Finite Group Databasebeta

← #20273240

Subgroups — #20273240

Order 6784.

242 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 81 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
2C4×C4⋊Dic53C211yesmaximal, characteristic, complemented
3#16810360C212yesmaximal, complemented
4#16810357C212yesmaximal
5C53×#529C211yesmaximal, characteristic
6C2×C4⋊Dic53—21no—
7C2×C4×Dic53—21no—
8C4⋊#88322—21no—
9C22⋊#88322C2212yes—
10#13133128—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—24no—
48C4×Dic53—42no—
49C2×#88322—22no—
50C23×C53—21no—
51C23×C53C2×D411yescharacteristic
52Q8×C53—41no—
53D4×C53—41no—
54D4×C53—41no—
55D4×C53—41no—
56C2×C4×C53—21no—
57C2×C4×C53—21no—
58C2×C4×C53—22no—
59C2×C4×C53—21no—
60C2×C4×C53—21no—
61C2×C4×C53—21no—
62C2×C4×C53—41no—
63C2×C4×C53—42no—
64C2×C4×C53C22⋊C411yescharacteristic
65C2×C4×C53C22⋊C411yescharacteristic
66C2×C4×C53C22⋊C411yescharacteristic
67C2×C4×C53C22⋊C411yescharacteristic
68C2×C4×C53C22×C411yescharacteristic
69C2×C4×C53C2×D411yescharacteristic
70C2×Dic53—24no—
71C2×Dic53—42no—
72C2×Dic53—42no—
73#88322—42no—
74C22×C53—21no—
75C22×C53—21no—
76C22×C53—41no—
77C22×C53—41no—
78C22×C53—41no—
79C22×C53C2×C22⋊C411yescharacteristic
80C22×C53AGL(1,ℤ/8)11yescharacteristic
81C22×C53#13711yescharacteristic
82C4×C53—21no—
83C4×C53—21no—
84C4×C53—21no—
85C4×C53—21no—
86C4×C53—21no—
87C4×C53—41no—
88C4×C53—41no—
89C4×C53—41no—
90C4×C53#10412yes—
91Dic53—44no—
92#1509—531nomaximal, Sylow
93C2×C53—21no—
94C2×C53—21no—
95C2×C53—81no—
96C2×C53#41011yescharacteristic
97C2×C53#41711yescharacteristic
98C2×C53C2×#10411yescharacteristic
99#529D5311yescharacteristic
100#432—531no—
101C4×C4⋊C4—531no—
102#328—532no—
103#327—532no—
104C53#150911yescharacteristic, complemented, Sylow
105#126—21no—
106#125Dic5311yescharacteristic, complemented
107#124Dic5311yescharacteristic, complemented
108#123—21no—
109C22⋊Q8C2×D5311yescharacteristic
110#121C2×D5311yescharacteristic
111C4×Q8Dic5311yescharacteristic, complemented
112C4×D4—21no—
113C4×D4Dic5311yescharacteristic, complemented
114#117—21no—
115C2×C4⋊C4—1061no—
116C2×C42—1061no—
117C2×C42C2×D5311yescharacteristic
118C4⋊C8—1061no—
119C22⋊C8—532no—
120#97—1061no—
121#95—532no—
122C2×Q8C2×Dic5311yescharacteristic
123C2×D4—21no—
124C2×D4C2×Dic5311yescharacteristic
125C22×C4—21no—
126C22×C4—1062no—
127C22×C4C22⋊D5311yescharacteristic
128C22×C4C22⋊D5311yescharacteristic
129C22×C4C22×D5311yescharacteristic
130C2×C8—1062no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—21no—
135C4⋊C4—1064no—
136C4⋊C4C2×Dic5311yescharacteristic
137C4⋊C4C2×Dic5311yescharacteristic
138C22⋊C4—21no—
139C22⋊C4—21no—
140C22⋊C4—21no—
141C22⋊C4—21no—
142C22⋊C4—21no—
143C42—21no—
144C42—2122no—
145C42C2×Dic5311yescharacteristic
146C42C2×Dic5311yescharacteristic
147C42C22⋊D5311yescharacteristic
148C42C22⋊D5311yescharacteristic
149C23—21no—
150C23C2×C22⋊D5311yescharacteristic
151Q8—41no—
152D4—41no—
153D4—41no—
154D4—41no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—41no—
162C2×C4—42no—
163C2×C4—1064no—
164C2×C4—2122no—
165C2×C4—2122no—
166C2×C4C22⋊Dic5311yescharacteristic
167C2×C4C22⋊Dic5311yescharacteristic
168C2×C4C22⋊Dic5311yescharacteristic
169C2×C4C22⋊Dic5311yescharacteristic
170C2×C4C22×Dic5311yescharacteristic
171C2×C4C2×C22⋊D5311yescharacteristic
172C8—2122no—
173C22—21no—
174C22—21no—
175C22—41no—
176C22—41no—
177C22—41no—
178C22C2×C22⋊Dic5311yescharacteristic
179C22#1313324211yescharacteristic
180C22#1313324711yescharacteristic
181C4—21no—
182C4—21no—
183C4—21no—
184C4—21no—
185C4—21no—
186C4—41no—
187C4—41no—
188C4—41no—
189C4—2124no—
190C4#1313314412yes—
191C2—21no—
192C2—21no—
193C2—81no—
194C2#1681059711yescharacteristic
195C2#1681062411yescharacteristic
196C2C2×#1313314411yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic53
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#529C2
socleC22×C53C2×C22⋊C4
hypercenter#529D53
hypocenterC53#1509
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. C4×C4⋊Dic53C2semidirect
3. #16810360C2semidirect
13. #125×C53C4semidirect
14. #124×C53C4semidirect
18. C4×Q8×C53C4semidirect
20. C4×D4×C53C4semidirect
104. C53#1509semidirect
106. #125Dic53semidirect
107. #124Dic53semidirect
111. C4×Q8Dic53semidirect
113. C4×D4Dic53semidirect