Finite Group Databasebeta

← #22599207

Subgroups — #22599207

Order 9088.

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

normal subgroups only

This group has 410 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
2#17986337C212yesmaximal, complemented
3#17986323C212yesmaximal, complemented
4#17986247C212yesmaximal, complemented
5#17986230C212yesmaximal, complemented
6#17986149C212yesmaximal, complemented
7#17986138C212yesmaximal, complemented
8#17985966C211yesmaximal, characteristic, complemented
9#536×C71C211yesmaximal, characteristic, complemented
10C22×D284C2211yescharacteristic
11C2×#11765852C2212yes—
12C2×#11765852C2212yes—
13C2×#11765849C2211yescharacteristic, complemented
14C22×C71⋊C8C2211yescharacteristic, complemented
15#14556124—21no—
16#14556117—21no—
17#14556102C2212yes—
18#14556102C2212yes—
19#14556096—21no—
20#14556096—21no—
21#14556087C2211yescharacteristic
22#14556061C2212yescomplemented
23#14556059C2212yescomplemented
24#14556052C2212yescomplemented
25#14556052C2212yescomplemented
26#14556048C2212yes—
27#14556048C2212yes—
28C22⋊(C71⋊C8)C2212yes—
29C22⋊(C71⋊C8)C2212yescomplemented
30C2×#41×C71C2211yescharacteristic
31C22×Q8×C71C2211yescharacteristic, complemented
32C4⋊Q8×C71—21no—
33#124×C71—21no—
34#123×C71—21no—
35#123×C71—21no—
36C22⋊Q8×C71C2212yes—
37C22⋊Q8×C71C2212yescomplemented
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#11765852—42no—
47#11765852—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—
57#41×C71—41no—
58#41×C71—41no—
59C2×Q8×C71—21no—
60C2×Q8×C71—41no—
61C2×Q8×C71—41no—
62C2×Q8×C71D411yescharacteristic
63C2×Q8×C71C2312yes—
64C2×D4×C71D411yescharacteristic
65C2×D4×C71D411yescharacteristic
66C2×D4×C71D411yescharacteristic
67C22×C4×C71—21no—
68C22×C4×C71C2311yescharacteristic
69C22×C4×C71C2311yescharacteristic
70C22×C4×C71C2311yescharacteristic
71C4⋊C4×C71—22no—
72C4⋊C4×C71—22no—
73C4⋊C4×C71C2312yes—
74C22⋊C4×C71—21no—
75C22⋊C4×C71—22no—
76C22⋊C4×C71—22no—
77C42×C71—21no—
78C23×C71C2×D411yescharacteristic
79C23×C71C2×D411yescharacteristic
80C23×C71C2×D411yescharacteristic
81C22×D71—22no—
82C22×D71—42no—
83C22×D71—82no—
84Q8×C71—22no—
85Q8×C71—22no—
86Q8×C71—41no—
87Q8×C71—42no—
88D4×C71—41no—
89D4×C71—41no—
90D4×C71—41no—
91C2×C4×C71—21no—
92C2×C4×C71—21no—
93C2×C4×C71—21no—
94C2×C4×C71—21no—
95C2×C4×C71—21no—
96C2×C4×C71—21no—
97C2×C4×C71—22no—
98C2×C4×C71—22no—
99C2×C4×C71—42no—
100C2×C4×C71C2×D411yescharacteristic
101C2×C4×C71C2×D411yescharacteristic
102C2×C4×C71C2×D411yescharacteristic
103C2×C4×C71C2411yescharacteristic
104C22⋊D71—82no—
105C22⋊D71—82no—
106C2×Dic71—22no—
107D284—22no—
108D284—22no—
109D284—42no—
110C71⋊C8—22no—
111C71⋊C8—22no—
112C22×C71—21no—
113C22×C71—21no—
114C22×C71—21no—
115C22×C71—22no—
116C22×C71—22no—
117C22×C71—22no—
118C22×C71C22×D411yescharacteristic
119C2×D71—42no—
120C2×D71—42no—
121C2×D71—42no—
122C2×D71—84no—
123C4×C71—21no—
124C4×C71—21no—
125C4×C71—21no—
126C4×C71—42no—
127C4×C71—42no—
128C4×C71H(5,𝔽2)11yescharacteristic
129C4×C71H(5,𝔽2)11yescharacteristic
130Dic71—42no—
131C2×C71—41no—
132C2×C71—41no—
133C2×C71—41no—
134C2×C71#53411yescharacteristic
135C2×C71#57712yes—
136D71—82no—
137#3200—711nomaximal, Sylow
138C71#320011yescharacteristic, complemented, Sylow
139#536D7111yescharacteristic, complemented
140#535—711no—
141#485—712no—
142#482—712no—
143#468—712no—
144#465—712no—
145#450—712no—
146#448—712no—
147#408—711no—
148C2×#41C2×D7111yescharacteristic
149C22×Q8C2×D7111yescharacteristic, complemented
150C22×D4—711no—
151C2×SD16—712no—
152C2×SD16—712no—
153C2×M16—711no—
154C22×C8—711no—
155C4⋊Q8—21no—
156#127—1421no—
157#124—21no—
158#124—1421no—
159#123—21no—
160#123—21no—
161C22⋊Q8C2×D7112yes—
162C22⋊Q8C2×D7112yescomplemented
163#121—712no—
164#121—712no—
165#120—1421no—
166#120—1421no—
167#117—711no—
168#117C2×D7111yescharacteristic, complemented
169#107—712no—
170#106—712no—
171#103—712no—
172#103—712no—
173#102—712no—
174#102—712no—
175C22⋊C8—712no—
176C22⋊C8—712no—
177C24—1421no—
178#41—41no—
179#41—41no—
180C2×Q8—21no—
181C2×Q8—41no—
182C2×Q8—41no—
183C2×Q8C22⋊D7111yescharacteristic
184C2×Q8C22×D7112yes—
185C2×D4—2841no—
186C2×D4—2841no—
187C2×D4—712no—
188C2×D4—1422no—
189C2×D4—1422no—
190C2×D4—1421no—
191C2×D4C22⋊D7111yescharacteristic
192C2×D4C22⋊D7111yescharacteristic
193C2×D4C22⋊D7111yescharacteristic
194C22×C4—21no—
195C22×C4C22×D7111yescharacteristic
196C22×C4C22×D7111yescharacteristic
197C22×C4C22×D7111yescharacteristic
198SD16—2842no—
199SD16—2842no—
200M16—2841no—
201C2×C8—2841no—
202C2×C8—712no—
203C2×C8—712no—
204C4⋊C4—22no—
205C4⋊C4—22no—
206C4⋊C4—712no—
207C4⋊C4C22×D7112yes—
208C22⋊C4—21no—
209C22⋊C4—22no—
210C22⋊C4—22no—
211C22⋊C4—1421no—
212C22⋊C4—1422no—
213C22⋊C4—1422no—
214C42—21no—
215C42—1421no—
216C23—2842no—
217C23—5682no—
218C23—1422no—
219C23C2×C22⋊D7111yescharacteristic
220C23C2×C22⋊D7111yescharacteristic
221C23C2×C22⋊D7111yescharacteristic
222Q8—22no—
223Q8—22no—
224Q8—41no—
225Q8—42no—
226D4—41no—
227D4—41no—
228D4—41no—
229D4—2842no—
230D4—5682no—
231D4—5682no—
232D4—1422no—
233D4—1422no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—42no—
243C2×C4—1422no—
244C2×C4C2×C22⋊D7111yescharacteristic
245C2×C4C2×C22⋊D7111yescharacteristic
246C2×C4C2×C22⋊D7111yescharacteristic
247C2×C4C23×D7111yescharacteristic
248C8—1422no—
249C8—1422no—
250C22—22no—
251C22—22no—
252C22—22no—
253C22—21no—
254C22—21no—
255C22—21no—
256C22—2842no—
257C22—2842no—
258C22—2842no—
259C22—5684no—
260C22C22×C22⋊D7111yescharacteristic
261C4—21no—
262C4—21no—
263C4—21no—
264C4—42no—
265C4—42no—
266C4—2842no—
267C4#1455617311yescharacteristic
268C4#1455617311yescharacteristic
269C2—41no—
270C2—41no—
271C2—41no—
272C2—5682no—
273C2#1798655811yescharacteristic
274C2#1798680712yes—
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#536×C71C2
socleC22×C71C22×D4
hypercenter#536D71
hypocenterC71#3200
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
2. #17986337C2semidirect
3. #17986323C2semidirect
4. #17986247C2semidirect
5. #17986230C2semidirect
6. #17986149C2semidirect
7. #17986138C2semidirect
8. #17985966C2semidirect
9. #536×C71C2semidirect
13. C2×#11765849C22semidirect
14. C22×C71⋊C8C22semidirect
22. #14556061C22semidirect
23. #14556059C22semidirect
24. #14556052C22semidirect
25. #14556052C22semidirect
29. C22⋊(C71⋊C8)C22semidirect
31. C22×Q8×C71C22semidirect
37. C22⋊Q8×C71C22semidirect
38. #117×C71C22semidirect
138. C71#3200semidirect
139. #536D71semidirect
149. C22×Q8C2×D71semidirect
162. C22⋊Q8C2×D71semidirect
168. #117C2×D71semidirect