Finite Group Databasebeta

← #20299235

Subgroups — #20299235

Order 7232.

334 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 107 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
2C22×C22⋊D113C211yesmaximal, characteristic, complemented
3#16813608C211yesmaximal, characteristic, complemented
4#16813606C211yesmaximal, characteristic, complemented
5#16813604C211yesmaximal, characteristic, complemented
6#16813599C211yesmaximal, characteristic, complemented
7#16813594C211yesmaximal, characteristic, complemented
8#16813594C211yesmaximal, characteristic, complemented
9#16813593C211yesmaximal, characteristic, complemented
10#16813590C211yesmaximal, characteristic, complemented
11D4×Dic113C211yesmaximal, characteristic, complemented
12#16813582C211yesmaximal, characteristic, complemented
13C2×C22⋊Dic113C211yesmaximal, characteristic, complemented
14C22⋊C4×D113C211yesmaximal, characteristic, complemented
15#120×C113C211yesmaximal, characteristic, complemented
16C23×D113C2211yescharacteristic
17#14268646—24no—
18C2×Dic226C2211yescharacteristic, complemented
19C2×C22⋊D113—24no—
20C2×C22⋊D113C2211yescharacteristic
21C2×C22⋊D113C2211yescharacteristic
22C2×C22⋊D113C2211yescharacteristic
23C2×C22⋊D113C2211yescharacteristic
24C2×C22⋊D113C2211yescharacteristic, complemented
25C2×C22⋊D113C2211yescharacteristic, complemented
26C22×Dic113C2211yescharacteristic
27C22×Dic113C2211yescharacteristic, complemented
28C22×Dic113C2211yescharacteristic, complemented
29C22×Dic113C2211yescharacteristic, complemented
30C2×C4×D113C2211yescharacteristic, complemented
31C2×C4×D113C2211yescharacteristic, complemented
32C4⋊Dic113C2211yescharacteristic, complemented
33C4⋊Dic113C2211yescharacteristic, complemented
34#14268626C2211yescharacteristic, complemented
35#14268626C2211yescharacteristic, complemented
36C22⋊Dic113C2211yescharacteristic
37C22⋊Dic113C2211yescharacteristic
38C22⋊Dic113C2211yescharacteristic, complemented
39C22⋊Dic113C2211yescharacteristic, complemented
40C22⋊Dic113C2211yescharacteristic, complemented
41#14268624C2211yescharacteristic, complemented
42#14268624C2211yescharacteristic, complemented
43#14268624C2211yescharacteristic, complemented
44#14268624C2211yescharacteristic, complemented
45C4×Dic113C2211yescharacteristic, complemented
46C24×C113C2211yescharacteristic
47C2×D4×C113C2211yescharacteristic
48C2×D4×C113C2211yescharacteristic, complemented
49C2×D4×C113C2211yescharacteristic, complemented
50C22⋊C4×C113C2211yescharacteristic
51C22⋊C4×C113C2211yescharacteristic, complemented
52C22⋊C4×C113C2211yescharacteristic, complemented
53C23×C113—22no—
54C23×C113—22no—
55C23×C113—41no—
56C23×C113C2311yescharacteristic
57C23×C113C2311yescharacteristic
58C23×C113C2311yescharacteristic
59C23×C113C2311yescharacteristic
60C22×D113—22no—
61C22×D113—22no—
62C22×D113—41no—
63C22×D113C2311yescharacteristic
64C22×D113C2311yescharacteristic
65D4×C113—22no—
66D4×C113—22no—
67D4×C113—41no—
68C2×C4×C113C2311yescharacteristic
69C2×C4×C113C2311yescharacteristic
70C2×C4×C113C2311yescharacteristic
71C22⋊D113—22no—
72C22⋊D113—22no—
73C22⋊D113—22no—
74C22⋊D113—22no—
75C22⋊D113—41no—
76C22⋊D113D414yescomplemented
77C2×Dic113—22no—
78C2×Dic113—22no—
79C2×Dic113—22no—
80C2×Dic113—41no—
81C2×Dic113C2311yescharacteristic
82C2×Dic113C2311yescharacteristic
83C2×Dic113C2311yescharacteristic
84C2×Dic113C2311yescharacteristic
85C2×Dic113C2311yescharacteristic
86C2×Dic113C2311yescharacteristic
87C4×D113—22no—
88C4×D113—41no—
89Dic226—22no—
90C22×C113—21no—
91C22×C113—21no—
92C22×C113—21no—
93C22×C113—21no—
94C22×C113—21no—
95C22×C113—21no—
96C22×C113—21no—
97C22×C113—21no—
98C22×C113—21no—
99C22×C113—21no—
100C22×C113—44no—
101C22×C113C2×D412yes—
102C22×C113#4112yescomplemented
103C22×C113C2411yescharacteristic
104C2×D113—21no—
105C2×D113—21no—
106C2×D113—21no—
107C2×D113—21no—
108C2×D113—21no—
109C2×D113—44no—
110C2×D113C2×D412yes—
111C4×C113—21no—
112C4×C113—21no—
113C4×C113—21no—
114Dic113—21no—
115Dic113—21no—
116Dic113—21no—
117Dic113—21no—
118Dic113—21no—
119Dic113C2×D412yes—
120C2×C113—22no—
121C2×C113—22no—
122C2×C113—41no—
123C2×C113—41no—
124C2×C113C22×D411yescharacteristic
125C2×C113C2×#4111yescharacteristic
126C2×C113H(5,𝔽2)11yescharacteristic
127D113—22no—
128D113—41no—
129C113#54611yescharacteristic, complemented, Sylow
130#546—1131nomaximal, Sylow
131C2×#41—1131no—
132C22×D4—1131no—
133#124—1131no—
134#123—1131no—
135#123—1131no—
136C22⋊Q8—1131no—
137#121—1131no—
138#121—1131no—
139#121—1131no—
140#120—1131no—
141#120D11311yescharacteristic, complemented
142C4×D4—1131no—
143C4×D4—1131no—
144C2×C22⋊C4—1131no—
145C2×C22⋊C4—1131no—
146C24—1131no—
147C24C2×D11311yescharacteristic
148#41—2264no—
149C2×Q8—1131no—
150C2×D4—1131no—
151C2×D4—1131no—
152C2×D4—1131no—
153C2×D4—1131no—
154C2×D4—1131no—
155C2×D4—1131no—
156C2×D4—2264no—
157C2×D4C2×D11311yescharacteristic
158C2×D4C2×D11311yescharacteristic, complemented
159C2×D4C2×D11311yescharacteristic, complemented
160C22×C4—1131no—
161C22×C4—1131no—
162C22×C4—1131no—
163C22×C4—1131no—
164C22×C4—1131no—
165C22×C4—1131no—
166C4⋊C4—1131no—
167C4⋊C4—1131no—
168C4⋊C4—1131no—
169C4⋊C4—1131no—
170C22⋊C4—1131no—
171C22⋊C4—1131no—
172C22⋊C4—1131no—
173C22⋊C4—1131no—
174C22⋊C4—1131no—
175C22⋊C4—1131no—
176C22⋊C4—1131no—
177C22⋊C4—1131no—
178C22⋊C4—1131no—
179C22⋊C4C2×D11311yescharacteristic
180C22⋊C4C2×D11311yescharacteristic, complemented
181C22⋊C4C2×D11311yescharacteristic, complemented
182C42—1131no—
183C23—22no—
184C23—22no—
185C23—41no—
186C23—1131no—
187C23—1131no—
188C23—4521no—
189C23—2262no—
190C23—2262no—
191C23C22×D11311yescharacteristic
192C23C22×D11311yescharacteristic
193C23C22×D11311yescharacteristic
194C23C22×D11311yescharacteristic
195Q8—2262no—
196D4—22no—
197D4—22no—
198D4—41no—
199D4—1134no—
200D4—4521no—
201D4—2262no—
202D4—2262no—
203D4—2262no—
204D4—2262no—
205C2×C4—1131no—
206C2×C4—1131no—
207C2×C4—1131no—
208C2×C4—1131no—
209C2×C4—1131no—
210C2×C4—1131no—
211C2×C4—4521no—
212C2×C4—4521no—
213C2×C4—2262no—
214C2×C4—2262no—
215C2×C4—2262no—
216C2×C4—2262no—
217C2×C4C22×D11311yescharacteristic
218C2×C4C22×D11311yescharacteristic
219C2×C4C22×D11311yescharacteristic
220C22—21no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—21no—
230C22—44no—
231C22—1132no—
232C22—4524no—
233C22—2261no—
234C22—2261no—
235C22—2261no—
236C22—2261no—
237C22—2261no—
238C22D4×D11312yes—
239C22#1426864612yescomplemented
240C22C23×D11311yescharacteristic
241C4—21no—
242C4—21no—
243C4—21no—
244C4—1132no—
245C4—2261no—
246C4—2261no—
247C4—2261no—
248C4—2261no—
249C4—2261no—
250C2—22no—
251C2—22no—
252C2—41no—
253C2—41no—
254C2—4521no—
255C2—2262no—
256C2C2×D4×D11311yescharacteristic
257C2C2×#1426864611yescharacteristic
258C2#1681366711yescharacteristic
subgroupG/H
center, FrattiniC22C23×D113
commutator, socleC22×C113C24
Fitting#120×C113C2
hypercenter#120D113
hypocenterC113#546
LCSG ▷ C22×C113 ▷ C113
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C2×#14268646C2semidirect
2. C22×C22⋊D113C2semidirect
3. #16813608C2semidirect
4. #16813606C2semidirect
5. #16813604C2semidirect
6. #16813599C2semidirect
7. #16813594C2semidirect
8. #16813594C2semidirect
9. #16813593C2semidirect
10. #16813590C2semidirect
11. D4×Dic113C2semidirect
12. #16813582C2semidirect
13. C2×C22⋊Dic113C2semidirect
14. C22⋊C4×D113C2semidirect
15. #120×C113C2semidirect
18. C2×Dic226C22semidirect
24. C2×C22⋊D113C22semidirect
25. C2×C22⋊D113C22semidirect
27. C22×Dic113C22semidirect
28. C22×Dic113C22semidirect
29. C22×Dic113C22semidirect
30. C2×C4×D113C22semidirect
31. C2×C4×D113C22semidirect
32. C4⋊Dic113C22semidirect
33. C4⋊Dic113C22semidirect
34. #14268626C22semidirect
35. #14268626C22semidirect
38. C22⋊Dic113C22semidirect
39. C22⋊Dic113C22semidirect
40. C22⋊Dic113C22semidirect
41. #14268624C22semidirect
42. #14268624C22semidirect
43. #14268624C22semidirect
44. #14268624C22semidirect
45. C4×Dic113C22semidirect
48. C2×D4×C113C22semidirect
49. C2×D4×C113C22semidirect
51. C22⋊C4×C113C22semidirect
52. C22⋊C4×C113C22semidirect
76. C22⋊D113D4semidirect
102. C22×C113#41semidirect
129. C113#546semidirect
141. #120D113semidirect
158. C2×D4C2×D113semidirect
159. C2×D4C2×D113semidirect
180. C22⋊C4C2×D113semidirect
181. C22⋊C4C2×D113semidirect
239. C22#14268646semidirect