Finite Group DatabasebetaFinite Group DatabasebetaOrder 960.
162 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 162 conjugacy classes of subgroups — too many to draw. Only its 24 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 | C22×SL(2,𝔽5) | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #91143 | C2 | 1 | 2 | yes | maximal, direct product factor |
| 3 | C2×SL(2,𝔽5) | C4 | 1 | 2 | yes | complemented |
| 4 | C2×SL(2,𝔽5) | C22 | 1 | 1 | yes | characteristic |
| 5 | C2×#859 | — | 5 | 1 | no | maximal, Hall |
| 6 | C22×#170 | — | 6 | 1 | no | maximal |
| 7 | SL(2,𝔽5) | C2×C4 | 1 | 1 | yes | characteristic, complemented |
| 8 | C22×SL(2,𝔽3) | — | 5 | 1 | no | — |
| 9 | C2×Dic3⋊C4 | — | 10 | 1 | no | maximal |
| 10 | #859 | — | 5 | 2 | no | — |
| 11 | C22×Dic5 | — | 6 | 1 | no | — |
| 12 | C2×#170 | — | 6 | 2 | no | — |
| 13 | C2×#170 | — | 6 | 4 | no | — |
| 14 | C2×#103 | — | 15 | 1 | no | Sylow |
| 15 | C22×C3×C4 | — | 10 | 1 | no | — |
| 16 | C22×Dic3 | — | 10 | 1 | no | — |
| 17 | C22×Dic3 | — | 10 | 1 | no | — |
| 18 | C2×SL(2,𝔽3) | — | 5 | 2 | no | — |
| 19 | C2×SL(2,𝔽3) | — | 5 | 1 | no | — |
| 20 | Dic3⋊C4 | — | 10 | 2 | no | — |
| 21 | Dic3⋊C4 | — | 10 | 2 | no | — |
| 22 | C23×C5 | — | 6 | 1 | no | — |
| 23 | C2×Dic5 | — | 6 | 2 | no | — |
| 24 | C2×Dic5 | — | 6 | 1 | no | — |
| 25 | C2×Dic5 | — | 6 | 2 | no | — |
| 26 | C2×Dic5 | — | 6 | 1 | no | — |
| 27 | #170 | — | 6 | 4 | no | — |
| 28 | C22×Q8 | — | 5 | 1 | no | — |
| 29 | C22×C8 | — | 15 | 1 | no | — |
| 30 | C2×C4⋊C4 | — | 15 | 1 | no | — |
| 31 | #103 | — | 15 | 2 | no | — |
| 32 | #103 | — | 15 | 2 | no | — |
| 33 | C23×C3 | — | 10 | 1 | no | — |
| 34 | C2×C3×C4 | — | 10 | 2 | no | — |
| 35 | C2×C3×C4 | — | 20 | 2 | no | — |
| 36 | C2×Dic3 | — | 10 | 2 | no | — |
| 37 | C2×Dic3 | — | 10 | 2 | no | — |
| 38 | C2×Dic3 | — | 10 | 1 | no | — |
| 39 | C2×Dic3 | — | 10 | 1 | no | — |
| 40 | C2×Dic3 | — | 10 | 2 | no | — |
| 41 | C2×Dic3 | — | 20 | 2 | no | — |
| 42 | SL(2,𝔽3) | — | 5 | 1 | no | — |
| 43 | C22×C5 | — | 6 | 2 | no | — |
| 44 | C22×C5 | — | 6 | 2 | no | — |
| 45 | C22×C5 | — | 6 | 2 | no | — |
| 46 | C22×C5 | — | 6 | 1 | no | — |
| 47 | Dic5 | — | 6 | 1 | no | — |
| 48 | Dic5 | — | 6 | 1 | no | — |
| 49 | Dic5 | — | 6 | 2 | no | — |
| 50 | C2×Q8 | — | 5 | 1 | no | — |
| 51 | C2×Q8 | — | 5 | 2 | no | — |
| 52 | C2×Q8 | — | 15 | 1 | no | — |
| 53 | C2×Q8 | — | 15 | 2 | no | — |
| 54 | C22×C4 | — | 10 | 1 | no | — |
| 55 | C22×C4 | — | 15 | 1 | no | — |
| 56 | C2×C8 | — | 15 | 2 | no | — |
| 57 | C2×C8 | — | 30 | 2 | no | — |
| 58 | C4⋊C4 | — | 15 | 2 | no | — |
| 59 | C4⋊C4 | — | 30 | 1 | no | — |
| 60 | C22×C3 | — | 10 | 2 | no | — |
| 61 | C22×C3 | — | 10 | 2 | no | — |
| 62 | C22×C3 | — | 10 | 2 | no | — |
| 63 | C22×C3 | — | 10 | 1 | no | — |
| 64 | C3×C4 | — | 20 | 2 | no | — |
| 65 | Dic3 | — | 10 | 1 | no | — |
| 66 | Dic3 | — | 10 | 2 | no | — |
| 67 | Dic3 | — | 10 | 1 | no | — |
| 68 | Dic3 | — | 20 | 2 | no | — |
| 69 | C2×C5 | — | 6 | 4 | no | — |
| 70 | C2×C5 | — | 6 | 1 | no | — |
| 71 | C2×C5 | — | 6 | 1 | no | — |
| 72 | C2×C5 | — | 6 | 1 | no | — |
| 73 | C23 | S5 | 1 | 1 | yes | characteristic |
| 74 | Q8 | — | 5 | 1 | no | — |
| 75 | Q8 | — | 15 | 1 | no | — |
| 76 | Q8 | — | 15 | 2 | no | — |
| 77 | Q8 | — | 30 | 1 | no | — |
| 78 | C2×C4 | — | 10 | 2 | no | — |
| 79 | C2×C4 | — | 15 | 2 | no | — |
| 80 | C2×C4 | — | 15 | 2 | no | — |
| 81 | C2×C4 | — | 15 | 1 | no | — |
| 82 | C2×C4 | — | 15 | 1 | no | — |
| 83 | C2×C4 | — | 20 | 2 | no | — |
| 84 | C8 | — | 30 | 2 | no | — |
| 85 | C2×C3 | — | 10 | 4 | no | — |
| 86 | C2×C3 | — | 10 | 1 | no | — |
| 87 | C2×C3 | — | 10 | 1 | no | — |
| 88 | C2×C3 | — | 10 | 1 | no | — |
| 89 | C5 | — | 6 | 1 | no | Sylow |
| 90 | C22 | #6728 | 1 | 2 | yes | — |
| 91 | C22 | #6729 | 1 | 2 | yes | — |
| 92 | C22 | A5⋊C4 | 1 | 2 | yes | — |
| 93 | C22 | C2×S5 | 1 | 1 | yes | characteristic |
| 94 | C4 | — | 15 | 1 | no | — |
| 95 | C4 | — | 15 | 1 | no | — |
| 96 | C4 | — | 15 | 2 | no | — |
| 97 | C4 | — | 20 | 2 | no | — |
| 98 | C3 | — | 10 | 1 | no | Sylow |
| 99 | C2 | #91143 | 1 | 4 | yes | direct product factor |
| 100 | C2 | C2×#6728 | 1 | 1 | yes | characteristic |
| 101 | C2 | C2×#6729 | 1 | 1 | yes | characteristic |
| 102 | C2 | C2×A5⋊C4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Fitting, socle, hypercenter | C23 | S5 |
| commutator, hypocenter | SL(2,𝔽5) | C2×C4 |
| Frattini | C22 | C2×S5 |
| prime radical | C22×SL(2,𝔽5) | C2 |
| cosocle | C2×SL(2,𝔽5) | C22 |
| LCS | G ▷ SL(2,𝔽5) | |
| UCS | C1 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #91143 | C2 | direct (N × H) |
| 3. C2×SL(2,𝔽5) | C4 | semidirect |
| 7. SL(2,𝔽5) | C2×C4 | semidirect |
| 99. C2 | #91143 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×SL(2,𝔽5) | 2 |
| #91143 | 2 |
| C2×#859 | 5 |
| C22×#170 | 6 |
| C2×Dic3⋊C4 | 10 |