Finite Group Databasebeta

← #20299158

Subgroups — #20299158

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×C22⋊D113C211yesmaximal, characteristic, complemented
2C2×D4×D113C211yesmaximal, characteristic, complemented
3#16813606C211yesmaximal, characteristic, complemented
4#16813604C212yesmaximal, complemented
5#16813603C211yesmaximal, characteristic, complemented
6#16813595C212yesmaximal, complemented
7#16813593C212yesmaximal, complemented
8#16813591C211yesmaximal, characteristic, complemented
9#16813590C212yesmaximal, complemented
10C22⋊C4×D113C211yesmaximal, characteristic, complemented
11#120×C113C211yesmaximal, characteristic, complemented
12C23×D113C2211yescharacteristic
13C23×D113C2211yescharacteristic, complemented
14C2×D452C2211yescharacteristic, complemented
15C2×C22⋊D113—24no—
16C2×C22⋊D113C2212yes—
17C2×C22⋊D113C2211yescharacteristic
18C2×C22⋊D113C2211yescharacteristic, complemented
19C2×C22⋊D113C2212yescomplemented
20C2×C22⋊D113C2212yescomplemented
21D4×D113—24no—
22C22×Dic113C2211yescharacteristic, complemented
23C2×C4×D113C2211yescharacteristic, complemented
24C2×C4×D113C2212yescomplemented
25C4⋊Dic113C2212yescomplemented
26#14268626C2212yescomplemented
27C22⋊Dic113C2211yescharacteristic, complemented
28C22⋊Dic113C2212yescomplemented
29C22⋊Dic113C2212yescomplemented
30#14268624C2212yescomplemented
31#14268624C2212yescomplemented
32C24×C113C2211yescharacteristic, complemented
33C2×D4×C113C2211yescharacteristic
34C2×D4×C113C2212yescomplemented
35C22⋊C4×C113C2211yescharacteristic, complemented
36C22⋊C4×C113C2212yescomplemented
37C23×C113—22no—
38C23×C113—42no—
39C23×C113C2312yes—
40C23×C113C2311yescharacteristic
41C23×C113C2311yescharacteristic
42C22×D113—22no—
43C22×D113—42no—
44C22×D113—42no—
45C22×D113D414yescomplemented
46C22×D113C2311yescharacteristic
47C22×D113C2311yescharacteristic
48C22×D113C2311yescharacteristic
49D4×C113—22no—
50D4×C113—42no—
51C2×C4×C113C2311yescharacteristic
52C2×C4×C113C2312yes—
53C22⋊D113—22no—
54C22⋊D113—22no—
55C22⋊D113—24no—
56C22⋊D113—24no—
57C22⋊D113—42no—
58C2×Dic113—22no—
59C2×Dic113C2312yes—
60C2×Dic113C2312yes—
61C2×Dic113C2311yescharacteristic
62D452—22no—
63C4×D113—22no—
64C4×D113—42no—
65C22×C113—22no—
66C22×C113—22no—
67C22×C113—21no—
68C22×C113—22no—
69C22×C113—22no—
70C22×C113—22no—
71C22×C113—44no—
72C22×C113C2×D412yes—
73C22×C113C2411yescharacteristic
74C2×D113—22no—
75C2×D113—22no—
76C2×D113—21no—
77C2×D113—22no—
78C2×D113—44no—
79C2×D113—44no—
80C2×D113C2×D412yes—
81C2×D113C2×D412yes—
82C4×C113—21no—
83C4×C113—22no—
84Dic113—21no—
85Dic113—22no—
86Dic113—22no—
87C2×C113—22no—
88C2×C113—41no—
89C2×C113—42no—
90C2×C113C22×D411yescharacteristic
91C2×C113H(5,𝔽2)12yes—
92D113—22no—
93D113—22no—
94D113—41no—
95C113#53411yescharacteristic, complemented, Sylow
96#534—1131nomaximal, Sylow
97C22×D4—1131no—
98C22×D4—1131no—
99#123—1131no—
100#123—1132no—
101#123—1131no—
102#121—1132no—
103#121—1132no—
104#120—1131no—
105#120—1132no—
106#120D11311yescharacteristic, complemented
107C2×C22⋊C4—1131no—
108C24—1131no—
109C24—1131no—
110C24C2×D11311yescharacteristic, complemented
111C2×D4—1132no—
112C2×D4—1132no—
113C2×D4—1131no—
114C2×D4—1131no—
115C2×D4—1132no—
116C2×D4—1131no—
117C2×D4—2264no—
118C2×D4—2264no—
119C2×D4C2×D11311yescharacteristic
120C2×D4C2×D11312yescomplemented
121C22×C4—1131no—
122C22×C4—1132no—
123C22×C4—1131no—
124C4⋊C4—1132no—
125C4⋊C4—1132no—
126C22⋊C4—1131no—
127C22⋊C4—1132no—
128C22⋊C4—1132no—
129C22⋊C4—1132no—
130C22⋊C4—1132no—
131C22⋊C4C2×D11311yescharacteristic, complemented
132C22⋊C4C2×D11312yescomplemented
133C23—22no—
134C23—42no—
135C23—1134no—
136C23—1131no—
137C23—1131no—
138C23—1131no—
139C23—4522no—
140C23—4522no—
141C23—2262no—
142C23C22×D11311yescharacteristic
143C23C22×D11312yes—
144C23C22×D11311yescharacteristic
145D4—22no—
146D4—42no—
147D4—4522no—
148D4—2262no—
149D4—2262no—
150D4—2264no—
151D4—2264no—
152D4—2262no—
153C2×C4—1132no—
154C2×C4—1131no—
155C2×C4—1132no—
156C2×C4—4522no—
157C2×C4—2262no—
158C2×C4—2262no—
159C2×C4C22×D11312yes—
160C2×C4C22×D11311yescharacteristic
161C22—22no—
162C22—21no—
163C22—22no—
164C22—22no—
165C22—22no—
166C22—22no—
167C22—44no—
168C22—1132no—
169C22—1132no—
170C22—4524no—
171C22—4524no—
172C22—2261no—
173C22—2262no—
174C22—2262no—
175C22—2262no—
176C22D4×D11312yes—
177C22C23×D11311yescharacteristic
178C4—21no—
179C4—22no—
180C4—2261no—
181C4—2262no—
182C4—2262no—
183C2—22no—
184C2—42no—
185C2—41no—
186C2—4521no—
187C2—2262no—
188C2—2262no—
189C2C2×D4×D11311yescharacteristic
190C2#1681366712yes—
subgroupG/H
center, FrattiniC22C23×D113
commutator, socleC22×C113C24
Fitting#120×C113C2
hypercenter#120D113
hypocenterC113#534
LCSG ▷ C22×C113 ▷ C113
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊D113C2semidirect
2. C2×D4×D113C2semidirect
3. #16813606C2semidirect
4. #16813604C2semidirect
5. #16813603C2semidirect
6. #16813595C2semidirect
7. #16813593C2semidirect
8. #16813591C2semidirect
9. #16813590C2semidirect
10. C22⋊C4×D113C2semidirect
11. #120×C113C2semidirect
13. C23×D113C22semidirect
14. C2×D452C22semidirect
18. C2×C22⋊D113C22semidirect
19. C2×C22⋊D113C22semidirect
20. C2×C22⋊D113C22semidirect
22. C22×Dic113C22semidirect
23. C2×C4×D113C22semidirect
24. C2×C4×D113C22semidirect
25. C4⋊Dic113C22semidirect
26. #14268626C22semidirect
27. C22⋊Dic113C22semidirect
28. C22⋊Dic113C22semidirect
29. C22⋊Dic113C22semidirect
30. #14268624C22semidirect
31. #14268624C22semidirect
32. C24×C113C22semidirect
34. C2×D4×C113C22semidirect
35. C22⋊C4×C113C22semidirect
36. C22⋊C4×C113C22semidirect
45. C22×D113D4semidirect
95. C113#534semidirect
106. #120D113semidirect
110. C24C2×D113semidirect
120. C2×D4C2×D113semidirect
131. C22⋊C4C2×D113semidirect
132. C22⋊C4C2×D113semidirect