Finite Group Databasebeta

← #20299242

Subgroups — #20299242

Order 7232.

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

normal subgroups only

This group has 334 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
1C2×#14268646C211yesmaximal, characteristic, complemented
2#16813609C211yesmaximal, characteristic, complemented
3#16813603C212yesmaximal, complemented
4#16813601C211yesmaximal, characteristic, complemented
5#16813594C212yesmaximal, complemented
6#16813593C211yesmaximal, characteristic, complemented
7#16813591C212yesmaximal, complemented
8D4×Dic113C211yesmaximal, characteristic, complemented
9C4×C22⋊D113C211yesmaximal, characteristic, complemented
10C2×C22⋊Dic113C212yesmaximal, complemented
11C22×D4×C113C211yesmaximal, characteristic, complemented
12#14268646—24no—
13C2×Dic226C2211yescharacteristic, complemented
14C2×C22⋊D113C2211yescharacteristic
15C2×C22⋊D113C2212yescomplemented
16C2×C22⋊D113C2212yescomplemented
17C22×Dic113C2212yescomplemented
18C22×Dic113C2212yescomplemented
19C2×C4×D113C2211yescharacteristic, complemented
20C4⋊Dic113C2211yescharacteristic, complemented
21#14268626C2211yescharacteristic, complemented
22#14268626C2212yescomplemented
23C22⋊Dic113C2211yescharacteristic
24C22⋊Dic113C2212yes—
25C22⋊Dic113C2212yescomplemented
26C22⋊Dic113C2212yescomplemented
27C22⋊Dic113C2212yescomplemented
28C22⋊Dic113C2212yescomplemented
29#14268624C2211yescharacteristic, complemented
30C4×Dic113C2211yescharacteristic, complemented
31C24×C113C2212yes—
32C2×D4×C113—24no—
33C2×D4×C113C2211yescharacteristic
34C2×D4×C113C2212yes—
35C2×D4×C113C2211yescharacteristic
36C22×C4×C113C2211yescharacteristic
37C23×C113—24no—
38C23×C113—24no—
39C23×C113—42no—
40C23×C113C2312yes—
41C23×C113C2312yes—
42C23×C113C2311yescharacteristic
43C22×D113C2311yescharacteristic
44D4×C113—24no—
45D4×C113—22no—
46D4×C113D414yescomplemented
47C2×C4×C113—22no—
48C2×C4×C113C2311yescharacteristic
49C2×C4×C113C2311yescharacteristic
50C22⋊D113—24no—
51C22⋊D113—22no—
52C22⋊D113—42no—
53C2×Dic113—24no—
54C2×Dic113—42no—
55C2×Dic113C2311yescharacteristic
56C2×Dic113C2311yescharacteristic
57C2×Dic113C2312yes—
58C2×Dic113C2311yescharacteristic
59C2×Dic113C2312yes—
60C4×D113—22no—
61Dic226—22no—
62C22×C113—21no—
63C22×C113—22no—
64C22×C113—22no—
65C22×C113—21no—
66C22×C113—22no—
67C22×C113—22no—
68C22×C113—22no—
69C22×C113—48no—
70C22×C113C2×D414yes—
71C22×C113#4112yescomplemented
72C22×C113C2411yescharacteristic
73C2×D113—21no—
74C2×D113—21no—
75C2×D113—21no—
76C4×C113—21no—
77C4×C113C2×D412yes—
78Dic113—22no—
79Dic113—21no—
80Dic113—21no—
81Dic113—21no—
82Dic113—22no—
83C2×C113—22no—
84C2×C113—24no—
85C2×C113—42no—
86C2×C113C22×D411yescharacteristic
87C2×C113C2×#4111yescharacteristic
88C2×C113H(5,𝔽2)11yescharacteristic
89D113—41no—
90C113#54611yescharacteristic, complemented, Sylow
91#546—1131nomaximal, Sylow
92C2×#41—1131no—
93C22×D4D11311yescharacteristic, complemented
94#124—1131no—
95#123—1132no—
96C22⋊Q8—1131no—
97#121—1131no—
98#121—1132no—
99#120—1132no—
100C4×D4—1131no—
101C4×D4—1131no—
102C2×C22⋊C4—1132no—
103C24C2×D11312yes—
104#41—2264no—
105C2×Q8—1131no—
106C2×D4—24no—
107C2×D4—1132no—
108C2×D4—1131no—
109C2×D4—1132no—
110C2×D4C2×D11312yes—
111C2×D4C2×D11311yescharacteristic
112C2×D4C2×D11311yescharacteristic
113C22×C4—1132no—
114C22×C4—1132no—
115C22×C4—1131no—
116C22×C4C2×D11311yescharacteristic
117C4⋊C4—1131no—
118C4⋊C4—1131no—
119C4⋊C4—1132no—
120C22⋊C4—1131no—
121C22⋊C4—1132no—
122C22⋊C4—1132no—
123C22⋊C4—1132no—
124C22⋊C4—1132no—
125C22⋊C4—1132no—
126C22⋊C4—1131no—
127C42—1131no—
128C23—24no—
129C23—24no—
130C23—42no—
131C23—1131no—
132C23C22×D11312yes—
133C23C22×D11312yes—
134C23C22×D11311yescharacteristic
135Q8—2262no—
136D4—24no—
137D4—22no—
138D4—4522no—
139D4—2264no—
140D4—2262no—
141D4C22⋊D11314yescomplemented
142C2×C4—22no—
143C2×C4—1131no—
144C2×C4—1131no—
145C2×C4—1132no—
146C2×C4—1131no—
147C2×C4—1132no—
148C2×C4—4522no—
149C2×C4—2264no—
150C2×C4—2262no—
151C2×C4C22×D11311yescharacteristic
152C2×C4C22×D11311yescharacteristic
153C22—21no—
154C22—22no—
155C22—22no—
156C22—21no—
157C22—22no—
158C22—22no—
159C22—22no—
160C22—48no—
161C22—2261no—
162C22—2261no—
163C22—2261no—
164C22C2×C22⋊D11314yes—
165C22#1426864612yescomplemented
166C22C23×D11311yescharacteristic
167C4—21no—
168C4—2261no—
169C4—2262no—
170C4—2261no—
171C4—2261no—
172C4—2262no—
173C4C2×C22⋊D11312yes—
174C2—22no—
175C2—24no—
176C2—42no—
177C2—4521no—
178C2C22×C22⋊D11311yescharacteristic
179C2C2×#1426864611yescharacteristic
180C2#1681366711yescharacteristic
subgroupG/H
center, FrattiniC22C23×D113
commutator, socleC22×C113C24
FittingC22×D4×C113C2
hypercenterC22×D4D113
hypocenterC113#546
LCSG ▷ C22×C113 ▷ C113
UCSC1 ◁ C22 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C2×#14268646C2semidirect
2. #16813609C2semidirect
3. #16813603C2semidirect
4. #16813601C2semidirect
5. #16813594C2semidirect
6. #16813593C2semidirect
7. #16813591C2semidirect
8. D4×Dic113C2semidirect
9. C4×C22⋊D113C2semidirect
10. C2×C22⋊Dic113C2semidirect
11. C22×D4×C113C2semidirect
13. C2×Dic226C22semidirect
15. C2×C22⋊D113C22semidirect
16. C2×C22⋊D113C22semidirect
17. C22×Dic113C22semidirect
18. C22×Dic113C22semidirect
19. C2×C4×D113C22semidirect
20. C4⋊Dic113C22semidirect
21. #14268626C22semidirect
22. #14268626C22semidirect
25. C22⋊Dic113C22semidirect
26. C22⋊Dic113C22semidirect
27. C22⋊Dic113C22semidirect
28. C22⋊Dic113C22semidirect
29. #14268624C22semidirect
30. C4×Dic113C22semidirect
46. D4×C113D4semidirect
71. C22×C113#41semidirect
90. C113#546semidirect
93. C22×D4D113semidirect
141. D4C22⋊D113semidirect
165. C22#14268646semidirect