Finite Group Databasebeta

← #20274038

Subgroups — #20274038

Order 6784.

196 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 49 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
1#16810993C211yesmaximal, characteristic, complemented
2#16810982C212yesmaximal, complemented
3#16810365C211yesmaximal, characteristic, complemented
4#16810363C212yesmaximal, complemented
5C53×#562C211yesmaximal, characteristic
6#13133217C2211yescharacteristic
7#13133141—22no—
8#13133140—22no—
9#13133138—22no—
10#13133128C2211yescharacteristic
11#13133128C2212yes—
12#125×C53C2211yescharacteristic
13#124×C53C2212yes—
14#123×C53—22no—
15C22⋊Q8×C53—22no—
16#121×C53—22no—
17C2×Q8×C53D412yes—
18C2×D4×C53—22no—
19C2×D4×C53D412yes—
20C22×C4×C53—22no—
21C2×Dic106—21no—
22C4⋊C4×C53—22no—
23C4⋊C4×C53—22no—
24C4⋊C4×C53D412yes—
25C22⋊C4×C53—22no—
26C22⋊C4×C53—22no—
27C22⋊C4×C53—22no—
28C42×C53C2311yescharacteristic
29C4⋊Dic53—22no—
30C2×#88322—22no—
31C2×#88322—21no—
32C23×C53—22no—
33Q8×C53—42no—
34D4×C53—42no—
35D4×C53—82no—
36C2×C4×C53—22no—
37C2×C4×C53—22no—
38C2×C4×C53—82no—
39C2×C4×C53C2×D411yescharacteristic
40C2×C4×C53C2×D412yes—
41C2×Dic53—41no—
42Dic106—42no—
43#88322—41no—
44#88322—42no—
45C22×C53—42no—
46C22×C53—42no—
47C22×C53—42no—
48C22×C53#12011yescharacteristic
49C4×C53—21no—
50C4×C53—22no—
51C4×C53—42no—
52C4×C53—42no—
53Dic53—81no—
54#1662—531nomaximal, Sylow
55C2×C53—82no—
56C2×C53#45411yescharacteristic
57C2×C53#45512yes—
58#562D5311yescharacteristic
59#491—531no—
60#489—532no—
61#330—531no—
62#329—532no—
63C53#166211yescharacteristic, complemented, Sylow
64C4⋊Q8—531no—
65#125C2×D5311yescharacteristic
66#124C2×D5312yes—
67#123—22no—
68C22⋊Q8—22no—
69#121—22no—
70#103—1062no—
71#103—1062no—
72#102—1062no—
73#97—531no—
74#97—532no—
75C2×Q8—1061no—
76C2×Q8C22⋊D5312yes—
77C2×D4—22no—
78C2×D4C22⋊D5312yes—
79C22×C4—22no—
80C2×C8—1062no—
81C2×C8—1061no—
82C4⋊C4—22no—
83C4⋊C4—22no—
84C4⋊C4—1062no—
85C4⋊C4C22⋊D5312yes—
86C22⋊C4—22no—
87C22⋊C4—22no—
88C22⋊C4—22no—
89C42C22×D5311yescharacteristic
90C23—22no—
91Q8—42no—
92Q8—2122no—
93D4—42no—
94D4—82no—
95C2×C4—22no—
96C2×C4—22no—
97C2×C4—82no—
98C2×C4—2121no—
99C2×C4C2×C22⋊D5311yescharacteristic
100C2×C4C2×C22⋊D5312yes—
101C8—2121no—
102C8—2122no—
103C22—42no—
104C22—42no—
105C22—42no—
106C22#1313318611yescharacteristic
107C4—21no—
108C4—22no—
109C4—42no—
110C4—42no—
111C4—4241no—
112C2—82no—
113C2#1681079611yescharacteristic
114C2#1681080112yes—
subgroupG/H
centerC22#13133186
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#562C2
socleC22×C53#120
hypercenter#562D53
hypocenterC53#1662
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #562

Complements

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

NH ≅ G/Ntype
1. #16810993C2semidirect
2. #16810982C2semidirect
3. #16810365C2semidirect
4. #16810363C2semidirect
63. C53#1662semidirect