Finite Group Databasebeta

← #20276501

Subgroups — #20276501

Order 6784.

156 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133157C211yesmaximal, characteristic
2#16810661C211yesmaximal, characteristic
3#16810450C212yesmaximal, complemented
4#16810448C212yesmaximal, complemented
5C2×#107×C53C211yesmaximal, characteristic
6#13133176—21no—
7#13133157C414yescomplemented
8C2×#11719908C2212yes—
9#13133151C2211yescharacteristic
10#13133151C2211yescharacteristic
11#13133149—21no—
12C22×C8×C53C2211yescharacteristic
13C2×C4⋊C4×C53—21no—
14#107×C53—21no—
15#107×C53C2212yes—
16C22×C4×C53—41no—
17C22×C4×C53D411yescharacteristic
18C2×C8×C53C2×C412yes—
19C2×C8×C53D411yescharacteristic
20C2×C8×C53Q812yes—
21C2×C8×C53C2311yescharacteristic
22C4⋊C4×C53—22no—
23C4⋊C4×C53—41no—
24C22⋊Dic53—41no—
25C4⋊Dic53—21no—
26C4⋊Dic53—21no—
27C4×Dic53—41no—
28#11719908C2×C414yescomplemented
29C23×C53D811yescharacteristic
30C2×C4×C53—42no—
31C2×C4×C53—82no—
32C2×C4×C53C4⋊C412yes—
33C2×C4×C53D811yescharacteristic
34C2×C4×C53Q1612yes—
35C2×C4×C53C2×D411yescharacteristic
36C2×Dic53—41no—
37C2×Dic53—41no—
38C8×C53C4⋊C412yes—
39C8×C53C22×C411yescharacteristic
40C8×C53C2×Q811yescharacteristic
41C22×C53—21no—
42C22×C53—21no—
43C22×C53#10712yes—
44C22×C53C2×D811yescharacteristic
45C4×C53—82no—
46C4×C53#10712yes—
47C4×C53C2×C4⋊C411yescharacteristic
48C4×C53C2×Q1611yescharacteristic
49Dic53—81no—
50Dic53—81no—
51#2159—531nomaximal, Sylow
52C2×C53—22no—
53C2×C53C2×#10711yescharacteristic
54C2×C53#50911yescharacteristic
55C2×C53#51011yescharacteristic
56C2×M32—531no—
57#427—531no—
58C2×#107D5311yescharacteristic
59#367—532no—
60#366—532no—
61C53#215911yescharacteristic, complemented, Sylow
62C22×C8C2×D5311yescharacteristic
63#117—1061no—
64C2×C4⋊C4—21no—
65M32—534no—
66C2×C16—532no—
67#107—21no—
68#107—531no—
69#107—531no—
70#107C2×D5312yes—
71#106—1061no—
72C22×C4—41no—
73C22×C4C22⋊D5311yescharacteristic
74C2×C8Dic10612yes—
75C2×C8C4×D5312yes—
76C2×C8C22⋊D5311yescharacteristic
77C2×C8C22×D5311yescharacteristic
78C4⋊C4—22no—
79C4⋊C4—41no—
80C4⋊C4—1061no—
81C4⋊C4—1061no—
82C22⋊C4—2121no—
83C42—2121no—
84C16—534no—
85C23#1171992211yescharacteristic
86C2×C4—42no—
87C2×C4—82no—
88C2×C4—2121no—
89C2×C4—2121no—
90C2×C4#1171991912yes—
91C2×C4#1171992211yescharacteristic
92C2×C4#1171992512yes—
93C2×C4C2×C22⋊D5311yescharacteristic
94C8#1171991912yes—
95C8C2×Dic10611yescharacteristic
96C8C2×C4×D5311yescharacteristic
97C22—21no—
98C22—21no—
99C22#1313315012yes—
100C22C2×#1171992211yescharacteristic
101C4—82no—
102C4—4241no—
103C4—4241no—
104C4#1313315012yes—
105C4C2×#1171991911yescharacteristic
106C4C2×#1171992511yescharacteristic
107C2—22no—
108C2C2×#1313315011yescharacteristic
109C2#1681105911yescharacteristic
110C2#1681106411yescharacteristic
subgroupG/H
centerC22C2×#11719922
commutatorC8×C53C22×C4
FrattiniC2×C8C22×D53
FittingC2×#107×C53C2
socleC22×C53C2×D8
hypercenterC2×#107D53
hypocenterC53#2159
LCSG ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#107

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #16810450C2semidirect
4. #16810448C2semidirect
7. #13133157C4semidirect
28. #11719908C2×C4semidirect
61. C53#2159semidirect