Finite Group Databasebeta

← #22593497

Subgroups — #22593497

Order 9088.

672 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 145 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×#14556105C211yesmaximal, characteristic, complemented
2C22×(C2×D71)⋊C4C211yesmaximal, characteristic
3#17985928C212yesmaximal, complemented
4#17985912C212yesmaximal, complemented
5#17985891C212yesmaximal, complemented
6#17985888C212yesmaximal, complemented
7#17985862C211yesmaximal, characteristic, complemented
8#17985854C211yesmaximal, characteristic
9#17985854C211yesmaximal, characteristic
10C22⋊C4×Dic71C211yesmaximal, characteristic, complemented
11#381×C71C211yesmaximal, characteristic, complemented
12C24×D71C2211yescharacteristic
13C22×Dic142C2211yescharacteristic, complemented
14C23×Dic71C2211yescharacteristic
15#14556105—24no—
16C2×C4⋊Dic71C2211yescharacteristic, complemented
17C2×Dic71⋊C4C2211yescharacteristic, complemented
18C2×Dic71⋊C4C2211yescharacteristic, complemented
19C2×C22⋊Dic71C2211yescharacteristic
20C2×C22⋊Dic71C2211yescharacteristic
21C2×(C2×D71)⋊C4—24no—
22C2×(C2×D71)⋊C4C2212yes—
23C2×(C2×D71)⋊C4C2212yes—
24C2×(C2×D71)⋊C4C2212yes—
25C2×(C2×D71)⋊C4C2212yes—
26C2×(C2×D71)⋊C4C2212yes—
27C2×(C2×D71)⋊C4C2212yes—
28C2×C4×Dic71C2212yescomplemented
29C71⋊#95C2212yescomplemented
30C71⋊#95C2212yescomplemented
31C71⋊#95C2212yescomplemented
32C23×C4×C71C2211yescharacteristic
33C2×C22⋊C4×C71C2211yescharacteristic
34C2×C22⋊C4×C71C2211yescharacteristic
35#95×C71C2212yescomplemented
36#95×C71C2212yescomplemented
37C23×D71—22no—
38C23×D71—24no—
39C23×D71—22no—
40C23×D71—22no—
41C23×D71—42no—
42C23×D71C2312yes—
43C2×Dic142—22no—
44C2×Dic142—42no—
45C22×Dic71—42no—
46C22×Dic71D414yes—
47C22×Dic71C2311yescharacteristic
48C22×Dic71C2312yes—
49C22×Dic71C2311yescharacteristic
50C22×Dic71C2311yescharacteristic
51C22×Dic71C2311yescharacteristic
52C4⋊Dic71—22no—
53Dic71⋊C4—22no—
54Dic71⋊C4—22no—
55C22⋊Dic71—22no—
56C22⋊Dic71—22no—
57(C2×D71)⋊C4—24no—
58(C2×D71)⋊C4—24no—
59(C2×D71)⋊C4—24no—
60(C2×D71)⋊C4—24no—
61(C2×D71)⋊C4—42no—
62(C2×D71)⋊C4—42no—
63C4×Dic71—42no—
64C24×C71C2311yescharacteristic
65C22×C4×C71—22no—
66C22×C4×C71—22no—
67C22×C4×C71—22no—
68C22×C4×C71C2311yescharacteristic
69C22×C4×C71C2312yes—
70C22×C4×C71C2311yescharacteristic
71C22×C4×C71C2312yes—
72C22⋊C4×C71—22no—
73C22⋊C4×C71—22no—
74C23×C71—22no—
75C23×C71—22no—
76C23×C71C2×D412yes—
77C23×C71#4112yes—
78C23×C71#4112yes—
79C23×C71C2411yescharacteristic
80C22×D71—28no—
81C22×D71—22no—
82C22×D71—22no—
83C22×D71—22no—
84C22×D71—22no—
85C22×D71—22no—
86C22×D71—22no—
87C22×D71—48no—
88C22×D71—48no—
89C22×D71—44no—
90C22×D71—44no—
91C22×D71#4114yes—
92C2×C4×C71—21no—
93C2×C4×C71—21no—
94C2×C4×C71—22no—
95C2×C4×C71—22no—
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×C71—44no—
103C2×Dic71—22no—
104C2×Dic71—22no—
105C2×Dic71—21no—
106C2×Dic71—21no—
107C2×Dic71—22no—
108C2×Dic71—22no—
109C2×Dic71—22no—
110C2×Dic71—22no—
111C2×Dic71—22no—
112C2×Dic71—44no—
113C2×Dic71C2×D412yes—
114C2×Dic71C2×D412yes—
115Dic142—44no—
116C22×C71—22no—
117C22×C71—22no—
118C22×C71—24no—
119C22×C71—24no—
120C22×C71#12414yes—
121C22×C71C22×D411yescharacteristic
122C22×C71C2×#4112yes—
123C22×C71C2×#4111yescharacteristic
124C22×C71C2×#4111yescharacteristic
125C22×C71H(5,𝔽2)12yes—
126C2×D71—24no—
127C2×D71—24no—
128C2×D71—24no—
129C2×D71—416no—
130C2×D71—42no—
131C2×D71—42no—
132C2×D71—42no—
133C2×D71—42no—
134C4×C71—41no—
135C4×C71—42no—
136C4×C71—41no—
137C4×C71—42no—
138Dic71—22no—
139Dic71—22no—
140Dic71—41no—
141Dic71—41no—
142Dic71—42no—
143C2×C71—24no—
144C2×C71#52511yescharacteristic
145C2×C71C2×#12411yescharacteristic
146C2×C71#53411yescharacteristic
147C2×C71#55112yes—
148C2×C71#55112yes—
149D71—44no—
150#2561—711nomaximal, Sylow
151C71#256111yescharacteristic, complemented, Sylow
152C2×C22⋊Q8—711no—
153C22×C22⋊C4—711no—
154#397—712no—
155#393—712no—
156#388—712no—
157#388—712no—
158#381—711no—
159#381D7111yescharacteristic, complemented
160#379—711no—
161#379—711no—
162C4×C22⋊C4—711no—
163C25—711no—
164C22×Q8—711no—
165C23×C4—711no—
166C23×C4C2×D7111yescharacteristic
167C22⋊Q8—1424no—
168C2×C4⋊C4—711no—
169C2×C4⋊C4—711no—
170C2×C4⋊C4—711no—
171C2×C22⋊C4—712no—
172C2×C22⋊C4—712no—
173C2×C22⋊C4—711no—
174C2×C22⋊C4—711no—
175C2×C22⋊C4—712no—
176C2×C22⋊C4—712no—
177C2×C22⋊C4—712no—
178C2×C22⋊C4—712no—
179C2×C22⋊C4—1424no—
180C2×C22⋊C4C2×D7111yescharacteristic
181C2×C22⋊C4C2×D7111yescharacteristic
182C2×C42—712no—
183#95—712no—
184#95—712no—
185#95—712no—
186#95C2×D7112yescomplemented
187#95C2×D7112yescomplemented
188C24—2842no—
189C24—712no—
190C24—1424no—
191C24—1422no—
192C24—1422no—
193C24—1422no—
194C24C22×D7111yescharacteristic
195C2×Q8—2842no—
196C2×Q8—1422no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—2842no—
201C22×C4—714no—
202C22×C4—711no—
203C22×C4—711no—
204C22×C4—712no—
205C22×C4—711no—
206C22×C4—711no—
207C22×C4C22×D7112yes—
208C22×C4C22×D7112yes—
209C22×C4C22×D7111yescharacteristic
210C22×C4C22×D7111yescharacteristic
211C4⋊C4—1422no—
212C4⋊C4—1422no—
213C4⋊C4—1422no—
214C22⋊C4—22no—
215C22⋊C4—22no—
216C22⋊C4—2842no—
217C22⋊C4—2842no—
218C22⋊C4—1424no—
219C22⋊C4—1424no—
220C22⋊C4—1424no—
221C22⋊C4—1424no—
222C22⋊C4—1422no—
223C22⋊C4—1422no—
224C42—2842no—
225C23—22no—
226C23—22no—
227C23—2844no—
228C23—2848no—
229C23—2848no—
230C23—2844no—
231C23—714no—
232C23—1428no—
233C23—1422no—
234C23—1422no—
235C23—1422no—
236C23—1422no—
237C23—1422no—
238C23—1422no—
239C23D4×D7112yes—
240C23#1176586412yes—
241C23#1176586512yes—
242C23C23×D7111yescharacteristic
243Q8—2844no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—44no—
255C2×C4—2844no—
256C2×C4—712no—
257C2×C4—712no—
258C2×C4—1421no—
259C2×C4—1421no—
260C2×C4—1422no—
261C2×C4—1422no—
262C2×C4—1422no—
263C2×C4—1422no—
264C2×C4—1422no—
265C2×C4—1422no—
266C2×C4—1422no—
267C22—22no—
268C22—24no—
269C22—24no—
270C22—22no—
271C22—28416no—
272C22—2842no—
273C22—2842no—
274C22—2842no—
275C22—2842no—
276C22—1424no—
277C22—1424no—
278C22—1424no—
279C22#1455611414yes—
280C22C2×D4×D7111yescharacteristic
281C22#41×D7112yes—
282C22C2×#1176586411yescharacteristic
283C22C2×#1176586511yescharacteristic
284C22#1455617412yes—
285C4—41no—
286C4—42no—
287C4—41no—
288C4—42no—
289C4—2841no—
290C4—2842no—
291C4—2841no—
292C4—1422no—
293C4—1422no—
294C2—24no—
295C2—2844no—
296C2#1798650711yescharacteristic
297C2C2×#1455611411yescharacteristic
298C2#1798656011yescharacteristic
299C2#1798667212yes—
300C2#1798667312yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#381×C71C2
hypercenter#381D71
hypocenterC71#2561
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
1. C2×#14556105C2semidirect
3. #17985928C2semidirect
4. #17985912C2semidirect
5. #17985891C2semidirect
6. #17985888C2semidirect
7. #17985862C2semidirect
10. C22⋊C4×Dic71C2semidirect
11. #381×C71C2semidirect
13. C22×Dic142C22semidirect
16. C2×C4⋊Dic71C22semidirect
17. C2×Dic71⋊C4C22semidirect
18. C2×Dic71⋊C4C22semidirect
28. C2×C4×Dic71C22semidirect
29. C71⋊#95C22semidirect
30. C71⋊#95C22semidirect
31. C71⋊#95C22semidirect
35. #95×C71C22semidirect
36. #95×C71C22semidirect
151. C71#2561semidirect
159. #381D71semidirect
186. #95C2×D71semidirect
187. #95C2×D71semidirect