Finite Group Databasebeta

← #22599213

Subgroups — #22599213

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#17986570C211yesmaximal, characteristic, complemented
2#17986563C211yesmaximal, characteristic
3#17986341C212yesmaximal, complemented
4#17986324C212yesmaximal, complemented
5#17986245C212yesmaximal, complemented
6#17986228C212yesmaximal, complemented
7#17986152C212yesmaximal, complemented
8#17986140C212yesmaximal, complemented
9#408×C71C211yesmaximal, characteristic, complemented
10C22×Dic142C2211yescharacteristic, complemented
11C22×D284C2211yescharacteristic
12C2×#11765854C2212yes—
13C2×#11765854C2212yes—
14#14556126—21no—
15#14556124—21no—
16#14556117—21no—
17#14556115—21no—
18#14556109—21no—
19#14556109—21no—
20#14556107C2212yes—
21#14556107C2212yescomplemented
22#14556102C2212yes—
23#14556102C2212yes—
24#14556096—21no—
25#14556096—21no—
26#14556087C2211yescharacteristic
27#14556087C2211yescharacteristic, complemented
28#14556062C2212yescomplemented
29#14556060C2212yescomplemented
30#14556053C2212yescomplemented
31#14556053C2212yescomplemented
32#14556050C2212yes—
33#14556050C2212yes—
34C2×#41×C71C2211yescharacteristic
35C2×M16×C71C2211yescharacteristic, complemented
36C22×C8×C71C2211yescharacteristic, complemented
37C22⋊C8×C71C2212yes—
38C22⋊C8×C71C2212yescomplemented
39C23×D71—21no—
40C2×Dic142—21no—
41C2×Dic142—41no—
42C2×Dic142—41no—
43C2×Dic142C2312yes—
44C2×D284—21no—
45C2×D284—41no—
46C2×D284—41no—
47C2×D284C2312yes—
48C2×C22⋊D71—22no—
49C2×C22⋊D71—22no—
50C22×Dic71—21no—
51#11765854—42no—
52#11765854—42no—
53C4⋊Dic71C2312yes—
54C4⋊Dic71C2312yes—
55Dic71⋊C4—22no—
56Dic71⋊C4—22no—
57C22⋊Dic71—21no—
58C22⋊Dic71—21no—
59C22⋊Dic71—22no—
60C22⋊Dic71—22no—
61(C2×D71)⋊C4—22no—
62(C2×D71)⋊C4—22no—
63C4×Dic71—21no—
64C4×Dic71—21no—
65#41×C71—41no—
66#41×C71—41no—
67C2×Q8×C71D411yescharacteristic
68C2×D4×C71D411yescharacteristic
69C2×D4×C71D411yescharacteristic
70C2×D4×C71D411yescharacteristic
71C22×C4×C71C2311yescharacteristic
72C22×C4×C71C2311yescharacteristic
73C22×C4×C71C2311yescharacteristic
74M16×C71—41no—
75C2×C8×C71—41no—
76C2×C8×C71C2312yes—
77C2×C8×C71C2312yes—
78C23×C71C2×D411yescharacteristic
79C23×C71C2×D411yescharacteristic
80C23×C71C2×D411yescharacteristic
81C22×D71—22no—
82C22×D71—42no—
83C22×D71—82no—
84Q8×C71—41no—
85D4×C71—41no—
86D4×C71—41no—
87D4×C71—41no—
88C2×C4×C71—21no—
89C2×C4×C71—21no—
90C2×C4×C71—21no—
91C2×C4×C71—21no—
92C2×C4×C71—21no—
93C2×C4×C71—21no—
94C2×C4×C71C2×D411yescharacteristic
95C2×C4×C71C2×D411yescharacteristic
96C2×C4×C71C2×D411yescharacteristic
97C2×C4×C71C2411yescharacteristic
98C22⋊D71—82no—
99C22⋊D71—82no—
100C2×Dic71—22no—
101C2×Dic71—22no—
102C2×Dic71—22no—
103C2×Dic71—42no—
104D284—22no—
105D284—22no—
106D284—42no—
107Dic142—22no—
108Dic142—22no—
109Dic142—42no—
110C8×C71—22no—
111C8×C71—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×C71H(5,𝔽2)11yescharacteristic
127C4×C71H(5,𝔽2)11yescharacteristic
128Dic71—42no—
129Dic71—42no—
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#536—711no—
140#535—711no—
141#485—712no—
142#482—712no—
143#468—712no—
144#465—712no—
145#450—712no—
146#448—712no—
147#408D7111yescharacteristic, complemented
148C2×#41C2×D7111yescharacteristic
149C22×Q8—711no—
150C22×D4—711no—
151C2×SD16—712no—
152C2×SD16—712no—
153C2×M16C2×D7111yescharacteristic, complemented
154C22×C8C2×D7111yescharacteristic, complemented
155C4⋊Q8—1421no—
156#127—1421no—
157#124—1421no—
158#124—1421no—
159#123—1421no—
160#123—1421no—
161C22⋊Q8—712no—
162C22⋊Q8—712no—
163#121—712no—
164#121—712no—
165#120—1421no—
166#120—1421no—
167#117—711no—
168#117—711no—
169#107—712no—
170#106—712no—
171#103—712no—
172#103—712no—
173#102—712no—
174#102—712no—
175C22⋊C8C2×D7112yes—
176C22⋊C8C2×D7112yescomplemented
177C24—1421no—
178#41—41no—
179#41—41no—
180C2×Q8—2841no—
181C2×Q8—2841no—
182C2×Q8—712no—
183C2×Q8—1421no—
184C2×Q8D28411yescharacteristic
185C2×D4—2841no—
186C2×D4—2841no—
187C2×D4—712no—
188C2×D4—1422no—
189C2×D4—1422no—
190C2×D4—1421no—
191C2×D4D28411yescharacteristic
192C2×D4D28411yescharacteristic
193C2×D4D28411yescharacteristic
194C22×C4—1421no—
195C22×C4C22×D7111yescharacteristic
196C22×C4C22×D7111yescharacteristic
197C22×C4C22×D7111yescharacteristic
198SD16—2842no—
199SD16—2842no—
200M16—41no—
201C2×C8—41no—
202C2×C8C22×D7112yes—
203C2×C8C22×D7112yes—
204C4⋊C4—712no—
205C4⋊C4—712no—
206C4⋊C4—1422no—
207C4⋊C4—1422no—
208C22⋊C4—1421no—
209C22⋊C4—1421no—
210C22⋊C4—1422no—
211C22⋊C4—1422no—
212C22⋊C4—1422no—
213C22⋊C4—1422no—
214C42—1421no—
215C42—1421no—
216C23—2842no—
217C23—5682no—
218C23—1422no—
219C23C2×D28411yescharacteristic
220C23C2×D28411yescharacteristic
221C23C2×D28411yescharacteristic
222Q8—41no—
223Q8—2842no—
224Q8—1422no—
225Q8—1422no—
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—2842no—
241C2×C4—1422no—
242C2×C4—1422no—
243C2×C4—1422no—
244C2×C4C2×D28411yescharacteristic
245C2×C4C2×D28411yescharacteristic
246C2×C4C2×D28411yescharacteristic
247C2×C4C23×D7111yescharacteristic
248C8—22no—
249C8—22no—
250C22—21no—
251C22—21no—
252C22—21no—
253C22—22no—
254C22—22no—
255C22—22no—
256C22—2842no—
257C22—2842no—
258C22—2842no—
259C22—5684no—
260C22C22×D28411yescharacteristic
261C4—21no—
262C4—21no—
263C4—21no—
264C4—2842no—
265C4—2842no—
266C4—2842no—
267C4#1455617311yescharacteristic
268C4#1455617311yescharacteristic
269C2—41no—
270C2—41no—
271C2—41no—
272C2—5682no—
273C2#1798656111yescharacteristic
274C2#1798681112yes—
subgroupG/H
centerC22C22×D284
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#408×C71C2
socleC22×C71C22×D4
hypercenter#408D71
hypocenterC71#3200
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
1. #17986570C2semidirect
3. #17986341C2semidirect
4. #17986324C2semidirect
5. #17986245C2semidirect
6. #17986228C2semidirect
7. #17986152C2semidirect
8. #17986140C2semidirect
9. #408×C71C2semidirect
10. C22×Dic142C22semidirect
21. #14556107C22semidirect
27. #14556087C22semidirect
28. #14556062C22semidirect
29. #14556060C22semidirect
30. #14556053C22semidirect
31. #14556053C22semidirect
35. C2×M16×C71C22semidirect
36. C22×C8×C71C22semidirect
38. C22⋊C8×C71C22semidirect
138. C71#3200semidirect
147. #408D71semidirect
153. C2×M16C2×D71semidirect
154. C22×C8C2×D71semidirect
176. C22⋊C8C2×D71semidirect