Finite Group Databasebeta

← #22599202

Subgroups — #22599202

Order 9088.

442 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 109 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#17986565C212yesmaximal, complemented
2#17986325C212yesmaximal, complemented
3#17986322C212yesmaximal, complemented
4#17986252C211yesmaximal, characteristic, complemented
5#17986233C212yesmaximal, complemented
6#17986134C212yesmaximal, complemented
7#17986131C212yesmaximal, complemented
8#17985967C211yesmaximal, characteristic, complemented
9#469×C71C211yesmaximal, characteristic, complemented
10C2×#11765865C2211yescharacteristic
11C2×D4×D71C2212yescomplemented
12C2×D568C2211yescharacteristic
13C2×#11765850C2212yes—
14C22×C71⋊C8C2211yescharacteristic, complemented
15#14556124—22no—
16#14556114—22no—
17#14556103C2212yes—
18#14556102C2212yescomplemented
19#14556099C2212yes—
20#14556098—22no—
21#14556096—22no—
22#14556086C2212yescomplemented
23#14556061C2212yescomplemented
24#14556060C2211yescharacteristic, complemented
25#14556050C2212yescomplemented
26#14556049C2212yescomplemented
27#14556048C2212yescomplemented
28(C2×D71)⋊C8C2212yescomplemented
29(C2×D71)⋊C8C2212yescomplemented
30C2×D8×C71C2211yescharacteristic
31C2×M16×C71C2211yescharacteristic, complemented
32#121×C71C2212yescomplemented
33#106×C71C2211yescharacteristic, complemented
34#102×C71C2212yescomplemented
35C23×D71—22no—
36#11765865—41no—
37#11765865—41no—
38C2×Dic142D411yescharacteristic
39C2×D284—22no—
40C2×D284D411yescharacteristic
41C2×D284C2312yes—
42C2×C22⋊D71—22no—
43C2×C22⋊D71—22no—
44C2×C22⋊D71—22no—
45C2×C22⋊D71D412yes—
46D4×D71—42no—
47D4×D71—42no—
48C2×C4×D71C2312yes—
49D568—41no—
50#11765850—42no—
51C2×C71⋊C8—41no—
52C2×C71⋊C8C2312yes—
53C4⋊Dic71C2312yes—
54C22⋊Dic71—22no—
55(C2×D71)⋊C4—22no—
56(C2×D71)⋊C4—22no—
57(C2×D71)⋊C4—22no—
58C4×Dic71—22no—
59C2×D4×C71—22no—
60C2×D4×C71C2312yes—
61C22×C4×C71C2311yescharacteristic
62D8×C71—41no—
63M16×C71—41no—
64C2×C8×C71C2311yescharacteristic
65C2×C8×C71C2311yescharacteristic
66C4⋊C4×C71C2312yes—
67C22⋊C4×C71—22no—
68C23×C71—22no—
69C23×C71C2×D411yescharacteristic
70C22×D71—22no—
71C22×D71—44no—
72C22×D71—82no—
73C22×D71—82no—
74C22×D71C2×D412yes—
75D4×C71—22no—
76D4×C71—22no—
77D4×C71—82no—
78C2×C4×C71—21no—
79C2×C4×C71—21no—
80C2×C4×C71—22no—
81C2×C4×C71C2×D411yescharacteristic
82C2×C4×C71C2411yescharacteristic
83C22⋊D71—42no—
84C22⋊D71—44no—
85C22⋊D71—82no—
86C22⋊D71—82no—
87C2×Dic71—22no—
88C2×Dic71C2×D412yes—
89D284—22no—
90D284—22no—
91D284—41no—
92D284—82no—
93C4×D71—22no—
94C4×D71—22no—
95Dic142—41no—
96C8×C71—21no—
97C8×C71—21no—
98C71⋊C8—22no—
99C22×C71—21no—
100C22×C71—22no—
101C22×C71—42no—
102C22×C71—42no—
103C22×C71—42no—
104C22×C71C22×D411yescharacteristic
105C2×D71—22no—
106C2×D71—22no—
107C2×D71—22no—
108C2×D71—42no—
109C2×D71—42no—
110C2×D71—42no—
111C2×D71—88no—
112C4×C71—21no—
113C4×C71—42no—
114C4×C71H(5,𝔽2)11yescharacteristic
115C4×C71H(5,𝔽2)11yescharacteristic
116Dic71—22no—
117Dic71—42no—
118C2×C71—41no—
119C2×C71—82no—
120C2×C71#53411yescharacteristic
121C2×C71#57612yes—
122D71—42no—
123D71—82no—
124#3199—711nomaximal, Sylow
125C71#319911yescharacteristic, complemented, Sylow
126#535—712no—
127#482—712no—
128#482—712no—
129#469—711no—
130#469D7111yescharacteristic, complemented
131#466—712no—
132#447—712no—
133#447—712no—
134#408—711no—
135C2×#41—711no—
136C22×D4—712no—
137C2×D8—712no—
138C2×D8—711no—
139C2×D8C2×D7111yescharacteristic
140C2×M16C2×D7111yescharacteristic, complemented
141C22×C8—711no—
142#127—1422no—
143#124—1422no—
144#121—712no—
145#121—712no—
146#121—712no—
147#121C2×D7112yescomplemented
148#120—1422no—
149#120—1422no—
150#117—712no—
151#107—712no—
152#106—711no—
153#106C2×D7111yescharacteristic, complemented
154#102—712no—
155#102—712no—
156#102—712no—
157#102C2×D7112yescomplemented
158C22⋊C8—712no—
159C22⋊C8—712no—
160C24—1422no—
161#41—2841no—
162#41—2841no—
163C2×Q8—711no—
164C2×D4—22no—
165C2×D4—2842no—
166C2×D4—2842no—
167C2×D4—712no—
168C2×D4—711no—
169C2×D4—712no—
170C2×D4—1422no—
171C2×D4—1422no—
172C2×D4—1422no—
173C2×D4—1422no—
174C2×D4C22×D7112yes—
175C22×C4—712no—
176C22×C4C22×D7111yescharacteristic
177D8—41no—
178D8—2841no—
179D8—2842no—
180M16—41no—
181C2×C8—2841no—
182C2×C8—712no—
183C2×C8C22×D7111yescharacteristic
184C2×C8C22×D7111yescharacteristic
185C4⋊C4—712no—
186C4⋊C4C22×D7112yes—
187C22⋊C4—22no—
188C22⋊C4—1422no—
189C22⋊C4—1422no—
190C22⋊C4—1422no—
191C22⋊C4—1422no—
192C42—1422no—
193C23—22no—
194C23—2844no—
195C23—5682no—
196C23—5682no—
197C23—712no—
198C23—1422no—
199C23D4×D7111yescharacteristic
200Q8—2841no—
201D4—22no—
202D4—22no—
203D4—82no—
204D4—2841no—
205D4—2842no—
206D4—2844no—
207D4—5682no—
208D4—5682no—
209D4—5682no—
210D4—1422no—
211D4—1422no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—712no—
216C2×C4—1422no—
217C2×C4—1422no—
218C2×C4—1422no—
219C2×C4D4×D7111yescharacteristic
220C2×C4C23×D7111yescharacteristic
221C8—21no—
222C8—21no—
223C8—1422no—
224C22—22no—
225C22—21no—
226C22—42no—
227C22—42no—
228C22—42no—
229C22—2842no—
230C22—2842no—
231C22—2842no—
232C22—5688no—
233C22—1422no—
234C22—1422no—
235C22—1422no—
236C22C2×D4×D7111yescharacteristic
237C4—21no—
238C4—42no—
239C4—2842no—
240C4—1422no—
241C4#1455617311yescharacteristic
242C4#1455617411yescharacteristic
243C2—41no—
244C2—82no—
245C2—2842no—
246C2—5682no—
247C2#1798655711yescharacteristic
248C2#1798680312yes—
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#469×C71C2
socleC22×C71C22×D4
hypercenter#469D71
hypocenterC71#3199
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #17986565C2semidirect
2. #17986325C2semidirect
3. #17986322C2semidirect
4. #17986252C2semidirect
5. #17986233C2semidirect
6. #17986134C2semidirect
7. #17986131C2semidirect
8. #17985967C2semidirect
9. #469×C71C2semidirect
11. C2×D4×D71C22semidirect
14. C22×C71⋊C8C22semidirect
18. #14556102C22semidirect
22. #14556086C22semidirect
23. #14556061C22semidirect
24. #14556060C22semidirect
25. #14556050C22semidirect
26. #14556049C22semidirect
27. #14556048C22semidirect
28. (C2×D71)⋊C8C22semidirect
29. (C2×D71)⋊C8C22semidirect
31. C2×M16×C71C22semidirect
32. #121×C71C22semidirect
33. #106×C71C22semidirect
34. #102×C71C22semidirect
125. C71#3199semidirect
130. #469D71semidirect
140. C2×M16C2×D71semidirect
147. #121C2×D71semidirect
153. #106C2×D71semidirect
157. #102C2×D71semidirect