Finite Group Databasebeta

← #22594716

Subgroups — #22594716

Order 9088.

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

normal subgroups only

This group has 688 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×#14556096C211yesmaximal, characteristic
2#17985927C212yesmaximal, complemented
3#17985914C212yesmaximal, complemented
4#17985910C212yesmaximal, complemented
5#17985890C212yesmaximal, complemented
6#17985879C211yesmaximal, characteristic, complemented
7#17985876C212yesmaximal, complemented
8#17985854C212yesmaximal, complemented
9C2×#124×C71C211yesmaximal, characteristic, complemented
10C24×D71C2211yescharacteristic
11C22×C22⋊D71C2212yes—
12C23×Dic71C2211yescharacteristic, complemented
13#14556096—24no—
14C2×Dic71⋊C4C2212yescomplemented
15C2×C22⋊Dic71C2211yescharacteristic
16C2×C22⋊Dic71C2212yes—
17C2×C22⋊Dic71C2211yescharacteristic, complemented
18C2×(C2×D71)⋊C4C2212yes—
19C2×(C2×D71)⋊C4C2212yes—
20C2×(C2×D71)⋊C4C2212yes—
21C2×(C2×D71)⋊C4C2212yes—
22C2×(C2×D71)⋊C4C2212yes—
23C71⋊#95C2212yescomplemented
24C71⋊#95C2212yescomplemented
25C71⋊#95C2212yescomplemented
26C71⋊#95C2212yescomplemented
27C22×Q8×C71C2211yescharacteristic, complemented
28C22×D4×C71C2211yescharacteristic
29#124×C71—24no—
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—41no—
43C22×Dic71—42no—
44C22×Dic71C2312yes—
45C22×Dic71C2312yes—
46C22×Dic71C2312yes—
47Dic71⋊C4—42no—
48C22⋊Dic71—22no—
49C22⋊Dic71—42no—
50C22⋊Dic71—41no—
51(C2×D71)⋊C4—24no—
52(C2×D71)⋊C4—24no—
53(C2×D71)⋊C4—42no—
54(C2×D71)⋊C4—42no—
55(C2×D71)⋊C4—42no—
56C24×C71C2311yescharacteristic
57C24×C71C2311yescharacteristic
58C2×Q8×C71—22no—
59C2×Q8×C71—41no—
60C2×Q8×C71—41no—
61C2×D4×C71—41no—
62C2×D4×C71—41no—
63C2×D4×C71D414yes—
64C22×C4×C71C2312yes—
65C22×C4×C71C2312yes—
66C22×C4×C71C2311yescharacteristic
67C22⋊C4×C71—24no—
68C22⋊C4×C71—24no—
69C42×C71—22no—
70C23×C71—22no—
71C23×C71—22no—
72C23×C71—21no—
73C23×C71—21no—
74C23×C71—21no—
75C23×C71—21no—
76C23×C71—22no—
77C23×C71—22no—
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—
93Q8×C71—44no—
94D4×C71—44no—
95C2×C4×C71—21no—
96C2×C4×C71—21no—
97C2×C4×C71—22no—
98C2×C4×C71—22no—
99C2×C4×C71—22no—
100C2×C4×C71—22no—
101C2×C4×C71—22no—
102C2×C4×C71—22no—
103C2×C4×C71C2×D412yes—
104C22⋊D71—48no—
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—22no—
114C2×Dic71—44no—
115C22×C71—22no—
116C22×C71—22no—
117C22×C71—24no—
118C22×C71—24no—
119C22×C71—42no—
120C22×C71—42no—
121C22×C71—41no—
122C22×C71—41no—
123C22×C71—41no—
124C22×C71—41no—
125C22×C71C22×D411yescharacteristic
126C22×C71C2×#4112yes—
127C22×C71H(5,𝔽2)11yescharacteristic
128C22×C71H(5,𝔽2)11yescharacteristic
129C22×C71H(5,𝔽2)12yes—
130C2×D71—24no—
131C2×D71—24no—
132C2×D71—24no—
133C2×D71—416no—
134C2×D71—42no—
135C2×D71—42no—
136C2×D71—42no—
137C2×D71—42no—
138C4×C71—22no—
139C4×C71—42no—
140C4×C71—42no—
141Dic71—42no—
142Dic71—42no—
143Dic71—42no—
144C2×C71—42no—
145C2×C71—42no—
146C2×C71#53411yescharacteristic
147C2×C71#53812yes—
148C2×C71#54612yes—
149C2×C71#55111yescharacteristic
150C2×C71H(3,𝔽4)11yescharacteristic
151D71—44no—
152#2684—711nomaximal, Sylow
153C71#268411yescharacteristic, complemented, Sylow
154C2×#124D7111yescharacteristic, complemented
155C2×#120—711no—
156#397—712no—
157#394—712no—
158#393—712no—
159#388—712no—
160#386—711no—
161#386—712no—
162#379—712no—
163C25—711no—
164C22×Q8C2×D7111yescharacteristic, complemented
165C22×D4—712no—
166C22×D4C2×D7111yescharacteristic
167C23×C4—711no—
168#124—24no—
169#120—1424no—
170C2×C4⋊C4—712no—
171C2×C22⋊C4—712no—
172C2×C22⋊C4—711no—
173C2×C22⋊C4—712no—
174C2×C22⋊C4—712no—
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—
185#95—712no—
186C24—2841no—
187C24—2841no—
188C24—2842no—
189C24—712no—
190C24—1424no—
191C24—1422no—
192C24C22×D7111yescharacteristic
193C24C22×D7111yescharacteristic
194C2×Q8—22no—
195C2×Q8—41no—
196C2×Q8—41no—
197C2×D4—41no—
198C2×D4—41no—
199C2×D4—2842no—
200C2×D4—1424no—
201C2×D4—1424no—
202C2×D4C22⋊D7114yes—
203C22×C4—2841no—
204C22×C4—2842no—
205C22×C4—712no—
206C22×C4—712no—
207C22×C4—712no—
208C22×C4C22×D7112yes—
209C22×C4C22×D7112yes—
210C22×C4C22×D7111yescharacteristic
211C4⋊C4—2842no—
212C22⋊C4—24no—
213C22⋊C4—24no—
214C22⋊C4—2842no—
215C22⋊C4—2842no—
216C22⋊C4—2842no—
217C22⋊C4—2841no—
218C22⋊C4—2842no—
219C22⋊C4—1424no—
220C22⋊C4—1422no—
221C22⋊C4—1424no—
222C42—22no—
223C23—22no—
224C23—22no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—22no—
230C23—22no—
231C23—2844no—
232C23—2848no—
233C23—2844no—
234C23—2848no—
235C23—2844no—
236C23—714no—
237C23—1422no—
238C23—1422no—
239C23—1422no—
240C23—1422no—
241C23—1422no—
242C23—1422no—
243C23C2×C22⋊D7112yes—
244C23C2×C22⋊D7112yes—
245C23C23×D7111yescharacteristic
246Q8—44no—
247D4—44no—
248D4—2848no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—2844no—
258C2×C4—1422no—
259C2×C4—1422no—
260C2×C4—1422no—
261C2×C4—1422no—
262C2×C4—1422no—
263C2×C4—1422no—
264C2×C4—1422no—
265C2×C4—1422no—
266C2×C4—1422no—
267C2×C4C2×C22⋊D7112yes—
268C22—24no—
269C22—24no—
270C22—22no—
271C22—22no—
272C22—42no—
273C22—42no—
274C22—41no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—28416no—
279C22—2842no—
280C22—2842no—
281C22—2842no—
282C22—2842no—
283C22—1424no—
284C22—1424no—
285C22—1424no—
286C22C22×C22⋊D7111yescharacteristic
287C22#41×D7112yes—
288C22#1455617311yescharacteristic
289C22#1455617311yescharacteristic
290C22#1455617412yes—
291C4—22no—
292C4—42no—
293C4—42no—
294C4—2842no—
295C4—2842no—
296C4—2842no—
297C2—42no—
298C2—42no—
299C2—2844no—
300C2#1798655811yescharacteristic
301C2#1798658312yes—
302C2#1798663712yes—
303C2#1798667911yescharacteristic
304C2#1798674111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
FittingC2×#124×C71C2
hypercenterC2×#124D71
hypocenterC71#2684
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
2. #17985927C2semidirect
3. #17985914C2semidirect
4. #17985910C2semidirect
5. #17985890C2semidirect
6. #17985879C2semidirect
7. #17985876C2semidirect
8. #17985854C2semidirect
9. C2×#124×C71C2semidirect
12. C23×Dic71C22semidirect
14. C2×Dic71⋊C4C22semidirect
17. C2×C22⋊Dic71C22semidirect
23. C71⋊#95C22semidirect
24. C71⋊#95C22semidirect
25. C71⋊#95C22semidirect
26. C71⋊#95C22semidirect
27. C22×Q8×C71C22semidirect
31. C2×C22⋊C4×C71C22semidirect
32. C2×C42×C71C22semidirect
153. C71#2684semidirect
154. C2×#124D71semidirect
164. C22×Q8C2×D71semidirect
180. C2×C22⋊C4C2×D71semidirect
181. C2×C42C2×D71semidirect