Finite Group Databasebeta

← #20299159

Subgroups — #20299159

Order 7232.

346 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 103 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
1C22×D452C211yesmaximal, characteristic, complemented
2C2×D4×D113C211yesmaximal, characteristic, complemented
3#16813607C212yesmaximal, complemented
4#16813605C211yesmaximal, characteristic, complemented
5#16813597C212yesmaximal, complemented
6#16813595C212yesmaximal, complemented
7#16813592C211yesmaximal, characteristic, complemented
8#16813592C212yesmaximal, complemented
9#16813590C211yesmaximal, characteristic, complemented
10C22⋊C4×D113C211yesmaximal, characteristic, complemented
11#123×C113C211yesmaximal, characteristic, complemented
12C23×D113C2211yescharacteristic
13C23×D113C2211yescharacteristic, complemented
14C23×D113C2211yescharacteristic, complemented
15C2×D452—24no—
16C2×D452C2212yes—
17C2×D452C2212yescomplemented
18C2×D452C2211yescharacteristic, complemented
19C2×D452C2212yescomplemented
20C2×C22⋊D113C2211yescharacteristic
21C2×C22⋊D113C2211yescharacteristic, complemented
22C2×C22⋊D113C2212yescomplemented
23D4×D113—24no—
24C2×C4×D113C2211yescharacteristic, complemented
25C2×C4×D113C2212yescomplemented
26#14268626C2212yescomplemented
27C22⋊Dic113C2211yescharacteristic, complemented
28#14268624C2212yescomplemented
29#14268624C2212yescomplemented
30#14268624C2212yescomplemented
31#14268624C2212yescomplemented
32C2×D4×C113C2211yescharacteristic
33C22×C4×C113C2211yescharacteristic, complemented
34C4⋊C4×C113C2212yescomplemented
35C22⋊C4×C113C2211yescharacteristic, complemented
36C22⋊C4×C113C2212yescomplemented
37C23×C113C2311yescharacteristic
38C23×C113C2311yescharacteristic
39C22×D113—22no—
40C22×D113—22no—
41C22×D113—42no—
42C22×D113—42no—
43C22×D113—42no—
44C22×D113D414yescomplemented
45C22×D113C2311yescharacteristic
46C22×D113C2312yes—
47C22×D113C2311yescharacteristic
48C22×D113C2311yescharacteristic
49D4×C113—22no—
50C2×C4×C113—22no—
51C2×C4×C113C2312yes—
52C2×C4×C113C2312yes—
53C2×C4×C113C2311yescharacteristic
54C22⋊D113—22no—
55C22⋊D113—22no—
56C22⋊D113—42no—
57C2×Dic113C2312yes—
58C2×Dic113C2311yescharacteristic
59D452—22no—
60D452—24no—
61D452—24no—
62D452—42no—
63C4×D113—22no—
64C4×D113—42no—
65C22×C113—22no—
66C22×C113—21no—
67C22×C113—22no—
68C22×C113C2×D412yes—
69C22×C113C2411yescharacteristic
70C2×D113—22no—
71C2×D113—22no—
72C2×D113—22no—
73C2×D113—21no—
74C2×D113—22no—
75C2×D113—22no—
76C2×D113—22no—
77C2×D113—44no—
78C2×D113—44no—
79C2×D113—44no—
80C2×D113C2×D412yes—
81C2×D113C2×D412yes—
82C4×C113—21no—
83C4×C113—22no—
84C4×C113—22no—
85Dic113—21no—
86Dic113—22no—
87C2×C113—22no—
88C2×C113—41no—
89C2×C113C22×D411yescharacteristic
90C2×C113H(5,𝔽2)12yes—
91D113—22no—
92D113—22no—
93D113—42no—
94D113—41no—
95C113#53411yescharacteristic, complemented, Sylow
96#534—1131nomaximal, Sylow
97C22×D4—1131no—
98C22×D4—1131no—
99#123—1131no—
100#123—1132no—
101#123D11311yescharacteristic, complemented
102#121—1132no—
103#121—1132no—
104#120—1131no—
105#120—1132no—
106#120—1131no—
107C2×C22⋊C4—1131no—
108C24—1131no—
109C24—1131no—
110C24—1131no—
111C2×D4—1132no—
112C2×D4—1131no—
113C2×D4—1131no—
114C2×D4—1132no—
115C2×D4—1132no—
116C2×D4—1131no—
117C2×D4—1132no—
118C2×D4—2264no—
119C2×D4—2264no—
120C2×D4C2×D11311yescharacteristic
121C22×C4—1132no—
122C22×C4—1131no—
123C22×C4C2×D11311yescharacteristic, complemented
124C4⋊C4—1132no—
125C4⋊C4C2×D11312yescomplemented
126C22⋊C4—1131no—
127C22⋊C4—1132no—
128C22⋊C4—1132no—
129C22⋊C4—1132no—
130C22⋊C4—1132no—
131C22⋊C4C2×D11311yescharacteristic, complemented
132C22⋊C4C2×D11312yescomplemented
133C23—1134no—
134C23—1131no—
135C23—1131no—
136C23—1132no—
137C23—1131no—
138C23—4522no—
139C23—4522no—
140C23—4522no—
141C23—2262no—
142C23—2262no—
143C23C22×D11311yescharacteristic
144C23C22×D11311yescharacteristic
145D4—22no—
146D4—4522no—
147D4—4522no—
148D4—2264no—
149D4—2264no—
150D4—2262no—
151D4—2262no—
152D4—2262no—
153C2×C4—22no—
154C2×C4—1132no—
155C2×C4—1131no—
156C2×C4—4522no—
157C2×C4—2262no—
158C2×C4C22×D11312yes—
159C2×C4C22×D11311yescharacteristic
160C2×C4C22×D11312yes—
161C22—22no—
162C22—21no—
163C22—22no—
164C22—1132no—
165C22—1132no—
166C22—4524no—
167C22—4524no—
168C22—4524no—
169C22—2262no—
170C22—2261no—
171C22—2262no—
172C22—2262no—
173C22—2262no—
174C22—2262no—
175C22—2262no—
176C22D4×D11312yes—
177C22C23×D11311yescharacteristic
178C4—22no—
179C4—21no—
180C4—22no—
181C4—2261no—
182C4—2262no—
183C2—22no—
184C2—41no—
185C2—4522no—
186C2—4521no—
187C2—2262no—
188C2—2262no—
189C2C2×D4×D11311yescharacteristic
190C2#1681366812yes—
subgroupG/H
center, FrattiniC22C23×D113
commutator, socleC22×C113C24
Fitting#123×C113C2
hypercenter#123D113
hypocenterC113#534
LCSG ▷ C22×C113 ▷ C113
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. C22×D452C2semidirect
2. C2×D4×D113C2semidirect
3. #16813607C2semidirect
4. #16813605C2semidirect
5. #16813597C2semidirect
6. #16813595C2semidirect
7. #16813592C2semidirect
8. #16813592C2semidirect
9. #16813590C2semidirect
10. C22⋊C4×D113C2semidirect
11. #123×C113C2semidirect
13. C23×D113C22semidirect
14. C23×D113C22semidirect
17. C2×D452C22semidirect
18. C2×D452C22semidirect
19. C2×D452C22semidirect
21. C2×C22⋊D113C22semidirect
22. C2×C22⋊D113C22semidirect
24. C2×C4×D113C22semidirect
25. C2×C4×D113C22semidirect
26. #14268626C22semidirect
27. C22⋊Dic113C22semidirect
28. #14268624C22semidirect
29. #14268624C22semidirect
30. #14268624C22semidirect
31. #14268624C22semidirect
33. C22×C4×C113C22semidirect
34. C4⋊C4×C113C22semidirect
35. C22⋊C4×C113C22semidirect
36. C22⋊C4×C113C22semidirect
44. C22×D113D4semidirect
95. C113#534semidirect
101. #123D113semidirect
123. C22×C4C2×D113semidirect
125. C4⋊C4C2×D113semidirect
131. C22⋊C4C2×D113semidirect
132. C22⋊C4C2×D113semidirect