Finite Group Databasebeta

← #22599308

Subgroups — #22599308

Order 9088.

410 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 113 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
1#17986535C211yesmaximal, characteristic, complemented
2C2×#14556115C211yesmaximal, characteristic, complemented
3#17986380C212yesmaximal, complemented
4#17986352C212yesmaximal, complemented
5#17986219C212yesmaximal, complemented
6#17986213C212yesmaximal, complemented
7#17986154C211yesmaximal, characteristic, complemented
8#17986140C211yesmaximal, characteristic, complemented
9#17986133C211yesmaximal, characteristic
10#17986049C211yesmaximal, characteristic, complemented
11#450×C71C211yesmaximal, characteristic, complemented
12C22×Dic142C2211yescharacteristic
13C2×#11765855C2212yescomplemented
14C2×#11765854C2212yescomplemented
15C2×#11765850C2212yes—
16#14556123—21no—
17#14556119C2211yescharacteristic, complemented
18#14556117C2211yescharacteristic, complemented
19#14556115—21no—
20#14556115—21no—
21#14556115C2212yes—
22#14556115C2212yes—
23#14556112—21no—
24#14556102C2211yescharacteristic
25Q8×Dic71C2211yescharacteristic, complemented
26#14556090C2211yescharacteristic, complemented
27(C22⋊D71)⋊C4—21no—
28#14556086—21no—
29C2×C22⋊Dic71—21no—
30C2×C22⋊Dic71—21no—
31C2×C4×Dic71C2211yescharacteristic
32Dic71⋊C8C2212yescomplemented
33#14556053C2212yescomplemented
34#14556051C2212yescomplemented
35#14556049C2212yes—
36C22⋊(C71⋊C8)C2211yescharacteristic
37C8×Dic71C2212yescomplemented
38C22×D4×C71C2211yescharacteristic
39C2×SD16×C71C2212yes—
40C22⋊Q8×C71C2211yescharacteristic
41#102×C71C2212yes—
42C22⋊C8×C71C2211yescharacteristic
43C2×Dic142—21no—
44C2×Dic142—42no—
45C2×Dic142C2312yes—
46C2×D284C2311yescharacteristic
47C2×C22⋊D71—21no—
48C22×Dic71—21no—
49C22×Dic71D411yescharacteristic
50C22×Dic71D411yescharacteristic
51C2×C4×D71—21no—
52#11765855—42no—
53#11765854—42no—
54#11765850—42no—
55C2×C71⋊C8C2312yes—
56C4⋊Dic71—21no—
57C4⋊Dic71—21no—
58C4⋊Dic71C2311yescharacteristic
59Dic71⋊C4—21no—
60C22⋊Dic71—22no—
61C22⋊Dic71—22no—
62C22⋊Dic71—22no—
63C22⋊Dic71—22no—
64(C2×D71)⋊C4—21no—
65(C2×D71)⋊C4—21no—
66(C2×D71)⋊C4—21no—
67(C2×D71)⋊C4—21no—
68C4×Dic71—21no—
69C4×Dic71D411yescharacteristic
70C4×Dic71D411yescharacteristic
71C4×Dic71C2311yescharacteristic
72C4×Dic71C2311yescharacteristic
73C24×C71—21no—
74C2×Q8×C71C2311yescharacteristic
75C2×D4×C71—21no—
76C2×D4×C71—42no—
77C2×D4×C71C2312yes—
78C22×C4×C71C2311yescharacteristic
79SD16×C71—42no—
80C2×C8×C71C2312yes—
81C4⋊C4×C71—21no—
82C4⋊C4×C71C2311yescharacteristic
83C22⋊C4×C71—21no—
84C23×C71—22no—
85C23×C71—42no—
86C23×C71—42no—
87C23×C71—42no—
88C23×C71C2×D411yescharacteristic
89C22×D71—21no—
90Q8×C71—22no—
91D4×C71—22no—
92D4×C71—22no—
93D4×C71—42no—
94C2×C4×C71—22no—
95C2×C4×C71—21no—
96C2×C4×C71—21no—
97C2×C4×C71C2×D411yescharacteristic
98C2×C4×C71C2411yescharacteristic
99C22⋊D71—42no—
100C2×Dic71—22no—
101C2×Dic71—21no—
102C2×Dic71—22no—
103C2×Dic71—41no—
104C2×Dic71—42no—
105C2×Dic71C2×D411yescharacteristic
106C2×Dic71C2×D411yescharacteristic
107C2×Dic71C2×D411yescharacteristic
108C2×Dic71C2×D411yescharacteristic
109D284—22no—
110C4×D71—42no—
111Dic142—22no—
112Dic142—22no—
113Dic142—42no—
114C8×C71—22no—
115C71⋊C8—22no—
116C22×C71—21no—
117C22×C71—21no—
118C22×C71—42no—
119C22×C71—42no—
120C22×C71—42no—
121C22×C71—84no—
122C22×C71#13512yes—
123C22×C71C22×D411yescharacteristic
124C2×D71—41no—
125C2×D71—41no—
126C2×D71—41no—
127C4×C71—21no—
128C4×C71—41no—
129C4×C71—41no—
130C4×C71H(5,𝔽2)12yes—
131Dic71—21no—
132Dic71—21no—
133Dic71—21no—
134Dic71—21no—
135Dic71—42no—
136Dic71—41no—
137C2×C71—22no—
138C2×C71—82no—
139C2×C71#53511yescharacteristic
140C2×C71C2×#13511yescharacteristic
141C2×C71#57511yescharacteristic
142D71—81no—
143#3211—711nomaximal, Sylow
144C71#321111yescharacteristic, complemented, Sylow
145#529—711no—
146C2×#124—711no—
147#495—712no—
148#488—712no—
149#464—712no—
150#463—712no—
151#451—711no—
152#450D7111yescharacteristic, complemented
153#448—711no—
154#447—711no—
155#431—711no—
156C22×Q8—711no—
157C22×D4C2×D7111yescharacteristic
158C2×Q16—712no—
159C2×SD16—712no—
160C2×SD16C2×D7112yes—
161C2×D8—712no—
162#126—1421no—
163#125—711no—
164#124—711no—
165#124—712no—
166#124—712no—
167#124—1421no—
168#124—1421no—
169#123—1421no—
170C22⋊Q8C2×D7111yescharacteristic
171#121—711no—
172C4×Q8—711no—
173C4×D4—711no—
174C4×D4—1421no—
175#117—1421no—
176C2×C22⋊C4—1421no—
177C2×C22⋊C4—1421no—
178C2×C42—711no—
179C4⋊C8—712no—
180#103—712no—
181#103—712no—
182#102—712no—
183#102C2×D7112yes—
184C22⋊C8—711no—
185C22⋊C8C2×D7111yescharacteristic
186C4×C8—712no—
187C24—21no—
188C2×Q8—2842no—
189C2×Q8—712no—
190C2×Q8—1421no—
191C2×Q8C22×D7111yescharacteristic
192C2×D4—21no—
193C2×D4—42no—
194C2×D4—711no—
195C2×D4—1421no—
196C2×D4C22×D7112yes—
197C22×C4—711no—
198C22×C4—711no—
199C22×C4—1421no—
200C22×C4—1421no—
201C22×C4C22×D7111yescharacteristic
202Q16—2842no—
203SD16—42no—
204SD16—2842no—
205D8—2842no—
206C2×C8—712no—
207C2×C8C22×D7112yes—
208C4⋊C4—21no—
209C4⋊C4—711no—
210C4⋊C4—1421no—
211C4⋊C4—1421no—
212C4⋊C4—1421no—
213C4⋊C4C22×D7111yescharacteristic
214C22⋊C4—21no—
215C22⋊C4—1421no—
216C22⋊C4—1421no—
217C22⋊C4—1421no—
218C22⋊C4—1422no—
219C22⋊C4—1422no—
220C22⋊C4—1422no—
221C22⋊C4—1422no—
222C22⋊C4—1421no—
223C42—711no—
224C42—711no—
225C42—711no—
226C42—711no—
227C42—1421no—
228C23—22no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23—1421no—
233C23D4×D7111yescharacteristic
234Q8—22no—
235Q8—2842no—
236Q8—1422no—
237Q8—1422no—
238D4—22no—
239D4—22no—
240D4—42no—
241D4—2842no—
242D4—1422no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—2842no—
247C2×C4—2842no—
248C2×C4—2841no—
249C2×C4—711no—
250C2×C4—711no—
251C2×C4—711no—
252C2×C4—711no—
253C2×C4—1421no—
254C2×C4—1422no—
255C2×C4—1422no—
256C2×C4D4×D7111yescharacteristic
257C2×C4C23×D7111yescharacteristic
258C8—22no—
259C8—1422no—
260C22—21no—
261C22—21no—
262C22—42no—
263C22—42no—
264C22—42no—
265C22—84no—
266C22—2841no—
267C22—2841no—
268C22—2841no—
269C22#1455614912yes—
270C22C2×D4×D7111yescharacteristic
271C4—21no—
272C4—41no—
273C4—41no—
274C4—2842no—
275C4—2841no—
276C4—1421no—
277C4—1421no—
278C4—1421no—
279C4—1421no—
280C4#1455617312yes—
281C2—22no—
282C2—82no—
283C2—5681no—
284C2#1798656411yescharacteristic
285C2C2×#1455614911yescharacteristic
286C2#1798679511yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#450×C71C2
socleC22×C71C22×D4
hypercenter#450D71
hypocenterC71#3211
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #17986535C2semidirect
2. C2×#14556115C2semidirect
3. #17986380C2semidirect
4. #17986352C2semidirect
5. #17986219C2semidirect
6. #17986213C2semidirect
7. #17986154C2semidirect
8. #17986140C2semidirect
10. #17986049C2semidirect
11. #450×C71C2semidirect
13. C2×#11765855C22semidirect
14. C2×#11765854C22semidirect
17. #14556119C22semidirect
18. #14556117C22semidirect
25. Q8×Dic71C22semidirect
26. #14556090C22semidirect
32. Dic71⋊C8C22semidirect
33. #14556053C22semidirect
34. #14556051C22semidirect
37. C8×Dic71C22semidirect
144. C71#3211semidirect
152. #450D71semidirect