Finite Group Databasebeta

← #22586878

Subgroups — #22586878

Order 9088.

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#17986056C211yesmaximal, characteristic, complemented
2#17985865C211yesmaximal, characteristic, complemented
3#17985736C212yesmaximal, complemented
4#17985733C212yesmaximal
5#529×C71C211yesmaximal, characteristic
6C2×C4⋊Dic71—21no—
7C4⋊(C71⋊C8)—21no—
8C22⋊(C71⋊C8)C2212yes—
9#14556039—21no—
10C71⋊#95—21no—
11C71⋊#95C2212yes—
12#126×C71—21no—
13#125×C71C411yescharacteristic, complemented
14#124×C71C411yescharacteristic, complemented
15#123×C71—21no—
16C22⋊Q8×C71C2211yescharacteristic
17#121×C71C2211yescharacteristic
18C4×Q8×C71C411yescharacteristic, complemented
19C4×D4×C71—21no—
20C4×D4×C71C411yescharacteristic, complemented
21#117×C71—21no—
22C2×C42×C71C2211yescharacteristic
23C22×Dic71—22no—
24C2×C71⋊C8—22no—
25C4⋊Dic71—42no—
26C2×Q8×C71C2×C411yescharacteristic
27C2×D4×C71—21no—
28C2×D4×C71C2×C411yescharacteristic
29C22×C4×C71—21no—
30C22×C4×C71D411yescharacteristic
31C22×C4×C71D411yescharacteristic
32C22×C4×C71C2311yescharacteristic
33C4⋊C4×C71—21no—
34C4⋊C4×C71—21no—
35C4⋊C4×C71—21no—
36C4⋊C4×C71—21no—
37C4⋊C4×C71C2×C411yescharacteristic
38C4⋊C4×C71C2×C411yescharacteristic
39C22⋊C4×C71—21no—
40C22⋊C4×C71—21no—
41C22⋊C4×C71—21no—
42C22⋊C4×C71—21no—
43C22⋊C4×C71—21no—
44C42×C71—21no—
45C42×C71C2×C411yescharacteristic
46C42×C71C2×C411yescharacteristic
47C42×C71D411yescharacteristic
48C42×C71D411yescharacteristic
49C23×C71—21no—
50C23×C71C2×D411yescharacteristic
51Q8×C71—41no—
52D4×C71—41no—
53D4×C71—41no—
54D4×C71—41no—
55C2×C4×C71—21no—
56C2×C4×C71—21no—
57C2×C4×C71—21no—
58C2×C4×C71—21no—
59C2×C4×C71—21no—
60C2×C4×C71—41no—
61C2×C4×C71—41no—
62C2×C4×C71—42no—
63C2×C4×C71C22⋊C411yescharacteristic
64C2×C4×C71C22⋊C411yescharacteristic
65C2×C4×C71C22⋊C411yescharacteristic
66C2×C4×C71C22⋊C411yescharacteristic
67C2×C4×C71C22×C411yescharacteristic
68C2×C4×C71C2×D411yescharacteristic
69C2×Dic71—42no—
70C2×Dic71—42no—
71C2×Dic71—42no—
72C71⋊C8—42no—
73C22×C71—21no—
74C22×C71—21no—
75C22×C71—41no—
76C22×C71—41no—
77C22×C71—41no—
78C22×C71C2×C22⋊C411yescharacteristic
79C22×C71AGL(1,ℤ/8)11yescharacteristic
80C22×C71#13711yescharacteristic
81C4×C71—21no—
82C4×C71—21no—
83C4×C71—21no—
84C4×C71—21no—
85C4×C71—21no—
86C4×C71—21no—
87C4×C71—41no—
88C4×C71—41no—
89C4×C71—41no—
90Dic71—82no—
91C2×C71—21no—
92C2×C71—21no—
93C2×C71—81no—
94C2×C71#41011yescharacteristic
95C2×C71#41711yescharacteristic
96C2×C71#42111yescharacteristic
97#1512—711nomaximal, Sylow
98C71#151211yescharacteristic, complemented, Sylow
99#529D7111yescharacteristic
100#432—711no—
101#382—711no—
102#328—712no—
103#327—712no—
104#126—21no—
105#125Dic7111yescharacteristic, complemented
106#124Dic7111yescharacteristic, complemented
107#123—21no—
108C22⋊Q8C2×D7111yescharacteristic
109#121C2×D7111yescharacteristic
110C4×Q8Dic7111yescharacteristic, complemented
111C4×D4—21no—
112C4×D4Dic7111yescharacteristic, complemented
113#117—21no—
114C2×C4⋊C4—1421no—
115C2×C42C2×D7111yescharacteristic
116C4⋊C8—1421no—
117C22⋊C8—712no—
118#97—1421no—
119#95—712no—
120#95—1421no—
121C2×Q8C2×Dic7111yescharacteristic
122C2×D4—21no—
123C2×D4C2×Dic7111yescharacteristic
124C22×C4—21no—
125C22×C4—1422no—
126C22×C4C22⋊D7111yescharacteristic
127C22×C4C22⋊D7111yescharacteristic
128C22×C4C22×D7111yescharacteristic
129C2×C8—1422no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—2842no—
135C4⋊C4C2×Dic7111yescharacteristic
136C4⋊C4C2×Dic7111yescharacteristic
137C22⋊C4—21no—
138C22⋊C4—21no—
139C22⋊C4—21no—
140C22⋊C4—21no—
141C22⋊C4—21no—
142C42—21no—
143C42C2×Dic7111yescharacteristic
144C42C2×Dic7111yescharacteristic
145C42C22⋊D7111yescharacteristic
146C42C22⋊D7111yescharacteristic
147C23—21no—
148C23C2×C22⋊D7111yescharacteristic
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—2842no—
162C2×C4—2842no—
163C2×C4—2842no—
164C2×C4C22⋊Dic7111yescharacteristic
165C2×C4C22⋊Dic7111yescharacteristic
166C2×C4C22⋊Dic7111yescharacteristic
167C2×C4C22⋊Dic7111yescharacteristic
168C2×C4C22×Dic7111yescharacteristic
169C2×C4C2×C22⋊D7111yescharacteristic
170C8—2842no—
171C22—21no—
172C22—21no—
173C22—41no—
174C22—41no—
175C22—41no—
176C22C2×C22⋊Dic7111yescharacteristic
177C22#1455615311yescharacteristic
178C22#1455615811yescharacteristic
179C4—21no—
180C4—21no—
181C4—21no—
182C4—21no—
183C4—21no—
184C4—21no—
185C4—41no—
186C4—41no—
187C4—41no—
188C4—5682no—
189C2—21no—
190C2—21no—
191C2—81no—
192C2#1798597311yescharacteristic
193C2#1798600011yescharacteristic
194C2#1798601611yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic71
commutatorC2×C4×C71C22×C4
FrattiniC22×C4C22×D71
Fitting#529×C71C2
socleC22×C71C2×C22⋊C4
hypercenter#529D71
hypocenterC71#1512
LCSG ▷ C2×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. #17986056C2semidirect
2. #17985865C2semidirect
3. #17985736C2semidirect
13. #125×C71C4semidirect
14. #124×C71C4semidirect
18. C4×Q8×C71C4semidirect
20. C4×D4×C71C4semidirect
98. C71#1512semidirect
105. #125Dic71semidirect
106. #124Dic71semidirect
110. C4×Q8Dic71semidirect
112. C4×D4Dic71semidirect