroups — too many to draw. Only its 76 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 | #11756944 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #11756724 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | D5⋊GU(2,𝔽3) | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #10617560 | — | 3 | 1 | no | maximal, Hall |
| 5 | C5×#994 | C4 | 1 | 1 | yes | characteristic, complemented |
| 6 | #91992 | — | 4 | 1 | no | maximal |
| 7 | #4150⋊C3 | C4 | 1 | 3 | yes | complemented |
| 8 | D5×#41⋊C3 | C22 | 1 | 3 | yes | — |
| 9 | #91885 | C22 | 1 | 1 | yes | characteristic, complemented |
| 10 | #91885 | C22 | 1 | 3 | yes | complemented |
| 11 | #85541 | — | 5 | 1 | no | maximal, Hall |
| 12 | #66285 | D3 | 1 | 1 | yes | characteristic |
| 13 | #41×AGL(1,𝔽5) | — | 3 | 1 | no | — |
| 14 | #66264 | — | 3 | 1 | no | — |
| 15 | #65795 | — | 3 | 3 | no | — |
| 16 | D5⋊#104 | — | 3 | 3 | no | — |
| 17 | #65769 | — | 3 | 3 | no | — |
| 18 | C2×#4008 | — | 3 | 3 | no | — |
| 19 | C3×#713 | — | 4 | 1 | no | — |
| 20 | C5×#41⋊C3 | C2×C4 | 1 | 3 | yes | complemented |
| 21 | #6758 | — | 4 | 3 | no | — |
| 22 | D5⋊(C3⋊C8) | — | 4 | 3 | no | — |
| 23 | D5×SL(2,𝔽3) | C2×C4 | 1 | 3 | yes | complemented |
| 24 | C5⋊(#41⋊C3) | C23 | 1 | 1 | yes | characteristic |
| 25 | #5945 | — | 5 | 1 | no | — |
| 26 | #5423 | — | 5 | 3 | no | — |
| 27 | C2×GU(2,𝔽3) | — | 5 | 3 | no | — |
| 28 | C5×H(5,𝔽2) | Dic3 | 1 | 1 | yes | characteristic |
| 29 | #4150 | — | 3 | 1 | no | — |
| 30 | #4150 | Dic3 | 1 | 3 | yes | — |
| 31 | D5×#41 | — | 3 | 1 | no | — |
| 32 | D5×#41 | C2×D3 | 1 | 3 | yes | — |
| 33 | #4144 | — | 3 | 3 | no | — |
| 34 | C2×#711 | — | 3 | 3 | no | — |
| 35 | Q8×AGL(1,𝔽5) | — | 6 | 1 | no | — |
| 36 | #4133 | — | 6 | 1 | no | — |
| 37 | D4×AGL(1,𝔽5) | — | 6 | 3 | no | — |
| 38 | #4131 | — | 6 | 3 | no | — |
| 39 | #4130 | — | 3 | 3 | no | — |
| 40 | C2×C4×AGL(1,𝔽5) | — | 3 | 3 | no | — |
| 41 | D5⋊M16 | — | 3 | 3 | no | — |
| 42 | C2×#700 | — | 3 | 3 | no | — |
| 43 | #4010 | — | 3 | 1 | no | — |
| 44 | #4010 | — | 3 | 3 | no | — |
| 45 | #4008 | — | 3 | 3 | no | — |
| 46 | #4008 | — | 3 | 3 | no | — |
| 47 | #4008 | — | 3 | 6 | no | — |
| 48 | #2901 | — | 15 | 1 | no | Sylow |
| 49 | C3×C5×Q8 | — | 4 | 1 | no | — |
| 50 | C3×D20 | — | 4 | 3 | no | — |
| 51 | C3×C4×D5 | — | 4 | 3 | no | — |
| 52 | C5×SL(2,𝔽3) | C22×C4 | 1 | 1 | yes | characteristic |
| 53 | C3⋊#170 | — | 4 | 1 | no | — |
| 54 | C3⋊#170 | — | 4 | 3 | no | — |
| 55 | 2−1+4⋊C3 | — | 5 | 3 | no | — |
| 56 | #994 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic, complemented |
| 57 | C2×#41⋊C3 | — | 5 | 3 | no | — |
| 58 | #948 | — | 20 | 1 | no | — |
| 59 | GU(2,𝔽3) | — | 5 | 1 | no | — |
| 60 | GU(2,𝔽3) | — | 5 | 3 | no | — |
| 61 | C22×AGL(1,𝔽5) | — | 6 | 3 | no | — |
| 62 | C5×#41 | — | 3 | 1 | no | — |
| 63 | C5×#41 | C2×Dic3 | 1 | 3 | yes | — |
| 64 | C2×C5×D4 | — | 3 | 3 | no | — |
| 65 | C2×C22⋊D5 | — | 6 | 3 | no | — |
| 66 | C22×Dic5 | — | 6 | 1 | no | — |
| 67 | #713 | S4 | 1 | 1 | yes | characteristic |
| 68 | #713 | C22×D3 | 1 | 1 | yes | characteristic |
| 69 | Q8×D5 | — | 3 | 1 | no | — |
| 70 | Q8×D5 | C2×Dic3 | 1 | 3 | yes | — |
| 71 | #711 | — | 3 | 3 | no | — |
| 72 | #711 | — | 3 | 6 | no | — |
| 73 | #711 | — | 3 | 3 | no | — |
| 74 | #711 | — | 6 | 3 | no | — |
| 75 | D4×D5 | — | 3 | 3 | no | — |
| 76 | D4×D5 | — | 3 | 3 | no | — |
| 77 | #709 | — | 3 | 3 | no | — |
| 78 | #709 | — | 3 | 3 | no | — |
| 79 | C2×C4×D5 | — | 3 | 3 | no | — |
| 80 | C2×Dic10 | — | 3 | 3 | no | — |
| 81 | C22⋊AGL(1,𝔽5) | — | 6 | 3 | no | — |
| 82 | #704 | — | 6 | 3 | no | — |
| 83 | C2×#170 | — | 6 | 3 | no | — |
| 84 | C4⋊AGL(1,𝔽5) | — | 6 | 3 | no | — |
| 85 | C4×AGL(1,𝔽5) | — | 3 | 1 | no | — |
| 86 | C4×AGL(1,𝔽5) | — | 3 | 3 | no | — |
| 87 | C4×AGL(1,𝔽5) | — | 6 | 3 | no | — |
| 88 | #700 | — | 3 | 3 | no | — |
| 89 | #700 | — | 3 | 1 | no | — |
| 90 | #700 | — | 6 | 3 | no | — |
| 91 | D5⋊C8 | — | 6 | 3 | no | — |
| 92 | #585 | — | 5 | 1 | no | — |
| 93 | #568 | — | 15 | 1 | no | — |
| 94 | C4×#41 | — | 15 | 1 | no | — |
| 95 | #421 | — | 15 | 3 | no | — |
| 96 | #421 | — | 15 | 3 | no | — |
| 97 | C2×#104 | — | 15 | 3 | no | — |
| 98 | C2×#104 | — | 15 | 3 | no | — |
| 99 | C2×C3×D5 | — | 4 | 3 | no | — |
| 100 | C3×C4×C5 | — | 4 | 3 | no | — |
| 101 | C3×Dic5 | — | 4 | 1 | no | — |
| 102 | C3×#41 | — | 20 | 1 | no | — |
| 103 | #41⋊C3 | — | 5 | 1 | no | — |
| 104 | #41⋊C3 | C2×AGL(1,𝔽5) | 1 | 3 | yes | complemented |
| 105 | C2×SL(2,𝔽3) | — | 5 | 3 | no | — |
| 106 | #208 | — | 20 | 3 | no | — |
| 107 | C2×C3⋊C8 | — | 20 | 3 | no | — |
| 108 | C23×C5 | — | 6 | 1 | no | — |
| 109 | C22×D5 | — | 3 | 3 | no | — |
| 110 | C2×AGL(1,𝔽5) | — | 6 | 1 | no | — |
| 111 | C2×AGL(1,𝔽5) | — | 6 | 3 | no | — |
| 112 | C2×AGL(1,𝔽5) | — | 12 | 6 | no | — |
| 113 | C5×Q8 | SL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 114 | C5×Q8 | C22×Dic3 | 1 | 1 | yes | characteristic |
| 115 | C5×D4 | — | 3 | 3 | no | — |
| 116 | C5×D4 | — | 3 | 3 | no | — |
| 117 | C2×C4×C5 | — | 3 | 3 | no | — |
| 118 | C22⋊D5 | — | 3 | 6 | no | — |
| 119 | C22⋊D5 | — | 6 | 3 | no | — |
| 120 | C2×Dic5 | — | 3 | 3 | no | — |
| 121 | C2×Dic5 | — | 6 | 3 | no | — |
| 122 | D20 | — | 3 | 1 | no | — |
| 123 | D20 | SL(2,ℤ/4) | 1 | 3 | yes | — |
| 124 | C4×D5 | — | 3 | 3 | no | — |
| 125 | C4×D5 | — | 3 | 3 | no | — |
| 126 | C4×D5 | — | 3 | 1 | no | — |
| 127 | C4×D5 | C2×S4 | 1 | 3 | yes | — |
| 128 | Dic10 | — | 3 | 3 | no | — |
| 129 | Dic10 | — | 3 | 3 | no | — |
| 130 | #170 | — | 6 | 1 | no | — |
| 131 | #170 | — | 6 | 3 | no | — |
| 132 | 2−1+4 | — | 5 | 3 | no | — |
| 133 | 2−1+4 | — | 15 | 1 | no | — |
| 134 | H(5,𝔽2) | — | 15 | 3 | no | — |
| 135 | H(5,𝔽2) | ΓL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 136 | C2×#41 | — | 5 | 3 | no | — |
| 137 | C2×#41 | — | 15 | 1 | no | — |
| 138 | C2×#41 | — | 15 | 3 | no | — |
| 139 | #131 | — | 30 | 1 | no | — |
| 140 | #131 | — | 30 | 3 | no | — |
| 141 | C2×M16 | — | 15 | 3 | no | — |
| 142 | C2×M16 | — | 15 | 3 | no | — |
| 143 | C4×Q8 | — | 30 | 1 | no | — |
| 144 | C4×D4 | — | 30 | 3 | no | — |
| 145 | #117 | — | 15 | 3 | no | — |
| 146 | C2×C42 | — | 15 | 3 | no | — |
| 147 | #104 | — | 15 | 1 | no | — |
| 148 | #104 | — | 15 | 3 | no | — |
| 149 | #104 | — | 15 | 3 | no | — |
| 150 | #104 | — | 15 | 3 | no | — |
| 151 | #104 | — | 15 | 6 | no | — |
| 152 | C2×C3×C5 | — | 4 | 1 | no | — |
| 153 | C3×D5 | — | 8 | 3 | no | — |
| 154 | C3×Q8 | — | 4 | 1 | no | — |
| 155 | C3×D4 | — | 20 | 3 | no | — |
| 156 | C2×C3×C4 | — | 20 | 3 | no | — |
| 157 | SL(2,𝔽3) | C22×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 158 | C3⋊C8 | — | 20 | 1 | no | — |
| 159 | C3⋊C8 | — | 20 | 3 | no | — |
| 160 | C22×C5 | — | 3 | 3 | no | — |
| 161 | C22×C5 | — | 24 | 1 | no | — |
| 162 | C2×D5 | — | 3 | 1 | no | — |
| 163 | C2×D5 | — | 12 | 3 | no | — |
| 164 | C2×D5 | C2×SL(2,ℤ/4) | 1 | 3 | yes | — |
| 165 | AGL(1,𝔽5) | — | 6 | 2 | no | — |
| 166 | AGL(1,𝔽5) | — | 12 | 3 | no | — |
| 167 | C4×C5 | — | 3 | 1 | no | — |
| 168 | C4×C5 | C2×SL(2,ℤ/4) | 1 | 3 | yes | — |
| 169 | Dic5 | — | 3 | 3 | no | — |
| 170 | Dic5 | C22×S4 | 1 | 1 | yes | characteristic |
| 171 | #41 | — | 3 | 1 | no | — |
| 172 | #41 | — | 5 | 1 | no | — |
| 173 | #41 | — | 5 | 1 | no | — |
| 174 | #41 | — | 15 | 3 | no | — |
| 175 | #41 | — | 15 | 3 | no | — |
| 176 | #41 | — | 15 | 3 | no | — |
| 177 | #41 | — | 15 | 3 | no | — |
| 178 | #41 | — | 15 | 6 | no | — |
| 179 | #41 | — | 30 | 3 | no | — |
| 180 | #41 | C2×ΓL(1,𝔽16) | 1 | 3 | yes | — |
| 181 | C2×Q8 | — | 5 | 3 | no | — |
| 182 | C2×Q8 | — | 15 | 3 | no | — |
| 183 | C2×Q8 | — | 15 | 1 | no | — |
| 184 | C2×D4 | — | 3 | 3 | no | — |
| 185 | C2×D4 | — | 15 | 3 | no | — |
| 186 | C2×D4 | — | 15 | 3 | no | — |
| 187 | C2×D4 | — | 30 | 3 | no | — |
| 188 | C22×C4 | — | 15 | 3 | no | — |
| 189 | C22×C4 | — | 30 | 1 | no | — |
| 190 | C22×C4 | — | 30 | 3 | no | — |
| 191 | M16 | — | 15 | 1 | no | — |
| 192 | M16 | — | 15 | 3 | no | — |
| 193 | M16 | — | 30 | 3 | no | — |
| 194 | M16 | — | 30 | 3 | no | — |
| 195 | C2×C8 | — | 30 | 3 | no | — |
| 196 | C2×C8 | — | 30 | 3 | no | — |
| 197 | C4⋊C4 | — | 30 | 3 | no | — |
| 198 | C22⋊C4 | — | 30 | 3 | no | — |
| 199 | C42 | — | 15 | 1 | no | — |
| 200 | C42 | — | 15 | 3 | no | — |
| 201 | C42 | — | 30 | 3 | no | — |
| 202 | C3×C5 | — | 4 | 1 | no | Hall |
| 203 | C22×C3 | — | 20 | 3 | no | — |
| 204 | C3×C4 | — | 4 | 3 | no | — |
| 205 | C3×C4 | — | 20 | 1 | no | — |
| 206 | C2×C5 | — | 6 | 3 | no | — |
| 207 | C2×C5 | C22×SL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 208 | D5 | — | 2 | 3 | no | — |
| 209 | D5 | — | 6 | 1 | no | — |
| 210 | C23 | — | 6 | 1 | no | — |
| 211 | C23 | — | 15 | 3 | no | — |
| 212 | Q8 | — | 15 | 3 | no | — |
| 213 | Q8 | — | 15 | 3 | no | — |
| 214 | Q8 | A4⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 215 | Q8 | C22×ΓL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 216 | D4 | — | 3 | 3 | no | — |
| 217 | D4 | — | 3 | 3 | no | — |
| 218 | D4 | — | 5 | 3 | no | — |
| 219 | D4 | — | 15 | 1 | no | — |
| 220 | D4 | — | 15 | 6 | no | — |
| 221 | D4 | — | 30 | 3 | no | — |
| 222 | C2×C4 | — | 3 | 3 | no | — |
| 223 | C2×C4 | — | 5 | 3 | no | — |
| 224 | C2×C4 | — | 15 | 1 | no | — |
| 225 | C2×C4 | — | 15 | 3 | no | — |
| 226 | C2×C4 | — | 15 | 3 | no | — |
| 227 | C2×C4 | — | 15 | 3 | no | — |
| 228 | C2×C4 | — | 30 | 3 | no | — |
| 229 | C2×C4 | — | 30 | 1 | no | — |
| 230 | C2×C4 | — | 30 | 3 | no | — |
| 231 | C2×C4 | — | 60 | 6 | no | — |
| 232 | C8 | — | 30 | 1 | no | — |
| 233 | C8 | — | 30 | 3 | no | — |
| 234 | C2×C3 | — | 4 | 1 | no | — |
| 235 | C2×C3 | — | 40 | 3 | no | — |
| 236 | C5 | #85541 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 237 | C22 | — | 3 | 3 | no | — |
| 238 | C22 | — | 5 | 3 | no | — |
| 239 | C22 | — | 15 | 1 | no | — |
| 240 | C22 | — | 24 | 1 | no | — |
| 241 | C22 | — | 60 | 3 | no | — |
| 242 | C4 | — | 3 | 1 | no | — |
| 243 | C4 | — | 5 | 1 | no | — |
| 244 | C4 | — | 15 | 3 | no | — |
| 245 | C4 | — | 30 | 2 | no | — |
| 246 | C4 | — | 60 | 3 | no | — |
| 247 | C4 | C2×A4⋊AGL(1,𝔽5) | 1 | 3 | yes | — |
| 248 | C3 | — | 4 | 1 | no | Sylow |
| 249 | C2 | — | 6 | 3 | no | — |
| 250 | C2 | — | 10 | 3 | no | — |
| 251 | C2 | — | 30 | 1 | no | — |
| 252 | C2 | C22×A4⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #11756724 | C2 | semidirect |
| 3. D5⋊GU(2,𝔽3) | C2 | semidirect |
| 5. C5×#994 | C4 | semidirect |
| 7. #4150⋊C3 | C4 | semidirect |
| 9. #91885 | C22 | semidirect |
| 10. #91885 | C22 | semidirect |
| 20. C5×#41⋊C3 | C2×C4 | semidirect |
| 23. D5×SL(2,𝔽3) | C2×C4 | semidirect |
| 56. #994 | AGL(1,𝔽5) | semidirect |
| 104. #41⋊C3 | C2×AGL(1,𝔽5) | semidirect |
| 236. C5 | #85541 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11756944 | 2 |
| #11756724 | 2 |
| D5⋊GU(2,𝔽3) | 2 |
| #10617560 | 3 |
| #91992 | 4 |
| #85541 | 5 |