Finite Group Databasebeta

← #20273292

Subgroups — #20273292

Order 6784.

218 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 77 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#16810676C211yesmaximal, characteristic, complemented
2#16810491C211yesmaximal, characteristic, complemented
3#16810360C212yesmaximal, complemented
4#16810357C212yesmaximal
5C53×#529C211yesmaximal, characteristic
6C4⋊#88322—21no—
7C22⋊#88322C2212yes—
8C4×#88322—21no—
9#13133124—21no—
10#13133124—21no—
11#13133124C2212yes—
12#126×C53—21no—
13#125×C53C411yescharacteristic, complemented
14#124×C53C411yescharacteristic, complemented
15#123×C53—21no—
16C22⋊Q8×C53C2211yescharacteristic
17#121×C53C2211yescharacteristic
18C4×Q8×C53C411yescharacteristic, complemented
19C4×D4×C53—21no—
20C4×D4×C53C411yescharacteristic, complemented
21#117×C53—21no—
22C2×C42×C53C2211yescharacteristic
23C2×Q8×C53C2×C411yescharacteristic
24C2×D4×C53—21no—
25C2×D4×C53C2×C411yescharacteristic
26C22×C4×C53—21no—
27C22×C4×C53D411yescharacteristic
28C22×C4×C53D411yescharacteristic
29C22×C4×C53C2311yescharacteristic
30C22×Dic53—22no—
31C4⋊C4×C53—21no—
32C4⋊C4×C53—21no—
33C4⋊C4×C53—21no—
34C4⋊C4×C53—21no—
35C4⋊C4×C53C2×C411yescharacteristic
36C4⋊C4×C53C2×C411yescharacteristic
37C22⋊C4×C53—21no—
38C22⋊C4×C53—21no—
39C22⋊C4×C53—21no—
40C22⋊C4×C53—21no—
41C22⋊C4×C53—21no—
42C42×C53—21no—
43C42×C53C2×C411yescharacteristic
44C42×C53C2×C411yescharacteristic
45C42×C53D411yescharacteristic
46C42×C53D411yescharacteristic
47C2×#88322—22no—
48C23×C53—21no—
49C23×C53C2×D411yescharacteristic
50Q8×C53—41no—
51D4×C53—41no—
52D4×C53—42no—
53C2×C4×C53—22no—
54C2×C4×C53—21no—
55C2×C4×C53—21no—
56C2×C4×C53—21no—
57C2×C4×C53—41no—
58C2×C4×C53—41no—
59C2×C4×C53—42no—
60C2×C4×C53C22⋊C411yescharacteristic
61C2×C4×C53C22⋊C411yescharacteristic
62C2×C4×C53C22⋊C411yescharacteristic
63C2×C4×C53C22⋊C411yescharacteristic
64C2×C4×C53C22×C411yescharacteristic
65C2×C4×C53C2×D411yescharacteristic
66C2×Dic53—42no—
67C2×Dic53—42no—
68C2×Dic53—42no—
69#88322—42no—
70C22×C53—21no—
71C22×C53—21no—
72C22×C53—41no—
73C22×C53—41no—
74C22×C53—41no—
75C22×C53C2×C22⋊C411yescharacteristic
76C22×C53#13512yes—
77C4×C53—21no—
78C4×C53—21no—
79C4×C53—21no—
80C4×C53—21no—
81C4×C53—21no—
82C4×C53—21no—
83C4×C53—41no—
84C4×C53—41no—
85C4×C53—41no—
86Dic53—82no—
87#1521—531nomaximal, Sylow
88C2×C53—22no—
89C2×C53—81no—
90C2×C53#41011yescharacteristic
91C2×C53#41611yescharacteristic
92C2×C53#42111yescharacteristic
93#529D5311yescharacteristic
94#431—531no—
95#383—531no—
96#328—532no—
97#327—532no—
98C53#152111yescharacteristic, complemented, Sylow
99#126—21no—
100#125Dic5311yescharacteristic, complemented
101#124Dic5311yescharacteristic, complemented
102#123—21no—
103C22⋊Q8C2×D5311yescharacteristic
104#121C2×D5311yescharacteristic
105C4×Q8Dic5311yescharacteristic, complemented
106C4×D4—21no—
107C4×D4Dic5311yescharacteristic, complemented
108#117—21no—
109C2×C42C2×D5311yescharacteristic
110C4⋊C8—1061no—
111C22⋊C8—532no—
112C4×C8—1061no—
113#95—532no—
114#95—1061no—
115#95—1061no—
116C2×Q8C2×Dic5311yescharacteristic
117C2×D4—21no—
118C2×D4C2×Dic5311yescharacteristic
119C22×C4—21no—
120C22×C4—1062no—
121C22×C4C22⋊D5311yescharacteristic
122C22×C4C22⋊D5311yescharacteristic
123C22×C4C22×D5311yescharacteristic
124C2×C8—1062no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4C2×Dic5311yescharacteristic
130C4⋊C4C2×Dic5311yescharacteristic
131C22⋊C4—21no—
132C22⋊C4—21no—
133C22⋊C4—21no—
134C22⋊C4—21no—
135C22⋊C4—21no—
136C42—21no—
137C42C2×Dic5311yescharacteristic
138C42C2×Dic5311yescharacteristic
139C42C22⋊D5311yescharacteristic
140C42C22⋊D5311yescharacteristic
141C23—21no—
142C23C2×C22⋊D5311yescharacteristic
143Q8—41no—
144D4—41no—
145D4—42no—
146C2×C4—22no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—42no—
153C2×C4—2122no—
154C2×C4—2122no—
155C2×C4—2122no—
156C2×C4C22⋊Dic5311yescharacteristic
157C2×C4C22⋊Dic5311yescharacteristic
158C2×C4C22⋊Dic5311yescharacteristic
159C2×C4C22⋊Dic5311yescharacteristic
160C2×C4C22×Dic5311yescharacteristic
161C2×C4C2×C22⋊D5311yescharacteristic
162C8—2122no—
163C22—21no—
164C22—21no—
165C22—41no—
166C22—41no—
167C22—41no—
168C22C2×C22⋊Dic5311yescharacteristic
169C22#1313323712yes—
170C4—21no—
171C4—21no—
172C4—21no—
173C4—21no—
174C4—21no—
175C4—21no—
176C4—41no—
177C4—41no—
178C4—41no—
179C4—4242no—
180C2—22no—
181C2—81no—
182C2#1681059711yescharacteristic
183C2#1681061911yescharacteristic
184C2#1681064011yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic53
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#529C2
socleC22×C53C2×C22⋊C4
hypercenter#529D53
hypocenterC53#1521
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. #16810676C2semidirect
2. #16810491C2semidirect
3. #16810360C2semidirect
13. #125×C53C4semidirect
14. #124×C53C4semidirect
18. C4×Q8×C53C4semidirect
20. C4×D4×C53C4semidirect
98. C53#1521semidirect
100. #125Dic53semidirect
101. #124Dic53semidirect
105. C4×Q8Dic53semidirect
107. C4×D4Dic53semidirect