Finite Group Databasebeta

← #20273253

Subgroups — #20273253

Order 6784.

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

normal subgroups only

This group has 222 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#16810680C211yesmaximal, characteristic, complemented
2#16810489C211yesmaximal, characteristic, complemented
3#16810360C212yesmaximal, complemented
4#16810357C212yesmaximal
5C53×#529C211yesmaximal, characteristic
6C2×C4⋊Dic53—21no—
7C4⋊#88322—21no—
8C22⋊#88322C2212yes—
9#13133128—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
47C4⋊Dic53—42no—
48C2×#88322—22no—
49C23×C53—21no—
50C23×C53C2×D411yescharacteristic
51Q8×C53—41no—
52D4×C53—41no—
53D4×C53—41no—
54D4×C53—41no—
55C2×C4×C53—21no—
56C2×C4×C53—21no—
57C2×C4×C53—21no—
58C2×C4×C53—21no—
59C2×C4×C53—21no—
60C2×C4×C53—41no—
61C2×C4×C53—41no—
62C2×C4×C53—42no—
63C2×C4×C53C22⋊C411yescharacteristic
64C2×C4×C53C22⋊C411yescharacteristic
65C2×C4×C53C22⋊C411yescharacteristic
66C2×C4×C53C22⋊C411yescharacteristic
67C2×C4×C53C22×C411yescharacteristic
68C2×C4×C53C2×D411yescharacteristic
69C2×Dic53—42no—
70C2×Dic53—42no—
71C2×Dic53—42no—
72#88322—42no—
73C22×C53—21no—
74C22×C53—21no—
75C22×C53—41no—
76C22×C53—41no—
77C22×C53—41no—
78C22×C53C2×C22⋊C411yescharacteristic
79C22×C53AGL(1,ℤ/8)11yescharacteristic
80C22×C53#13711yescharacteristic
81C4×C53—21no—
82C4×C53—21no—
83C4×C53—21no—
84C4×C53—21no—
85C4×C53—21no—
86C4×C53—21no—
87C4×C53—41no—
88C4×C53—41no—
89C4×C53—41no—
90Dic53—82no—
91#1512—531nomaximal, Sylow
92C2×C53—21no—
93C2×C53—21no—
94C2×C53—81no—
95C2×C53#41011yescharacteristic
96C2×C53#41711yescharacteristic
97C2×C53#42111yescharacteristic
98#529D5311yescharacteristic
99#432—531no—
100#382—531no—
101#328—532no—
102#327—532no—
103C53#151211yescharacteristic, complemented, Sylow
104#126—21no—
105#125Dic5311yescharacteristic, complemented
106#124Dic5311yescharacteristic, complemented
107#123—21no—
108C22⋊Q8C2×D5311yescharacteristic
109#121C2×D5311yescharacteristic
110C4×Q8Dic5311yescharacteristic, complemented
111C4×D4—21no—
112C4×D4Dic5311yescharacteristic, complemented
113#117—21no—
114C2×C4⋊C4—1061no—
115C2×C42C2×D5311yescharacteristic
116C4⋊C8—1061no—
117C22⋊C8—532no—
118#97—1061no—
119#95—532no—
120#95—1061no—
121C2×Q8C2×Dic5311yescharacteristic
122C2×D4—21no—
123C2×D4C2×Dic5311yescharacteristic
124C22×C4—21no—
125C22×C4—1062no—
126C22×C4C22⋊D5311yescharacteristic
127C22×C4C22⋊D5311yescharacteristic
128C22×C4C22×D5311yescharacteristic
129C2×C8—1062no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—2122no—
135C4⋊C4C2×Dic5311yescharacteristic
136C4⋊C4C2×Dic5311yescharacteristic
137C22⋊C4—21no—
138C22⋊C4—21no—
139C22⋊C4—21no—
140C22⋊C4—21no—
141C22⋊C4—21no—
142C42—21no—
143C42C2×Dic5311yescharacteristic
144C42C2×Dic5311yescharacteristic
145C42C22⋊D5311yescharacteristic
146C42C22⋊D5311yescharacteristic
147C23—21no—
148C23C2×C22⋊D5311yescharacteristic
149Q8—41no—
150D4—41no—
151D4—41no—
152D4—41no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—42no—
161C2×C4—2122no—
162C2×C4—2122no—
163C2×C4—2122no—
164C2×C4C22⋊Dic5311yescharacteristic
165C2×C4C22⋊Dic5311yescharacteristic
166C2×C4C22⋊Dic5311yescharacteristic
167C2×C4C22⋊Dic5311yescharacteristic
168C2×C4C22×Dic5311yescharacteristic
169C2×C4C2×C22⋊D5311yescharacteristic
170C8—2122no—
171C22—21no—
172C22—21no—
173C22—41no—
174C22—41no—
175C22—41no—
176C22C2×C22⋊Dic5311yescharacteristic
177C22#1313324211yescharacteristic
178C22#1313324711yescharacteristic
179C4—21no—
180C4—21no—
181C4—21no—
182C4—21no—
183C4—21no—
184C4—21no—
185C4—41no—
186C4—41no—
187C4—41no—
188C4—4242no—
189C2—21no—
190C2—21no—
191C2—81no—
192C2#1681059711yescharacteristic
193C2#1681062411yescharacteristic
194C2#1681064011yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic53
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#529C2
socleC22×C53C2×C22⋊C4
hypercenter#529D53
hypocenterC53#1512
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. #16810680C2semidirect
2. #16810489C2semidirect
3. #16810360C2semidirect
13. #125×C53C4semidirect
14. #124×C53C4semidirect
18. C4×Q8×C53C4semidirect
20. C4×D4×C53C4semidirect
103. C53#1512semidirect
105. #125Dic53semidirect
106. #124Dic53semidirect
110. C4×Q8Dic53semidirect
112. C4×D4Dic53semidirect