Finite Group Databasebeta

← #22599176

Subgroups — #22599176

Order 9088.

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

normal subgroups only

This group has 418 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#17986509C212yesmaximal, complemented
2#17986343C212yesmaximal, complemented
3#17986322C212yesmaximal, complemented
4#17986257C211yesmaximal, characteristic, complemented
5#17986243C212yesmaximal, complemented
6#17986162C212yesmaximal, complemented
7#17986141C212yesmaximal, complemented
8#17985962C211yesmaximal, characteristic, complemented
9#469×C71C211yesmaximal, characteristic, complemented
10C2×#11765864C2212yes—
11C22×C4×D71C2211yescharacteristic
12C2×Dic284C2211yescharacteristic, complemented
13C2×#11765853C2212yescomplemented
14C2×#11765849C2211yescharacteristic
15#14556110—22no—
16#14556108C2212yes—
17#14556107C2212yes—
18#14556099C2212yes—
19#14556096—22no—
20D4×Dic71—22no—
21(C22⋊D71)⋊C4—22no—
22#14556086C2212yes—
23#14556061C2212yescomplemented
24#14556060C2211yescharacteristic, complemented
25#14556053C2212yescomplemented
26#14556051C2212yescomplemented
27#14556049C2212yescomplemented
28(C2×D71)⋊C8C2212yes—
29(C2×D71)⋊C8C2212yes—
30C2×D8×C71C2211yescharacteristic, complemented
31C2×M16×C71C2211yescharacteristic
32#121×C71C2212yes—
33#106×C71C2211yescharacteristic, complemented
34#102×C71C2212yescomplemented
35C23×D71D411yescharacteristic
36#11765864—42no—
37#11765864—42no—
38C2×Dic142C2312yes—
39C2×C22⋊D71—22no—
40C2×C22⋊D71—22no—
41C22×Dic71—22no—
42C22×Dic71D411yescharacteristic
43C2×C4×D71—41no—
44C2×C4×D71—41no—
45C2×C4×D71D412yes—
46C2×C4×D71C2312yes—
47Dic284—41no—
48#11765853—42no—
49#11765849—41no—
50C2×C71⋊C8C2312yes—
51C4⋊Dic71—22no—
52C4⋊Dic71C2312yes—
53Dic71⋊C4—22no—
54C22⋊Dic71—22no—
55C22⋊Dic71—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—41no—
71C22×D71—42no—
72C22×D71C2×D412yes—
73D4×C71—22no—
74D4×C71—22no—
75D4×C71—44no—
76C2×C4×C71—21no—
77C2×C4×C71—21no—
78C2×C4×C71—22no—
79C2×C4×C71C2×D411yescharacteristic
80C2×C4×C71C2411yescharacteristic
81C22⋊D71—44no—
82C22⋊D71—44no—
83C2×Dic71—22no—
84C2×Dic71—22no—
85C2×Dic71—41no—
86C2×Dic71—44no—
87C2×Dic71C2×D412yes—
88C4×D71—22no—
89C4×D71—22no—
90C4×D71—42no—
91Dic142—22no—
92Dic142—22no—
93C8×C71—21no—
94C8×C71—21no—
95C71⋊C8—22no—
96C22×C71—21no—
97C22×C71—22no—
98C22×C71—42no—
99C22×C71—42no—
100C22×C71—42no—
101C22×C71C22×D411yescharacteristic
102C2×D71—22no—
103C2×D71—22no—
104C2×D71—22no—
105C2×D71—44no—
106C4×C71—21no—
107C4×C71—42no—
108C4×C71H(5,𝔽2)11yescharacteristic
109C4×C71H(5,𝔽2)11yescharacteristic
110Dic71—22no—
111Dic71—42no—
112Dic71—42no—
113C2×C71—41no—
114C2×C71—82no—
115C2×C71#53411yescharacteristic
116C2×C71#57512yes—
117D71—42no—
118#3197—711nomaximal, Sylow
119C71#319711yescharacteristic, complemented, Sylow
120#525—712no—
121#485—712no—
122#482—712no—
123#470—711no—
124#469D7111yescharacteristic, complemented
125#468—712no—
126#452—712no—
127#449—712no—
128#407—711no—
129C2×#41—712no—
130C23×C4—711no—
131C2×Q16—711no—
132C2×SD16—712no—
133C2×D8C2×D7111yescharacteristic, complemented
134C2×M16—711no—
135C2×M16C2×D7111yescharacteristic
136#123—1422no—
137C22⋊Q8—712no—
138C22⋊Q8—712no—
139#121—712no—
140#121C2×D7112yes—
141#120—1422no—
142C4×D4—1422no—
143C4×D4—1422no—
144#117—712no—
145#107—712no—
146#106—711no—
147#106C2×D7111yescharacteristic, complemented
148#103—712no—
149#103—712no—
150#102—712no—
151#102C2×D7112yescomplemented
152C22⋊C8—712no—
153C22⋊C8—712no—
154C24—711no—
155#41—2842no—
156#41—2842no—
157C2×Q8—712no—
158C2×D4—22no—
159C2×D4—1422no—
160C2×D4—1422no—
161C2×D4C22×D7112yes—
162C22×C4—2841no—
163C22×C4—2841no—
164C22×C4—711no—
165C22×C4—712no—
166C22×C4—712no—
167C22×C4—1422no—
168C22×C4C22×D7111yescharacteristic
169Q16—2841no—
170SD16—2842no—
171D8—41no—
172M16—41no—
173M16—2841no—
174C2×C8—712no—
175C2×C8C22×D7111yescharacteristic
176C2×C8C22×D7111yescharacteristic
177C4⋊C4—712no—
178C4⋊C4—1422no—
179C4⋊C4—1422no—
180C4⋊C4C22×D7112yes—
181C22⋊C4—22no—
182C22⋊C4—1422no—
183C22⋊C4—1422no—
184C22⋊C4—1422no—
185C22⋊C4—1422no—
186C42—1422no—
187C23—22no—
188C23—2841no—
189C23—2842no—
190C23—712no—
191C23D4×D7111yescharacteristic
192Q8—1422no—
193Q8—1422no—
194D4—22no—
195D4—22no—
196D4—44no—
197D4—2844no—
198D4—2844no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—2842no—
203C2×C4—2844no—
204C2×C4—2841no—
205C2×C4—712no—
206C2×C4—1422no—
207C2×C4—1422no—
208C2×C4—1422no—
209C2×C4—1422no—
210C2×C4D4×D7111yescharacteristic
211C2×C4C23×D7111yescharacteristic
212C8—21no—
213C8—21no—
214C8—1422no—
215C22—22no—
216C22—21no—
217C22—42no—
218C22—42no—
219C22—42no—
220C22—2844no—
221C22—1422no—
222C22—1422no—
223C22—1422no—
224C22C2×D4×D7111yescharacteristic
225C4—21no—
226C4—42no—
227C4—2842no—
228C4—2842no—
229C4—1422no—
230C4#1455617311yescharacteristic
231C4#1455617411yescharacteristic
232C2—41no—
233C2—82no—
234C2—2842no—
235C2#1798655711yescharacteristic
236C2#1798679512yes—
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#469×C71C2
socleC22×C71C22×D4
hypercenter#469D71
hypocenterC71#3197
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. #17986509C2semidirect
2. #17986343C2semidirect
3. #17986322C2semidirect
4. #17986257C2semidirect
5. #17986243C2semidirect
6. #17986162C2semidirect
7. #17986141C2semidirect
8. #17985962C2semidirect
9. #469×C71C2semidirect
12. C2×Dic284C22semidirect
13. C2×#11765853C22semidirect
23. #14556061C22semidirect
24. #14556060C22semidirect
25. #14556053C22semidirect
26. #14556051C22semidirect
27. #14556049C22semidirect
30. C2×D8×C71C22semidirect
33. #106×C71C22semidirect
34. #102×C71C22semidirect
119. C71#3197semidirect
124. #469D71semidirect
133. C2×D8C2×D71semidirect
147. #106C2×D71semidirect
151. #102C2×D71semidirect