Finite Group Databasebeta

← #20299232

Subgroups — #20299232

Order 7232.

428 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 115 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×D113C212yesmaximal, complemented
3C2×D4×D113C211yesmaximal, characteristic, complemented
4#16813618C211yesmaximal, characteristic, complemented
5#16813597C211yesmaximal, characteristic, complemented
6#16813595C212yesmaximal, complemented
7#16813592C212yesmaximal, complemented
8#16813590C212yesmaximal, complemented
9#16813584C211yesmaximal, characteristic, complemented
10C4×C22⋊D113C211yesmaximal, characteristic, complemented
11#121×C113C211yesmaximal, characteristic, complemented
12C23×D113C2212yescomplemented
13C23×D113C2212yescomplemented
14C2×D452—24no—
15C2×D452C2211yescharacteristic
16C2×D452C2212yescomplemented
17C2×D452C2212yescomplemented
18C2×D452C2211yescharacteristic, complemented
19C2×C22⋊D113C2211yescharacteristic
20C2×C22⋊D113C2212yescomplemented
21C2×C22⋊D113C2212yescomplemented
22C2×C22⋊D113C2212yescomplemented
23D4×D113—28no—
24D4×D113—24no—
25C2×C4×D113C2211yescharacteristic, complemented
26C2×C4×D113C2212yescomplemented
27#14268626C2211yescharacteristic, complemented
28C22⋊Dic113C2211yescharacteristic, complemented
29#14268624C2211yescharacteristic, complemented
30#14268624C2212yescomplemented
31#14268624C2212yescomplemented
32C4×Dic113C2211yescharacteristic, complemented
33C2×D4×C113C2212yescomplemented
34C2×D4×C113C2211yescharacteristic, complemented
35C22×C4×C113C2211yescharacteristic, complemented
36C4⋊C4×C113C2211yescharacteristic, complemented
37C22⋊C4×C113C2212yescomplemented
38C23×C113C2312yes—
39C23×C113C2311yescharacteristic
40C22×D113—24no—
41C22×D113—24no—
42C22×D113—24no—
43C22×D113—24no—
44C22×D113—42no—
45C22×D113—42no—
46C22×D113C2312yes—
47C22×D113C2311yescharacteristic
48C22×D113C2312yes—
49D4×C113—24no—
50D4×C113—22no—
51C2×C4×C113—22no—
52C2×C4×C113C2311yescharacteristic
53C2×C4×C113C2312yes—
54C2×C4×C113C2311yescharacteristic
55C22⋊D113—24no—
56C22⋊D113—24no—
57C22⋊D113—42no—
58C22⋊D113D414yescomplemented
59C2×Dic113C2311yescharacteristic
60C2×Dic113C2311yescharacteristic
61C2×Dic113C2311yescharacteristic
62D452—24no—
63D452—24no—
64D452—22no—
65D452D414yescomplemented
66C4×D113—22no—
67C4×D113—24no—
68C22×C113—21no—
69C22×C113—21no—
70C22×C113—22no—
71C22×C113—22no—
72C22×C113—22no—
73C22×C113C2×D412yescomplemented
74C22×C113C2411yescharacteristic
75C2×D113—21no—
76C2×D113—22no—
77C2×D113—21no—
78C2×D113—22no—
79C2×D113—22no—
80C2×D113—22no—
81C2×D113—22no—
82C2×D113—48no—
83C2×D113—48no—
84C2×D113C2×D412yescomplemented
85C2×D113C2×D414yescomplemented
86C4×C113—22no—
87C4×C113—21no—
88C4×C113C2×D412yescomplemented
89Dic113—21no—
90Dic113—21no—
91Dic113C2×D412yescomplemented
92C2×C113—22no—
93C2×C113—42no—
94C2×C113C22×D411yescharacteristic
95C2×C113C22×D411yescharacteristic
96C2×C113H(5,𝔽2)11yescharacteristic
97D113—22no—
98D113—24no—
99D113—42no—
100C113D4211yescharacteristic, complemented, Sylow
101D42—1131nomaximal, Sylow
102C22×D4—1132no—
103C22×D4—1131no—
104C22×D4—1131no—
105#127—1131no—
106#121—1131no—
107#121—1132no—
108#121D11311yescharacteristic, complemented
109#120—1132no—
110#120—1132no—
111C4×D4—1131no—
112C4×D4—1131no—
113C24—1132no—
114C24—1132no—
115C2×D4—1131no—
116C2×D4—1132no—
117C2×D4—1132no—
118C2×D4—1132no—
119C2×D4—1131no—
120C2×D4—1132no—
121C2×D4—1132no—
122C2×D4—1131no—
123C2×D4—2264no—
124C2×D4—2268no—
125C2×D4—2264no—
126C2×D4C2×D11312yescomplemented
127C2×D4C2×D11311yescharacteristic, complemented
128C22×C4—1132no—
129C22×C4—1131no—
130C22×C4C2×D11311yescharacteristic, complemented
131C4⋊C4—1131no—
132C4⋊C4C2×D11311yescharacteristic, complemented
133C22⋊C4—1131no—
134C22⋊C4—1132no—
135C22⋊C4—1131no—
136C22⋊C4—1132no—
137C22⋊C4C2×D11312yescomplemented
138C42—1131no—
139C23—1131no—
140C23—1132no—
141C23—1132no—
142C23—4522no—
143C23—4522no—
144C23—2264no—
145C23—2264no—
146C23—2264no—
147C23—2264no—
148C23C22×D11312yes—
149C23C22×D11311yescharacteristic
150D4—24no—
151D4—22no—
152D4—1134no—
153D4—1134no—
154D4—4522no—
155D4—2264no—
156D4—2264no—
157D4—2264no—
158D4—2264no—
159D4—2262no—
160C2×C4—22no—
161C2×C4—1131no—
162C2×C4—1131no—
163C2×C4—1131no—
164C2×C4—2264no—
165C2×C4—2262no—
166C2×C4C22×D11312yes—
167C2×C4C22×D11311yescharacteristic
168C2×C4C22×D11311yescharacteristic
169C22—21no—
170C22—21no—
171C22—22no—
172C22—22no—
173C22—22no—
174C22—1132no—
175C22—1134no—
176C22—4528no—
177C22—4528no—
178C22—2262no—
179C22—2262no—
180C22—2262no—
181C22—2262no—
182C22—2261no—
183C22—2262no—
184C22—2261no—
185C22D4×D11312yescomplemented
186C22C23×D11311yescharacteristic
187C4—21no—
188C4—22no—
189C4—1132no—
190C4—2261no—
191C4—2261no—
192C4D4×D11312yescomplemented
193C2—22no—
194C2—42no—
195C2—4522no—
196C2—2264no—
197C2—2262no—
198C2C2×D4×D11311yescharacteristic
199C2C2×D4×D11311yescharacteristic
200C2#1681366811yescharacteristic
subgroupG/H
center, FrattiniC22C23×D113
commutator, socleC22×C113C24
Fitting#121×C113C2
hypercenter#121D113
hypocenterC113D42
LCSG ▷ C22×C113 ▷ C113
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C22×D452C2semidirect
2. C2×D4×D113C2semidirect
3. C2×D4×D113C2semidirect
4. #16813618C2semidirect
5. #16813597C2semidirect
6. #16813595C2semidirect
7. #16813592C2semidirect
8. #16813590C2semidirect
9. #16813584C2semidirect
10. C4×C22⋊D113C2semidirect
11. #121×C113C2semidirect
12. C23×D113C22semidirect
13. C23×D113C22semidirect
16. C2×D452C22semidirect
17. C2×D452C22semidirect
18. C2×D452C22semidirect
20. C2×C22⋊D113C22semidirect
21. C2×C22⋊D113C22semidirect
22. C2×C22⋊D113C22semidirect
25. C2×C4×D113C22semidirect
26. C2×C4×D113C22semidirect
27. #14268626C22semidirect
28. C22⋊Dic113C22semidirect
29. #14268624C22semidirect
30. #14268624C22semidirect
31. #14268624C22semidirect
32. C4×Dic113C22semidirect
33. C2×D4×C113C22semidirect
34. C2×D4×C113C22semidirect
35. C22×C4×C113C22semidirect
36. C4⋊C4×C113C22semidirect
37. C22⋊C4×C113C22semidirect
58. C22⋊D113D4semidirect
65. D452D4semidirect
73. C22×C113C2×D4semidirect
84. C2×D113C2×D4semidirect
85. C2×D113C2×D4semidirect
88. C4×C113C2×D4semidirect
91. Dic113C2×D4semidirect
100. C113D42semidirect
108. #121D113semidirect
126. C2×D4C2×D113semidirect
127. C2×D4C2×D113semidirect
130. C22×C4C2×D113semidirect
132. C4⋊C4C2×D113semidirect
137. C22⋊C4C2×D113semidirect
185. C22D4×D113semidirect
192. C4D4×D113semidirect