Finite Group DatabasebetaFinite Group DatabasebetaOrder 1600.
503 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 503 conjugacy classes of subgroups — too many to draw. Only its 89 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 | #11717477 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | D4×#1044 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #11717400 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | C2×#87836 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #11717391 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | C4×D52⋊C2 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #11717380 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #11717378 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | C22×#1044 | C22 | 1 | 2 | yes | — |
| 10 | C2×D52⋊C2 | C22 | 1 | 1 | yes | characteristic |
| 11 | D4×C5⋊D5 | C22 | 1 | 1 | yes | characteristic |
| 12 | C5×#711 | — | 2 | 2 | no | — |
| 13 | #87886 | C22 | 1 | 4 | yes | — |
| 14 | C2×#6036 | C22 | 1 | 4 | yes | — |
| 15 | #87879 | — | 2 | 4 | no | — |
| 16 | C4×D52 | C22 | 1 | 2 | yes | — |
| 17 | #87874 | — | 2 | 2 | no | — |
| 18 | #87872 | C22 | 1 | 2 | yes | — |
| 19 | C22⋊#1044 | C22 | 1 | 2 | yes | — |
| 20 | C4⋊#1044 | C22 | 1 | 1 | yes | characteristic |
| 21 | C4×#1044 | C22 | 1 | 1 | yes | characteristic |
| 22 | #87836 | C22 | 1 | 4 | yes | — |
| 23 | #87836 | C22 | 1 | 2 | yes | — |
| 24 | #87836 | C22 | 1 | 2 | yes | — |
| 25 | #87836 | C22 | 1 | 1 | yes | characteristic |
| 26 | #87835 | C22 | 1 | 4 | yes | — |
| 27 | #87835 | C22 | 1 | 2 | yes | — |
| 28 | C22×C5⋊D5 | C23 | 1 | 2 | yes | — |
| 29 | C2×D52 | C23 | 1 | 2 | yes | — |
| 30 | C2×#1044 | — | 2 | 4 | no | — |
| 31 | C2×#1044 | C23 | 1 | 1 | yes | characteristic |
| 32 | C2×#1044 | C23 | 1 | 2 | yes | — |
| 33 | C2×#1044 | C23 | 1 | 1 | yes | characteristic |
| 34 | D52⋊C2 | — | 2 | 2 | no | — |
| 35 | C52×D4 | D4 | 1 | 1 | yes | characteristic, complemented |
| 36 | C22⋊(C5⋊D5) | D4 | 1 | 2 | yes | complemented |
| 37 | C4⋊(C5⋊D5) | D4 | 1 | 1 | yes | characteristic, complemented |
| 38 | C4×C5⋊D5 | C23 | 1 | 1 | yes | characteristic |
| 39 | C5×C22⋊D5 | — | 2 | 4 | no | — |
| 40 | C2×C5×Dic5 | — | 2 | 4 | no | — |
| 41 | C4×C5×D5 | — | 2 | 2 | no | — |
| 42 | C5×Dic10 | — | 2 | 2 | no | — |
| 43 | #6039 | — | 2 | 2 | no | — |
| 44 | #6038 | — | 2 | 4 | no | — |
| 45 | #6036 | — | 2 | 4 | no | — |
| 46 | #6036 | C23 | 1 | 4 | yes | — |
| 47 | #6036 | C23 | 1 | 2 | yes | — |
| 48 | D5×Dic5 | — | 2 | 2 | no | — |
| 49 | D5×#41 | — | 10 | 2 | no | — |
| 50 | D4×AGL(1,𝔽5) | — | 10 | 1 | no | — |
| 51 | C22×C52 | C2×D4 | 1 | 2 | yes | — |
| 52 | C2×C5⋊D5 | — | 2 | 2 | no | — |
| 53 | C2×C5⋊D5 | — | 2 | 2 | no | — |
| 54 | C2×C5⋊D5 | C2×D4 | 1 | 2 | yes | — |
| 55 | C2×C5⋊D5 | C24 | 1 | 1 | yes | characteristic |
| 56 | C2×C5×D5 | — | 2 | 2 | no | — |
| 57 | D52 | — | 2 | 2 | no | — |
| 58 | D52 | — | 2 | 2 | no | — |
| 59 | #1044 | — | 2 | 2 | no | — |
| 60 | #1044 | — | 2 | 1 | no | — |
| 61 | #1044 | #41 | 1 | 2 | yes | complemented |
| 62 | C4×C52 | C2×D4 | 1 | 1 | yes | characteristic |
| 63 | #1039 | C2×D4 | 1 | 1 | yes | characteristic |
| 64 | C5×Dic5 | — | 2 | 2 | no | — |
| 65 | C5×Dic5 | — | 2 | 4 | no | — |
| 66 | C22×AGL(1,𝔽5) | — | 10 | 2 | no | — |
| 67 | C5×#41 | — | 10 | 2 | no | — |
| 68 | #713 | — | 10 | 2 | no | — |
| 69 | Q8×D5 | — | 10 | 2 | no | — |
| 70 | #711 | — | 2 | 2 | no | — |
| 71 | #711 | — | 10 | 4 | no | — |
| 72 | D4×D5 | — | 10 | 4 | no | — |
| 73 | D4×D5 | — | 10 | 1 | no | — |
| 74 | D4×D5 | — | 10 | 2 | no | — |
| 75 | #709 | — | 10 | 2 | no | — |
| 76 | #709 | — | 10 | 4 | no | — |
| 77 | C2×C4×D5 | — | 10 | 2 | no | — |
| 78 | C2×C4×D5 | — | 10 | 4 | no | — |
| 79 | C22⋊AGL(1,𝔽5) | — | 10 | 2 | no | — |
| 80 | C4⋊AGL(1,𝔽5) | — | 10 | 1 | no | — |
| 81 | C4×AGL(1,𝔽5) | — | 10 | 1 | no | — |
| 82 | #547 | — | 25 | 1 | no | maximal, Sylow |
| 83 | C2×C52 | — | 2 | 2 | no | — |
| 84 | C2×C52 | C22×D4 | 1 | 1 | yes | characteristic |
| 85 | C5⋊D5 | — | 2 | 2 | no | — |
| 86 | C5⋊D5 | C2×#41 | 1 | 1 | yes | characteristic |
| 87 | C5⋊D5 | 2−1+4 | 1 | 1 | yes | characteristic |
| 88 | C5×D5 | — | 4 | 2 | no | — |
| 89 | C22×D5 | — | 10 | 2 | no | — |
| 90 | C22×D5 | — | 10 | 2 | no | — |
| 91 | C22×D5 | — | 10 | 4 | no | — |
| 92 | C2×AGL(1,𝔽5) | — | 10 | 1 | no | — |
| 93 | C2×AGL(1,𝔽5) | — | 10 | 1 | no | — |
| 94 | C2×AGL(1,𝔽5) | — | 10 | 2 | no | — |
| 95 | C2×AGL(1,𝔽5) | — | 20 | 4 | no | — |
| 96 | C5×Q8 | — | 10 | 2 | no | — |
| 97 | C5×D4 | — | 2 | 1 | no | — |
| 98 | C5×D4 | — | 2 | 2 | no | — |
| 99 | C5×D4 | — | 10 | 4 | no | — |
| 100 | C2×C4×C5 | — | 10 | 2 | no | — |
| 101 | C2×C4×C5 | — | 10 | 4 | no | — |
| 102 | C22⋊D5 | — | 2 | 4 | no | — |
| 103 | C22⋊D5 | — | 10 | 4 | no | — |
| 104 | C22⋊D5 | — | 10 | 2 | no | — |
| 105 | C22⋊D5 | — | 10 | 4 | no | — |
| 106 | C2×Dic5 | — | 2 | 4 | no | — |
| 107 | C2×Dic5 | — | 10 | 2 | no | — |
| 108 | D20 | — | 10 | 2 | no | — |
| 109 | D20 | — | 10 | 1 | no | — |
| 110 | D20 | — | 10 | 4 | no | — |
| 111 | C4×D5 | — | 2 | 2 | no | — |
| 112 | C4×D5 | — | 10 | 2 | no | — |
| 113 | C4×D5 | — | 10 | 4 | no | — |
| 114 | C4×D5 | — | 10 | 1 | no | — |
| 115 | C4×D5 | — | 10 | 2 | no | — |
| 116 | C4×D5 | — | 10 | 4 | no | — |
| 117 | C4×D5 | — | 10 | 4 | no | — |
| 118 | C4×D5 | — | 10 | 2 | no | — |
| 119 | Dic10 | — | 2 | 2 | no | — |
| 120 | Dic10 | — | 10 | 4 | no | — |
| 121 | C2×#41 | — | 25 | 2 | no | — |
| 122 | C4⋊Q8 | — | 25 | 1 | no | — |
| 123 | #123 | — | 25 | 4 | no | — |
| 124 | C22⋊Q8 | — | 25 | 2 | no | — |
| 125 | #121 | — | 25 | 2 | no | — |
| 126 | C4×D4 | — | 25 | 1 | no | — |
| 127 | C4×D4 | — | 25 | 1 | no | — |
| 128 | C2×C4⋊C4 | — | 25 | 2 | no | — |
| 129 | C52 | #547 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 130 | C22×C5 | — | 2 | 4 | no | — |
| 131 | C22×C5 | — | 2 | 2 | no | — |
| 132 | C22×C5 | — | 10 | 2 | no | — |
| 133 | C2×D5 | — | 2 | 2 | no | — |
| 134 | C2×D5 | — | 10 | 2 | no | — |
| 135 | C2×D5 | — | 10 | 1 | no | — |
| 136 | C2×D5 | — | 10 | 4 | no | — |
| 137 | C2×D5 | — | 10 | 2 | no | — |
| 138 | C2×D5 | — | 20 | 2 | no | — |
| 139 | C2×D5 | — | 20 | 2 | no | — |
| 140 | C2×D5 | — | 20 | 2 | no | — |
| 141 | C2×D5 | — | 20 | 4 | no | — |
| 142 | C2×D5 | — | 20 | 2 | no | — |
| 143 | C2×D5 | — | 20 | 4 | no | — |
| 144 | AGL(1,𝔽5) | — | 10 | 2 | no | — |
| 145 | AGL(1,𝔽5) | — | 20 | 1 | no | — |
| 146 | AGL(1,𝔽5) | — | 20 | 2 | no | — |
| 147 | C4×C5 | — | 2 | 2 | no | — |
| 148 | C4×C5 | — | 2 | 1 | no | — |
| 149 | C4×C5 | — | 10 | 4 | no | — |
| 150 | C4×C5 | — | 10 | 2 | no | — |
| 151 | Dic5 | — | 2 | 4 | no | — |
| 152 | Dic5 | — | 2 | 2 | no | — |
| 153 | Dic5 | — | 10 | 2 | no | — |
| 154 | Dic5 | — | 10 | 1 | no | — |
| 155 | #41 | — | 10 | 2 | no | — |
| 156 | #41 | — | 50 | 4 | no | — |
| 157 | #41 | — | 50 | 2 | no | — |
| 158 | C2×Q8 | — | 25 | 2 | no | — |
| 159 | C2×D4 | — | 25 | 4 | no | — |
| 160 | C2×D4 | — | 25 | 1 | no | — |
| 161 | C2×D4 | — | 25 | 1 | no | — |
| 162 | C22×C4 | — | 25 | 4 | no | — |
| 163 | C22×C4 | — | 25 | 2 | no | — |
| 164 | C22×C4 | — | 25 | 2 | no | — |
| 165 | C4⋊C4 | — | 25 | 1 | no | — |
| 166 | C4⋊C4 | — | 25 | 2 | no | — |
| 167 | C4⋊C4 | — | 25 | 4 | no | — |
| 168 | C4⋊C4 | — | 25 | 2 | no | — |
| 169 | C4⋊C4 | — | 25 | 1 | no | — |
| 170 | C22⋊C4 | — | 25 | 2 | no | — |
| 171 | C22⋊C4 | — | 25 | 4 | no | — |
| 172 | C22⋊C4 | — | 25 | 2 | no | — |
| 173 | C42 | — | 25 | 1 | no | — |
| 174 | C2×C5 | — | 2 | 2 | no | — |
| 175 | C2×C5 | — | 2 | 1 | no | — |
| 176 | C2×C5 | — | 4 | 4 | no | — |
| 177 | C2×C5 | — | 4 | 2 | no | — |
| 178 | C2×C5 | — | 20 | 2 | no | — |
| 179 | D5 | — | 4 | 2 | no | — |
| 180 | D5 | — | 10 | 1 | no | — |
| 181 | D5 | — | 10 | 2 | no | — |
| 182 | D5 | — | 10 | 1 | no | — |
| 183 | D5 | — | 10 | 2 | no | — |
| 184 | D5 | — | 20 | 4 | no | — |
| 185 | D5 | — | 20 | 2 | no | — |
| 186 | C23 | — | 25 | 2 | no | — |
| 187 | C23 | — | 25 | 2 | no | — |
| 188 | Q8 | — | 10 | 2 | no | — |
| 189 | Q8 | — | 50 | 2 | no | — |
| 190 | D4 | — | 10 | 4 | no | — |
| 191 | D4 | — | 25 | 2 | no | — |
| 192 | D4 | — | 25 | 1 | no | — |
| 193 | D4 | — | 50 | 4 | no | — |
| 194 | D4 | — | 50 | 2 | no | — |
| 195 | D4 | D52⋊C2 | 1 | 1 | yes | characteristic, complemented |
| 196 | C2×C4 | — | 10 | 4 | no | — |
| 197 | C2×C4 | — | 10 | 2 | no | — |
| 198 | C2×C4 | — | 25 | 2 | no | — |
| 199 | C2×C4 | — | 25 | 4 | no | — |
| 200 | C2×C4 | — | 25 | 1 | no | — |
| 201 | C2×C4 | — | 25 | 1 | no | — |
| 202 | C2×C4 | — | 25 | 1 | no | — |
| 203 | C2×C4 | — | 25 | 2 | no | — |
| 204 | C2×C4 | — | 50 | 4 | no | — |
| 205 | C2×C4 | — | 50 | 2 | no | — |
| 206 | C2×C4 | — | 50 | 4 | no | — |
| 207 | C5 | — | 2 | 2 | no | — |
| 208 | C5 | — | 2 | 1 | no | — |
| 209 | C22 | — | 10 | 2 | no | — |
| 210 | C22 | — | 25 | 1 | no | — |
| 211 | C22 | — | 25 | 2 | no | — |
| 212 | C22 | — | 50 | 2 | no | — |
| 213 | C22 | — | 50 | 2 | no | — |
| 214 | C22 | — | 50 | 2 | no | — |
| 215 | C22 | — | 50 | 2 | no | — |
| 216 | C22 | C2×D52⋊C2 | 1 | 2 | yes | — |
| 217 | C4 | — | 10 | 2 | no | — |
| 218 | C4 | — | 10 | 4 | no | — |
| 219 | C4 | — | 25 | 2 | no | — |
| 220 | C4 | — | 25 | 1 | no | — |
| 221 | C4 | — | 50 | 2 | no | — |
| 222 | C4 | — | 50 | 1 | no | — |
| 223 | C4 | C2×D52⋊C2 | 1 | 1 | yes | characteristic |
| 224 | C2 | — | 2 | 2 | no | — |
| 225 | C2 | — | 20 | 2 | no | — |
| 226 | C2 | — | 25 | 1 | no | — |
| 227 | C2 | — | 25 | 1 | no | — |
| 228 | C2 | — | 50 | 2 | no | — |
| 229 | C2 | C22×D52⋊C2 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D52⋊C2 |
| commutator, cosocle | C2×C5⋊D5 | C24 |
| Fitting | C52×D4 | D4 |
| socle | C2×C52 | C22×D4 |
| hypercenter | D4 | D52⋊C2 |
| hypocenter | C52 | #547 |
| LCS | G ▷ C2×C5⋊D5 ▷ C52 | |
| UCS | C1 ◁ C2 ◁ D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11717477 | C2 | semidirect |
| 2. D4×#1044 | C2 | semidirect |
| 3. #11717400 | C2 | semidirect |
| 4. C2×#87836 | C2 | semidirect |
| 5. #11717391 | C2 | semidirect |
| 6. C4×D52⋊C2 | C2 | semidirect |
| 7. #11717380 | C2 | semidirect |
| 8. #11717378 | C2 | semidirect |
| 35. C52×D4 | D4 | semidirect |
| 36. C22⋊(C5⋊D5) | D4 | semidirect |
| 37. C4⋊(C5⋊D5) | D4 | semidirect |
| 61. #1044 | #41 | semidirect |
| 129. C52 | #547 | semidirect |
| 195. D4 | D52⋊C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11717477 | 2 |
| D4×#1044 | 2 |
| #11717400 | 2 |
| C2×#87836 | 2 |
| #11717391 | 2 |
| C4×D52⋊C2 | 2 |
| #11717380 | 2 |
| #11717378 | 2 |
| #547 | 25 |