Finite Group Databasebeta

← #22599182

Subgroups — #22599182

Order 9088.

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

normal subgroups only

This group has 418 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#17986510C212yesmaximal, complemented
2#17986341C212yesmaximal, complemented
3#17986324C212yesmaximal, complemented
4#17986257C211yesmaximal, characteristic, complemented
5#17986252C211yesmaximal, characteristic, complemented
6#17986245C212yesmaximal, complemented
7#17986163C212yesmaximal, complemented
8#17986146C212yesmaximal, complemented
9#407×C71C211yesmaximal, characteristic, complemented
10C2×#11765865C2212yes—
11C2×Dic284C2211yescharacteristic, complemented
12C2×#11765854C2212yescomplemented
13C2×D568C2211yescharacteristic, complemented
14#14556111—22no—
15#14556107C2212yes—
16#14556107C2212yes—
17#14556102C2212yes—
18#14556102C2212yes—
19#14556097—22no—
20C4×C22⋊D71—22no—
21C4×C22⋊D71—22no—
22#14556087C2212yes—
23#14556062C2212yescomplemented
24#14556060C2211yescharacteristic, complemented
25#14556060C2211yescharacteristic, complemented
26#14556053C2212yescomplemented
27#14556053C2212yescomplemented
28#14556050C2212yescomplemented
29#14556050C2212yescomplemented
30C23×C4×C71C2211yescharacteristic
31C2×M16×C71C2211yescharacteristic
32C2×M16×C71C2211yescharacteristic
33C22⋊C8×C71C2212yes—
34C22⋊C8×C71C2212yes—
35#11765865—42no—
36#11765865—42no—
37C2×Dic142C2312yes—
38C2×D284C2312yes—
39C2×C22⋊D71—22no—
40C2×C22⋊D71—22no—
41C2×C22⋊D71—22no—
42C2×C4×D71—22no—
43Dic284—41no—
44#11765854—42no—
45D568—41no—
46C4⋊Dic71C2312yes—
47C4⋊Dic71C2312yes—
48Dic71⋊C4—22no—
49Dic71⋊C4—22no—
50C22⋊Dic71—22no—
51C22⋊Dic71—22no—
52C22⋊Dic71—22no—
53(C2×D71)⋊C4—22no—
54(C2×D71)⋊C4—22no—
55C4×Dic71—22no—
56C24×C71D411yescharacteristic
57C22×C4×C71—41no—
58C22×C4×C71—41no—
59C22×C4×C71D411yescharacteristic
60C22×C4×C71D412yes—
61C22×C4×C71C2312yes—
62C22×C4×C71C2311yescharacteristic
63M16×C71—41no—
64M16×C71—41no—
65C2×C8×C71C2311yescharacteristic
66C2×C8×C71C2312yes—
67C2×C8×C71C2311yescharacteristic
68C23×C71—41no—
69C23×C71—42no—
70C23×C71C2×D412yes—
71C23×C71C2×D411yescharacteristic
72C22×D71—22no—
73C2×C4×C71—21no—
74C2×C4×C71—21no—
75C2×C4×C71—22no—
76C2×C4×C71—22no—
77C2×C4×C71—41no—
78C2×C4×C71—42no—
79C2×C4×C71C2×D412yes—
80C2×C4×C71C2×D411yescharacteristic
81C2×C4×C71C2411yescharacteristic
82C22⋊D71—44no—
83C22⋊D71—44no—
84C22⋊D71—44no—
85C2×Dic71—22no—
86C2×Dic71—22no—
87C2×Dic71—22no—
88D284—22no—
89D284—22no—
90C4×D71—44no—
91Dic142—22no—
92Dic142—22no—
93C8×C71—21no—
94C8×C71—22no—
95C8×C71—21no—
96C22×C71—22no—
97C22×C71—21no—
98C22×C71—22no—
99C22×C71—22no—
100C22×C71—22no—
101C22×C71—44no—
102C22×C71C22×D411yescharacteristic
103C2×D71—42no—
104C2×D71—42no—
105C2×D71—42no—
106C4×C71—22no—
107C4×C71—21no—
108C4×C71H(5,𝔽2)11yescharacteristic
109C4×C71H(5,𝔽2)11yescharacteristic
110Dic71—42no—
111Dic71—42no—
112Dic71—42no—
113C2×C71—42no—
114C2×C71—41no—
115C2×C71#53411yescharacteristic
116C2×C71#57512yes—
117D71—82no—
118#3197—711nomaximal, Sylow
119C71#319711yescharacteristic, complemented, Sylow
120#525—712no—
121#485—712no—
122#482—712no—
123#470—711no—
124#469—711no—
125#468—712no—
126#452—712no—
127#449—712no—
128#407D7111yescharacteristic, complemented
129C2×#41—712no—
130C23×C4C2×D7111yescharacteristic
131C2×Q16—711no—
132C2×SD16—712no—
133C2×D8—711no—
134C2×M16C2×D7111yescharacteristic
135C2×M16C2×D7111yescharacteristic
136#123—1422no—
137C22⋊Q8—712no—
138C22⋊Q8—712no—
139#121—712no—
140#121—712no—
141#120—1422no—
142C4×D4—1422no—
143C4×D4—1422no—
144#117—712no—
145#107—712no—
146#106—711no—
147#106—711no—
148#103—712no—
149#103—712no—
150#102—712no—
151#102—712no—
152C22⋊C8C2×D7112yes—
153C22⋊C8C2×D7112yes—
154C24D28411yescharacteristic
155#41—2842no—
156#41—2842no—
157C2×Q8—712no—
158C2×D4—712no—
159C2×D4—1422no—
160C2×D4—1422no—
161C2×D4—1422no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—1422no—
165C22×C4D28411yescharacteristic
166C22×C4D28412yes—
167C22×C4C22×D7112yes—
168C22×C4C22×D7111yescharacteristic
169Q16—2841no—
170SD16—2842no—
171D8—2841no—
172M16—41no—
173M16—41no—
174C2×C8C22×D7111yescharacteristic
175C2×C8C22×D7112yes—
176C2×C8C22×D7111yescharacteristic
177C4⋊C4—712no—
178C4⋊C4—712no—
179C4⋊C4—1422no—
180C4⋊C4—1422no—
181C22⋊C4—1422no—
182C22⋊C4—1422no—
183C22⋊C4—1422no—
184C22⋊C4—1422no—
185C22⋊C4—1422no—
186C42—1422no—
187C23—41no—
188C23—42no—
189C23—1422no—
190C23C2×D28412yes—
191C23C2×D28411yescharacteristic
192Q8—1422no—
193Q8—1422no—
194D4—2844no—
195D4—2844no—
196D4—2844no—
197D4—1422no—
198D4—1422no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—41no—
204C2×C4—42no—
205C2×C4—2844no—
206C2×C4—1422no—
207C2×C4—1422no—
208C2×C4—1422no—
209C2×C4C2×D28412yes—
210C2×C4C2×D28411yescharacteristic
211C2×C4C23×D7111yescharacteristic
212C8—21no—
213C8—22no—
214C8—21no—
215C22—22no—
216C22—21no—
217C22—22no—
218C22—22no—
219C22—22no—
220C22—44no—
221C22—2842no—
222C22—2842no—
223C22—2842no—
224C22C22×D28411yescharacteristic
225C4—22no—
226C4—21no—
227C4—2842no—
228C4—2842no—
229C4—2842no—
230C4#1455617311yescharacteristic
231C4#1455617311yescharacteristic
232C2—42no—
233C2—41no—
234C2—5682no—
235C2#1798656111yescharacteristic
236C2#1798679912yes—
subgroupG/H
centerC22C22×D284
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#407×C71C2
socleC22×C71C22×D4
hypercenter#407D71
hypocenterC71#3197
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. #17986510C2semidirect
2. #17986341C2semidirect
3. #17986324C2semidirect
4. #17986257C2semidirect
5. #17986252C2semidirect
6. #17986245C2semidirect
7. #17986163C2semidirect
8. #17986146C2semidirect
9. #407×C71C2semidirect
11. C2×Dic284C22semidirect
12. C2×#11765854C22semidirect
13. C2×D568C22semidirect
23. #14556062C22semidirect
24. #14556060C22semidirect
25. #14556060C22semidirect
26. #14556053C22semidirect
27. #14556053C22semidirect
28. #14556050C22semidirect
29. #14556050C22semidirect
119. C71#3197semidirect
128. #407D71semidirect