Finite Group Databasebeta

← #22599201

Subgroups — #22599201

Order 9088.

442 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 117 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#17986563C211yesmaximal, characteristic, complemented
2#17986323C212yesmaximal, complemented
3#17986320C212yesmaximal, complemented
4#17986250C212yesmaximal, complemented
5#17986233C212yesmaximal, complemented
6#17986133C212yesmaximal, complemented
7#17986132C212yesmaximal, complemented
8#17985966C211yesmaximal, characteristic, complemented
9#535×C71C211yesmaximal, characteristic, complemented
10C22×D284C2211yescharacteristic, complemented
11C2×#11765850C2212yes—
12C2×#11765850C2212yes—
13C2×#11765849C2211yescharacteristic, complemented
14C22×C71⋊C8C2211yescharacteristic, complemented
15#14556124—21no—
16#14556117—21no—
17#14556102C2212yes—
18#14556102C2212yescomplemented
19#14556096—21no—
20#14556096—21no—
21#14556087C2211yescharacteristic, complemented
22#14556061C2212yescomplemented
23#14556059C2212yescomplemented
24#14556049C2212yescomplemented
25#14556049C2212yescomplemented
26#14556048C2212yescomplemented
27#14556048C2212yescomplemented
28C22⋊(C71⋊C8)C2212yescomplemented
29C22⋊(C71⋊C8)C2212yescomplemented
30C2×#41×C71C2211yescharacteristic
31C22×D4×C71C2211yescharacteristic, complemented
32#127×C71—21no—
33#124×C71—21no—
34#121×C71C2212yes—
35#121×C71C2212yescomplemented
36#120×C71—21no—
37#120×C71—21no—
38#117×C71C2211yescharacteristic, complemented
39C23×D71—21no—
40C2×D284—21no—
41C2×D284—41no—
42C2×D284—41no—
43C2×D284C2312yes—
44C2×C22⋊D71—22no—
45C2×C22⋊D71—22no—
46#11765850—42no—
47#11765850—42no—
48#11765849—41no—
49C2×C71⋊C8—41no—
50C2×C71⋊C8C2312yes—
51C2×C71⋊C8C2312yes—
52C4⋊Dic71C2312yes—
53C22⋊Dic71—21no—
54(C2×D71)⋊C4—22no—
55(C2×D71)⋊C4—22no—
56C4×Dic71—21no—
57C24×C71—21no—
58#41×C71—41no—
59#41×C71—41no—
60C2×Q8×C71D411yescharacteristic
61C2×D4×C71—22no—
62C2×D4×C71—22no—
63C2×D4×C71—21no—
64C2×D4×C71—41no—
65C2×D4×C71—41no—
66C2×D4×C71D411yescharacteristic
67C2×D4×C71D411yescharacteristic
68C2×D4×C71D411yescharacteristic
69C2×D4×C71C2312yes—
70C22×C4×C71C2311yescharacteristic
71C22×C4×C71C2311yescharacteristic
72C22×C4×C71C2311yescharacteristic
73C4⋊C4×C71C2312yes—
74C22⋊C4×C71—21no—
75C22⋊C4×C71—22no—
76C22⋊C4×C71—22no—
77C42×C71—21no—
78C23×C71—22no—
79C23×C71—42no—
80C23×C71—82no—
81C23×C71C2×D411yescharacteristic
82C23×C71C2×D411yescharacteristic
83C23×C71C2×D411yescharacteristic
84C22×D71—22no—
85C22×D71—42no—
86C22×D71—82no—
87Q8×C71—41no—
88D4×C71—22no—
89D4×C71—22no—
90D4×C71—41no—
91D4×C71—41no—
92D4×C71—41no—
93D4×C71—42no—
94D4×C71—82no—
95D4×C71—82no—
96C2×C4×C71—21no—
97C2×C4×C71—21no—
98C2×C4×C71—21no—
99C2×C4×C71—21no—
100C2×C4×C71—21no—
101C2×C4×C71—21no—
102C2×C4×C71—22no—
103C2×C4×C71C2×D411yescharacteristic
104C2×C4×C71C2×D411yescharacteristic
105C2×C4×C71C2×D411yescharacteristic
106C2×C4×C71C2411yescharacteristic
107C22⋊D71—82no—
108C22⋊D71—82no—
109C2×Dic71—22no—
110D284—22no—
111D284—22no—
112D284—42no—
113C71⋊C8—22no—
114C71⋊C8—22no—
115C22×C71—21no—
116C22×C71—21no—
117C22×C71—21no—
118C22×C71—22no—
119C22×C71—22no—
120C22×C71—22no—
121C22×C71—42no—
122C22×C71—42no—
123C22×C71—42no—
124C22×C71—84no—
125C22×C71C22×D411yescharacteristic
126C2×D71—42no—
127C2×D71—42no—
128C2×D71—42no—
129C2×D71—84no—
130C4×C71—21no—
131C4×C71—21no—
132C4×C71—21no—
133C4×C71—42no—
134C4×C71H(5,𝔽2)11yescharacteristic
135C4×C71H(5,𝔽2)11yescharacteristic
136Dic71—42no—
137C2×C71—41no—
138C2×C71—41no—
139C2×C71—41no—
140C2×C71—82no—
141C2×C71#53411yescharacteristic
142C2×C71#57612yes—
143D71—82no—
144#3199—711nomaximal, Sylow
145C71#319911yescharacteristic, complemented, Sylow
146#535—711no—
147#535D7111yescharacteristic, complemented
148#482—712no—
149#482—712no—
150#469—712no—
151#466—712no—
152#447—712no—
153#447—712no—
154#408—711no—
155C2×#41C2×D7111yescharacteristic
156C22×D4—711no—
157C22×D4C2×D7111yescharacteristic, complemented
158C2×D8—712no—
159C2×D8—712no—
160C2×M16—711no—
161C22×C8—711no—
162#127—21no—
163#127—1421no—
164#124—21no—
165#124—1421no—
166#121—712no—
167#121—712no—
168#121C2×D7112yes—
169#121C2×D7112yescomplemented
170#120—21no—
171#120—21no—
172#120—1421no—
173#120—1421no—
174#117—711no—
175#117C2×D7111yescharacteristic, complemented
176#107—712no—
177#106—712no—
178#102—712no—
179#102—712no—
180#102—712no—
181#102—712no—
182C22⋊C8—712no—
183C22⋊C8—712no—
184C24—21no—
185C24—1421no—
186#41—41no—
187#41—41no—
188C2×Q8C22⋊D7111yescharacteristic
189C2×D4—22no—
190C2×D4—22no—
191C2×D4—21no—
192C2×D4—41no—
193C2×D4—41no—
194C2×D4—2841no—
195C2×D4—2841no—
196C2×D4—712no—
197C2×D4—1422no—
198C2×D4—1422no—
199C2×D4—1421no—
200C2×D4C22⋊D7111yescharacteristic
201C2×D4C22⋊D7111yescharacteristic
202C2×D4C22⋊D7111yescharacteristic
203C2×D4C22×D7112yes—
204C22×C4C22×D7111yescharacteristic
205C22×C4C22×D7111yescharacteristic
206C22×C4C22×D7111yescharacteristic
207D8—2842no—
208D8—2842no—
209M16—2841no—
210C2×C8—2841no—
211C2×C8—712no—
212C2×C8—712no—
213C4⋊C4—712no—
214C4⋊C4C22×D7112yes—
215C22⋊C4—21no—
216C22⋊C4—22no—
217C22⋊C4—22no—
218C22⋊C4—1421no—
219C22⋊C4—1422no—
220C22⋊C4—1422no—
221C42—21no—
222C42—1421no—
223C23—22no—
224C23—42no—
225C23—82no—
226C23—2842no—
227C23—5682no—
228C23—1422no—
229C23C2×C22⋊D7111yescharacteristic
230C23C2×C22⋊D7111yescharacteristic
231C23C2×C22⋊D7111yescharacteristic
232Q8—41no—
233D4—22no—
234D4—22no—
235D4—41no—
236D4—41no—
237D4—41no—
238D4—42no—
239D4—82no—
240D4—82no—
241D4—2842no—
242D4—5682no—
243D4—5682no—
244D4—1422no—
245D4—1422no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—1422no—
254C2×C4C2×C22⋊D7111yescharacteristic
255C2×C4C2×C22⋊D7111yescharacteristic
256C2×C4C2×C22⋊D7111yescharacteristic
257C2×C4C23×D7111yescharacteristic
258C8—1422no—
259C8—1422no—
260C22—22no—
261C22—22no—
262C22—22no—
263C22—21no—
264C22—21no—
265C22—21no—
266C22—42no—
267C22—42no—
268C22—42no—
269C22—84no—
270C22—2842no—
271C22—2842no—
272C22—2842no—
273C22—5684no—
274C22C22×C22⋊D7111yescharacteristic
275C4—21no—
276C4—21no—
277C4—21no—
278C4—42no—
279C4—2842no—
280C4#1455617311yescharacteristic
281C4#1455617311yescharacteristic
282C2—41no—
283C2—41no—
284C2—41no—
285C2—82no—
286C2—5682no—
287C2#1798655811yescharacteristic
288C2#1798680212yes—
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#535×C71C2
socleC22×C71C22×D4
hypercenter#535D71
hypocenterC71#3199
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #17986563C2semidirect
2. #17986323C2semidirect
3. #17986320C2semidirect
4. #17986250C2semidirect
5. #17986233C2semidirect
6. #17986133C2semidirect
7. #17986132C2semidirect
8. #17985966C2semidirect
9. #535×C71C2semidirect
10. C22×D284C22semidirect
13. C2×#11765849C22semidirect
14. C22×C71⋊C8C22semidirect
18. #14556102C22semidirect
21. #14556087C22semidirect
22. #14556061C22semidirect
23. #14556059C22semidirect
24. #14556049C22semidirect
25. #14556049C22semidirect
26. #14556048C22semidirect
27. #14556048C22semidirect
28. C22⋊(C71⋊C8)C22semidirect
29. C22⋊(C71⋊C8)C22semidirect
31. C22×D4×C71C22semidirect
35. #121×C71C22semidirect
38. #117×C71C22semidirect
145. C71#3199semidirect
147. #535D71semidirect
157. C22×D4C2×D71semidirect
169. #121C2×D71semidirect
175. #117C2×D71semidirect