Finite Group DatabasebetaFinite Group DatabasebetaOrder 1600.
266 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 266 conjugacy classes of subgroups — too many to draw. Only its 35 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 | C2×#87862 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×#87837 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #11717030 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #11716801 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #11716797 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | C2×C4×C5⋊D5 | C22 | 1 | 1 | yes | characteristic |
| 7 | C2×#6038 | — | 2 | 2 | no | — |
| 8 | C2×#6037 | C22 | 1 | 2 | yes | — |
| 9 | C2×#6034 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×#6034 | C22 | 1 | 1 | yes | characteristic |
| 11 | #87862 | — | 2 | 2 | no | — |
| 12 | #87837 | — | 2 | 2 | no | — |
| 13 | C5×#685 | — | 2 | 2 | no | — |
| 14 | #87781 | C22 | 1 | 2 | yes | — |
| 15 | C22×C5⋊D5 | D4 | 1 | 1 | yes | characteristic |
| 16 | C22×C5×D5 | — | 2 | 2 | no | — |
| 17 | C2×C4×C52 | D4 | 1 | 1 | yes | characteristic |
| 18 | C2×#1039 | C23 | 1 | 1 | yes | characteristic |
| 19 | C4×C5⋊D5 | — | 2 | 1 | no | — |
| 20 | C4×C5⋊D5 | — | 2 | 1 | no | — |
| 21 | C2×C5×Dic5 | — | 2 | 2 | no | — |
| 22 | #6038 | — | 4 | 4 | no | — |
| 23 | #6037 | — | 2 | 2 | no | — |
| 24 | #6037 | — | 2 | 2 | no | — |
| 25 | #6034 | — | 2 | 1 | no | — |
| 26 | #6034 | D4 | 1 | 2 | yes | complemented |
| 27 | C2×D5⋊C8 | — | 10 | 1 | no | — |
| 28 | #4030 | — | 10 | 2 | no | — |
| 29 | C22×C52 | C2×D4 | 1 | 1 | yes | characteristic |
| 30 | C2×C5⋊D5 | — | 2 | 1 | no | — |
| 31 | C2×C5⋊D5 | — | 2 | 1 | no | — |
| 32 | C2×C5⋊D5 | D8 | 1 | 2 | yes | complemented |
| 33 | C2×C5×D5 | — | 4 | 2 | no | — |
| 34 | C2×C5×D5 | — | 4 | 2 | no | — |
| 35 | C2×C5×D5 | — | 4 | 2 | no | — |
| 36 | C4×C52 | — | 2 | 1 | no | — |
| 37 | #1039 | C2×D4 | 1 | 1 | yes | characteristic |
| 38 | #1039 | #41 | 1 | 1 | yes | characteristic |
| 39 | C5×Dic5 | — | 4 | 2 | no | — |
| 40 | C2×C22⋊D5 | — | 10 | 2 | no | — |
| 41 | C2×C22⋊D5 | — | 10 | 2 | no | — |
| 42 | C2×D20 | — | 10 | 2 | no | — |
| 43 | C2×C4×D5 | — | 10 | 1 | no | — |
| 44 | C2×C4×D5 | — | 10 | 2 | no | — |
| 45 | C2×#170 | — | 10 | 1 | no | — |
| 46 | C2×#170 | — | 10 | 1 | no | — |
| 47 | D5⋊C8 | — | 10 | 4 | no | — |
| 48 | C5×C22⋊C4 | — | 10 | 2 | no | — |
| 49 | #685 | — | 2 | 2 | no | — |
| 50 | #683 | — | 10 | 2 | no | — |
| 51 | #466 | — | 25 | 1 | no | maximal, Sylow |
| 52 | C2×C52 | #121 | 1 | 1 | yes | characteristic |
| 53 | C2×C52 | C2×D8 | 1 | 1 | yes | characteristic |
| 54 | C2×C52 | #135 | 1 | 1 | yes | characteristic |
| 55 | C5⋊D5 | — | 2 | 2 | no | — |
| 56 | C5×D5 | — | 8 | 2 | no | — |
| 57 | C23×C5 | — | 10 | 2 | no | — |
| 58 | C22×D5 | — | 2 | 2 | no | — |
| 59 | C22×D5 | — | 10 | 1 | no | — |
| 60 | C22×D5 | — | 10 | 2 | no | — |
| 61 | C2×C4×C5 | — | 2 | 1 | no | — |
| 62 | C2×C4×C5 | — | 2 | 2 | no | — |
| 63 | C2×C4×C5 | — | 10 | 2 | no | — |
| 64 | C22⋊D5 | — | 20 | 2 | no | — |
| 65 | C22⋊D5 | — | 20 | 2 | no | — |
| 66 | C22⋊D5 | — | 20 | 4 | no | — |
| 67 | C2×Dic5 | — | 2 | 2 | no | — |
| 68 | C2×Dic5 | — | 10 | 2 | no | — |
| 69 | C2×Dic5 | — | 10 | 1 | no | — |
| 70 | D20 | — | 20 | 4 | no | — |
| 71 | C4×D5 | — | 10 | 2 | no | — |
| 72 | C4×D5 | — | 10 | 2 | no | — |
| 73 | C4×D5 | — | 20 | 2 | no | — |
| 74 | C4×D5 | — | 20 | 2 | no | — |
| 75 | #170 | — | 10 | 2 | no | — |
| 76 | #170 | — | 10 | 2 | no | — |
| 77 | C2×D8 | — | 25 | 1 | no | — |
| 78 | C22×C8 | — | 25 | 1 | no | — |
| 79 | #121 | — | 25 | 2 | no | — |
| 80 | #107 | — | 25 | 1 | no | — |
| 81 | #102 | — | 25 | 2 | no | — |
| 82 | C52 | #466 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 83 | C22×C5 | — | 2 | 2 | no | — |
| 84 | C22×C5 | — | 2 | 1 | no | — |
| 85 | C22×C5 | — | 20 | 2 | no | — |
| 86 | C22×C5 | — | 20 | 2 | no | — |
| 87 | C22×C5 | — | 20 | 2 | no | — |
| 88 | C2×D5 | — | 4 | 2 | no | — |
| 89 | C2×D5 | — | 4 | 2 | no | — |
| 90 | C2×D5 | — | 4 | 2 | no | — |
| 91 | C2×D5 | — | 10 | 2 | no | — |
| 92 | C2×D5 | — | 10 | 4 | no | — |
| 93 | C2×D5 | — | 10 | 2 | no | — |
| 94 | C2×D5 | — | 10 | 2 | no | — |
| 95 | C2×D5 | — | 20 | 2 | no | — |
| 96 | C2×D5 | — | 20 | 2 | no | — |
| 97 | C4×C5 | — | 2 | 2 | no | — |
| 98 | C4×C5 | — | 4 | 2 | no | — |
| 99 | C4×C5 | — | 20 | 2 | no | — |
| 100 | Dic5 | — | 4 | 2 | no | — |
| 101 | Dic5 | — | 10 | 1 | no | — |
| 102 | Dic5 | — | 10 | 2 | no | — |
| 103 | Dic5 | — | 10 | 1 | no | — |
| 104 | Dic5 | — | 10 | 2 | no | — |
| 105 | C2×D4 | — | 25 | 2 | no | — |
| 106 | C2×D4 | — | 50 | 2 | no | — |
| 107 | C22×C4 | — | 25 | 1 | no | — |
| 108 | D8 | — | 50 | 2 | no | — |
| 109 | C2×C8 | — | 25 | 1 | no | — |
| 110 | C2×C8 | — | 25 | 1 | no | — |
| 111 | C2×C8 | — | 50 | 2 | no | — |
| 112 | C4⋊C4 | — | 25 | 2 | no | — |
| 113 | C22⋊C4 | — | 10 | 2 | no | — |
| 114 | C2×C5 | — | 2 | 2 | no | — |
| 115 | C2×C5 | — | 2 | 1 | no | — |
| 116 | C2×C5 | — | 2 | 2 | no | — |
| 117 | C2×C5 | — | 2 | 1 | no | — |
| 118 | C2×C5 | — | 2 | 1 | no | — |
| 119 | C2×C5 | — | 2 | 2 | no | — |
| 120 | C2×C5 | — | 40 | 2 | no | — |
| 121 | D5 | — | 8 | 2 | no | — |
| 122 | D5 | — | 10 | 4 | no | — |
| 123 | D5 | — | 20 | 4 | no | — |
| 124 | C23 | — | 10 | 2 | no | — |
| 125 | C23 | — | 25 | 1 | no | — |
| 126 | D4 | — | 50 | 2 | no | — |
| 127 | D4 | — | 50 | 2 | no | — |
| 128 | D4 | — | 100 | 4 | no | — |
| 129 | C2×C4 | — | 10 | 2 | no | — |
| 130 | C2×C4 | — | 25 | 1 | no | — |
| 131 | C2×C4 | — | 50 | 1 | no | — |
| 132 | C2×C4 | — | 50 | 1 | no | — |
| 133 | C2×C4 | D52⋊C2 | 1 | 1 | yes | characteristic |
| 134 | C8 | — | 25 | 2 | no | — |
| 135 | C8 | — | 50 | 1 | no | — |
| 136 | C5 | — | 2 | 2 | no | — |
| 137 | C5 | — | 2 | 1 | no | — |
| 138 | C22 | — | 20 | 2 | no | — |
| 139 | C22 | — | 20 | 2 | no | — |
| 140 | C22 | — | 20 | 2 | no | — |
| 141 | C22 | — | 25 | 2 | no | — |
| 142 | C22 | — | 50 | 1 | no | — |
| 143 | C22 | — | 50 | 1 | no | — |
| 144 | C22 | C2×D52⋊C2 | 1 | 1 | yes | characteristic |
| 145 | C4 | — | 2 | 1 | no | — |
| 146 | C4 | — | 20 | 2 | no | — |
| 147 | C4 | — | 25 | 1 | no | — |
| 148 | C4 | — | 25 | 1 | no | — |
| 149 | C2 | — | 40 | 2 | no | — |
| 150 | C2 | — | 50 | 2 | no | — |
| 151 | C2 | #11717381 | 1 | 1 | yes | characteristic |
| 152 | C2 | #11717383 | 1 | 1 | yes | characteristic |
| 153 | C2 | #11717400 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×D52⋊C2 |
| commutator, cosocle | C2×#1039 | C23 |
| Fitting | C2×C4×C52 | D4 |
| socle | C22×C52 | C2×D4 |
| hypercenter | C2×C4 | D52⋊C2 |
| hypocenter | C52 | #466 |
| LCS | G ▷ C2×#1039 ▷ C2×C52 ▷ C52 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#87862 | C2 | semidirect |
| 2. C2×#87837 | C2 | semidirect |
| 3. #11717030 | C2 | semidirect |
| 4. #11716801 | C2 | semidirect |
| 5. #11716797 | C2 | semidirect |
| 26. #6034 | D4 | semidirect |
| 32. C2×C5⋊D5 | D8 | semidirect |
| 82. C52 | #466 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#87862 | 2 |
| C2×#87837 | 2 |
| #11717030 | 2 |
| #11716801 | 2 |
| #11716797 | 2 |
| #466 | 25 |