Finite Group Databasebeta

← #22594902

Subgroups — #22594902

Order 9088.

680 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 680 conjugacy classes of subgroups — too many to draw. Only its 137 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×#14556109C211yesmaximal, characteristic, complemented
2C2×#14556096C211yesmaximal, characteristic
3#17985924C212yesmaximal, complemented
4#17985921C212yesmaximal, complemented
5#17985915C212yesmaximal, complemented
6#17985890C212yesmaximal, complemented
7#17985876C212yesmaximal, complemented
8#17985854C212yesmaximal, complemented
9#390×C71C211yesmaximal, characteristic, complemented
10C24×D71C2211yescharacteristic
11C22×C22⋊D71C2212yes—
12C23×Dic71C2211yescharacteristic, complemented
13#14556109—24no—
14#14556096—24no—
15C2×C4⋊Dic71C2212yescomplemented
16C2×Dic71⋊C4C2212yescomplemented
17C2×C22⋊Dic71C2212yes—
18C2×C22⋊Dic71C2211yescharacteristic
19C2×C22⋊Dic71C2211yescharacteristic, complemented
20C2×(C2×D71)⋊C4C2212yes—
21C2×(C2×D71)⋊C4C2212yes—
22C2×(C2×D71)⋊C4C2212yes—
23C2×(C2×D71)⋊C4C2212yes—
24C2×(C2×D71)⋊C4C2212yes—
25C71⋊#95C2212yescomplemented
26C71⋊#95C2212yescomplemented
27C71⋊#95C2212yescomplemented
28C22×D4×C71C2211yescharacteristic
29C2×C4⋊C4×C71C2211yescharacteristic, complemented
30C2×C22⋊C4×C71C2212yes—
31C2×C22⋊C4×C71C2212yescomplemented
32C2×C42×C71C2211yescharacteristic, complemented
33C23×D71—22no—
34C23×D71—24no—
35C23×D71—41no—
36C23×D71—41no—
37C23×D71—42no—
38C23×D71C2312yes—
39C2×C22⋊D71—24no—
40C2×C22⋊D71—24no—
41C2×C22⋊D71—42no—
42C22×Dic71—22no—
43C22×Dic71—41no—
44C22×Dic71—41no—
45C22×Dic71C2312yes—
46C22×Dic71C2312yes—
47C22×Dic71C2312yes—
48C4⋊Dic71—42no—
49Dic71⋊C4—24no—
50C22⋊Dic71—22no—
51C22⋊Dic71—24no—
52C22⋊Dic71—22no—
53(C2×D71)⋊C4—24no—
54(C2×D71)⋊C4—24no—
55(C2×D71)⋊C4—42no—
56(C2×D71)⋊C4—42no—
57(C2×D71)⋊C4—42no—
58C24×C71C2311yescharacteristic
59C24×C71C2311yescharacteristic
60C2×D4×C71—41no—
61C2×D4×C71—41no—
62C2×D4×C71D414yes—
63C22×C4×C71C2312yes—
64C22×C4×C71C2312yes—
65C22×C4×C71C2311yescharacteristic
66C4⋊C4×C71—41no—
67C22⋊C4×C71—42no—
68C22⋊C4×C71—42no—
69C42×C71—41no—
70C23×C71—22no—
71C23×C71—22no—
72C23×C71—22no—
73C23×C71—22no—
74C23×C71—21no—
75C23×C71—21no—
76C23×C71—21no—
77C23×C71—21no—
78C23×C71C2×D412yes—
79C23×C71C2×D412yes—
80C23×C71C2411yescharacteristic
81C22×D71—22no—
82C22×D71—22no—
83C22×D71—22no—
84C22×D71—22no—
85C22×D71—22no—
86C22×D71—22no—
87C22×D71—44no—
88C22×D71—48no—
89C22×D71—48no—
90C22×D71—44no—
91C22×D71—44no—
92C22×D71#4114yes—
93D4×C71—44no—
94C2×C4×C71—21no—
95C2×C4×C71—21no—
96C2×C4×C71—22no—
97C2×C4×C71—22no—
98C2×C4×C71—22no—
99C2×C4×C71—22no—
100C2×C4×C71—22no—
101C2×C4×C71—22no—
102C2×C4×C71C2×D412yes—
103C22⋊D71—48no—
104C2×Dic71—22no—
105C2×Dic71—22no—
106C2×Dic71—22no—
107C2×Dic71—22no—
108C2×Dic71—22no—
109C2×Dic71—22no—
110C2×Dic71—22no—
111C2×Dic71—22no—
112C2×Dic71—22no—
113C2×Dic71—44no—
114C22×C71—22no—
115C22×C71—22no—
116C22×C71—24no—
117C22×C71—24no—
118C22×C71—42no—
119C22×C71—42no—
120C22×C71—41no—
121C22×C71—41no—
122C22×C71—41no—
123C22×C71—41no—
124C22×C71C22×D411yescharacteristic
125C22×C71C2×#4112yes—
126C22×C71H(5,𝔽2)11yescharacteristic
127C22×C71H(5,𝔽2)11yescharacteristic
128C22×C71H(5,𝔽2)12yes—
129C2×D71—24no—
130C2×D71—24no—
131C2×D71—24no—
132C2×D71—416no—
133C2×D71—42no—
134C2×D71—42no—
135C2×D71—42no—
136C2×D71—42no—
137C4×C71—22no—
138C4×C71—42no—
139C4×C71—42no—
140Dic71—42no—
141Dic71—42no—
142Dic71—42no—
143C2×C71—42no—
144C2×C71—42no—
145C2×C71#53411yescharacteristic
146C2×C71#53812yes—
147C2×C71#54612yes—
148C2×C71#55111yescharacteristic
149C2×C71#56011yescharacteristic
150D71—44no—
151#2697—711nomaximal, Sylow
152C71#269711yescharacteristic, complemented, Sylow
153C2×#123—711no—
154C2×#120—711no—
155#397—712no—
156#396—712no—
157#394—712no—
158#390D7111yescharacteristic, complemented
159#388—712no—
160#386—712no—
161#379—712no—
162C25—711no—
163C22×D4—712no—
164C22×D4C2×D7111yescharacteristic
165C23×C4—711no—
166#123—1424no—
167#120—1424no—
168C2×C4⋊C4—712no—
169C2×C4⋊C4—712no—
170C2×C4⋊C4C2×D7111yescharacteristic, complemented
171C2×C22⋊C4—712no—
172C2×C22⋊C4—712no—
173C2×C22⋊C4—712no—
174C2×C22⋊C4—711no—
175C2×C22⋊C4—712no—
176C2×C22⋊C4—711no—
177C2×C22⋊C4—712no—
178C2×C22⋊C4—712no—
179C2×C22⋊C4C2×D7112yes—
180C2×C22⋊C4C2×D7112yescomplemented
181C2×C42C2×D7111yescharacteristic, complemented
182#95—712no—
183#95—712no—
184#95—712no—
185C24—2842no—
186C24—2841no—
187C24—2841no—
188C24—712no—
189C24—1424no—
190C24—1422no—
191C24C22×D7111yescharacteristic
192C24C22×D7111yescharacteristic
193C2×D4—41no—
194C2×D4—41no—
195C2×D4—2842no—
196C2×D4—1424no—
197C2×D4—1424no—
198C2×D4D28414yes—
199C22×C4—2841no—
200C22×C4—2841no—
201C22×C4—712no—
202C22×C4—712no—
203C22×C4—712no—
204C22×C4—1422no—
205C22×C4C22×D7112yes—
206C22×C4C22×D7112yes—
207C22×C4C22×D7111yescharacteristic
208C4⋊C4—41no—
209C4⋊C4—2842no—
210C4⋊C4—1424no—
211C22⋊C4—42no—
212C22⋊C4—42no—
213C22⋊C4—2842no—
214C22⋊C4—2842no—
215C22⋊C4—2842no—
216C22⋊C4—1424no—
217C22⋊C4—1422no—
218C22⋊C4—1422no—
219C22⋊C4—1424no—
220C22⋊C4—1424no—
221C42—41no—
222C23—22no—
223C23—22no—
224C23—22no—
225C23—22no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—2844no—
231C23—2848no—
232C23—2844no—
233C23—2848no—
234C23—2844no—
235C23—714no—
236C23—1422no—
237C23—1422no—
238C23—1422no—
239C23—1422no—
240C23—1422no—
241C23—1422no—
242C23C2×D28412yes—
243C23C2×D28412yes—
244C23C23×D7111yescharacteristic
245D4—44no—
246D4—2848no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—2844no—
256C2×C4—1422no—
257C2×C4—1422no—
258C2×C4—1422no—
259C2×C4—1422no—
260C2×C4—1422no—
261C2×C4—1422no—
262C2×C4—1422no—
263C2×C4—1422no—
264C2×C4—1422no—
265C2×C4C2×D28412yes—
266C22—22no—
267C22—22no—
268C22—24no—
269C22—24no—
270C22—42no—
271C22—42no—
272C22—41no—
273C22—41no—
274C22—41no—
275C22—41no—
276C22—28416no—
277C22—2842no—
278C22—2842no—
279C22—2842no—
280C22—2842no—
281C22—1424no—
282C22—1424no—
283C22—1424no—
284C22C22×D28411yescharacteristic
285C22#41×D7112yes—
286C22#1455617311yescharacteristic
287C22#1455617311yescharacteristic
288C22#1455617312yes—
289C4—22no—
290C4—42no—
291C4—42no—
292C4—2842no—
293C4—2842no—
294C4—2842no—
295C2—42no—
296C2—42no—
297C2—2844no—
298C2#1798656111yescharacteristic
299C2#1798658112yes—
300C2#1798664412yes—
301C2#1798667911yescharacteristic
302C2#1798673611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#390×C71C2
hypercenter#390D71
hypocenterC71#2697
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#14556109C2semidirect
3. #17985924C2semidirect
4. #17985921C2semidirect
5. #17985915C2semidirect
6. #17985890C2semidirect
7. #17985876C2semidirect
8. #17985854C2semidirect
9. #390×C71C2semidirect
12. C23×Dic71C22semidirect
15. C2×C4⋊Dic71C22semidirect
16. C2×Dic71⋊C4C22semidirect
19. C2×C22⋊Dic71C22semidirect
25. C71⋊#95C22semidirect
26. C71⋊#95C22semidirect
27. C71⋊#95C22semidirect
29. C2×C4⋊C4×C71C22semidirect
31. C2×C22⋊C4×C71C22semidirect
32. C2×C42×C71C22semidirect
152. C71#2697semidirect
158. #390D71semidirect
170. C2×C4⋊C4C2×D71semidirect
180. C2×C22⋊C4C2×D71semidirect
181. C2×C42C2×D71semidirect