Finite Group Databasebeta

← #22591512

Subgroups — #22591512

Order 9088.

520 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 173 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
1C2×#14556117C211yesmaximal, characteristic
2#17985906C211yesmaximal, characteristic, complemented
3#17985890C212yesmaximal, complemented
4#17985888C212yesmaximal, complemented
5#17985877C212yesmaximal
6#17985873C212yesmaximal, complemented
7#17985870C212yesmaximal, complemented
8#17985863C212yesmaximal, complemented
9#391×C71C211yesmaximal, characteristic, complemented
10C22×D284C2211yescharacteristic
11C22×C4×D71C2212yes—
12#14556117C418yescomplemented
13C2×C4⋊Dic71C2211yescharacteristic
14C2×Dic71⋊C4C2212yes—
15C2×Dic71⋊C4C2212yes—
16C2×(C2×D71)⋊C4C2212yes—
17C2×(C2×D71)⋊C4C2212yes—
18C2×(C2×D71)⋊C4C2212yes—
19C2×(C2×D71)⋊C4C2212yes—
20C2×(C2×D71)⋊C4C2212yes—
21C2×C4×Dic71C2211yescharacteristic
22C2×C4×Dic71C2211yescharacteristic
23C71⋊#95C2212yes—
24C71⋊#95C2212yes—
25C71⋊#95C2212yes—
26C71⋊#95C2212yes—
27C22×Q8×C71C2211yescharacteristic
28C2×C4⋊C4×C71C2211yescharacteristic
29C2×C42×C71C2211yescharacteristic
30#95×C71C2212yes—
31#95×C71C2212yes—
32C23×D71C2312yes—
33C2×D284—41no—
34C2×D284—41no—
35C2×D284C2×C414yes—
36C22×Dic71C2312yes—
37C22×Dic71C2312yes—
38C22×Dic71C2312yes—
39C2×C4×D71—42no—
40C2×C4×D71—42no—
41C2×C4×D71—42no—
42C4⋊Dic71—41no—
43Dic71⋊C4—42no—
44Dic71⋊C4—42no—
45(C2×D71)⋊C4—42no—
46(C2×D71)⋊C4—42no—
47(C2×D71)⋊C4—42no—
48(C2×D71)⋊C4C2×C418yescomplemented
49(C2×D71)⋊C4C2×C418yescomplemented
50C4×Dic71—41no—
51C4×Dic71C2×C414yescomplemented
52C2×Q8×C71—41no—
53C2×Q8×C71—41no—
54C2×Q8×C71C2×C414yescomplemented
55C22×C4×C71C2312yes—
56C22×C4×C71C2312yes—
57C22×C4×C71C2311yescharacteristic
58C22×C4×C71C2311yescharacteristic
59C22×C4×C71C2311yescharacteristic
60C4⋊C4×C71—41no—
61C42×C71—41no—
62C23×C71C2411yescharacteristic
63C22×D71—22no—
64C22×D71—22no—
65C22×D71—22no—
66C22×D71—22no—
67C22×D71—22no—
68C22×D71—22no—
69C22×D71C22×C414yes—
70Q8×C71—44no—
71C2×C4×C71—21no—
72C2×C4×C71—21no—
73C2×C4×C71—21no—
74C2×C4×C71—21no—
75C2×C4×C71—21no—
76C2×C4×C71—21no—
77C2×C4×C71—22no—
78C2×C4×C71—22no—
79C2×C4×C71—22no—
80C2×C4×C71—22no—
81C2×C4×C71—22no—
82C2×C4×C71—22no—
83C2×C4×C71C22×C412yes—
84C2×C4×C71C22×C412yes—
85C2×C4×C71C22×C412yes—
86C2×Dic71—22no—
87C2×Dic71—22no—
88C2×Dic71—22no—
89C2×Dic71—22no—
90C2×Dic71—22no—
91C2×Dic71—22no—
92C2×Dic71—22no—
93C2×Dic71—22no—
94C2×Dic71C22×C414yes—
95D284—44no—
96C4×D71—48no—
97C22×C71C23×C411yescharacteristic
98C22×C71H(5,𝔽2)11yescharacteristic
99C22×C71H(5,𝔽2)11yescharacteristic
100C22×C71H(5,𝔽2)12yes—
101C22×C712−1+412yes—
102C2×D71—24no—
103C2×D71—24no—
104C2×D71—24no—
105C2×D71—42no—
106C2×D71—42no—
107C2×D71—42no—
108C2×D71—42no—
109C4×C71—22no—
110C4×C71—22no—
111C4×C71—22no—
112C4×C71—42no—
113C4×C71—42no—
114Dic71—24no—
115Dic71—42no—
116Dic71—42no—
117C2×C71#51811yescharacteristic
118C2×C71#52012yes—
119C2×C71H(3,𝔽4)11yescharacteristic
120C2×C71#56212yes—
121C2×C71#56311yescharacteristic
122D71—44no—
123#2379—711nomaximal, Sylow
124C71#237911yescharacteristic, complemented, Sylow
125C2×#124—711no—
126#391—711no—
127#391D7111yescharacteristic, complemented
128#388—712no—
129#388—712no—
130#386—712no—
131#385—712no—
132#385—712no—
133#382—712no—
134C22×Q8C2×D7111yescharacteristic
135C22×D4—711no—
136C23×C4—712no—
137#124—718no—
138C2×C4⋊C4—712no—
139C2×C4⋊C4—712no—
140C2×C4⋊C4—711no—
141C2×C4⋊C4C2×D7111yescharacteristic
142C2×C22⋊C4—712no—
143C2×C22⋊C4—712no—
144C2×C22⋊C4—712no—
145C2×C22⋊C4—712no—
146C2×C22⋊C4—712no—
147C2×C42—711no—
148C2×C42—711no—
149C2×C42C2×D7111yescharacteristic
150#95—712no—
151#95—712no—
152#95—712no—
153#95—712no—
154#95C2×D7112yes—
155#95C2×D7112yes—
156C24—712no—
157C2×Q8—41no—
158C2×Q8—41no—
159C2×Q8C4×D7114yescomplemented
160C2×D4—2841no—
161C2×D4—2841no—
162C2×D4—714no—
163C22×C4—2842no—
164C22×C4—2842no—
165C22×C4—2842no—
166C22×C4—712no—
167C22×C4—712no—
168C22×C4—712no—
169C22×C4C22×D7112yes—
170C22×C4C22×D7112yes—
171C22×C4C22×D7111yescharacteristic
172C22×C4C22×D7111yescharacteristic
173C22×C4C22×D7111yescharacteristic
174C4⋊C4—41no—
175C4⋊C4—2842no—
176C4⋊C4—2842no—
177C4⋊C4—2841no—
178C22⋊C4—2842no—
179C22⋊C4—2842no—
180C22⋊C4—2842no—
181C22⋊C4—718no—
182C22⋊C4—718no—
183C42—41no—
184C42—2841no—
185C42—714no—
186C23—714no—
187C23—1422no—
188C23—1422no—
189C23—1422no—
190C23—1422no—
191C23—1422no—
192C23—1422no—
193C23C23×D7111yescharacteristic
194Q8—44no—
195D4—2844no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—2848no—
209C2×C4—714no—
210C2×C4—1422no—
211C2×C4—1422no—
212C2×C4—1422no—
213C2×C4—1422no—
214C2×C4—1422no—
215C2×C4—1422no—
216C2×C4—1422no—
217C2×C4—1422no—
218C2×C4C2×C4×D7112yes—
219C2×C4C2×C4×D7112yes—
220C2×C4C2×C4×D7112yes—
221C22—2842no—
222C22—2842no—
223C22—2842no—
224C22—2842no—
225C22—1424no—
226C22—1424no—
227C22—1424no—
228C22C22×C4×D7111yescharacteristic
229C22#1455617311yescharacteristic
230C22#1455617311yescharacteristic
231C22#1455617412yes—
232C22#1455617612yes—
233C4—22no—
234C4—22no—
235C4—22no—
236C4—42no—
237C4—42no—
238C4—2842no—
239C4—2842no—
240C4—1424no—
241C2—2844no—
242C2#1798647211yescharacteristic
243C2#1798648112yes—
244C2#1798674111yescharacteristic
245C2#1798674512yes—
246C2#1798674811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D71
commutatorC22×C71C23×C4
Fitting#391×C71C2
socleC23×C71C24
hypercenter#391D71
hypocenterC71#2379
LCSG ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
2. #17985906C2semidirect
3. #17985890C2semidirect
4. #17985888C2semidirect
6. #17985873C2semidirect
7. #17985870C2semidirect
8. #17985863C2semidirect
9. #391×C71C2semidirect
12. #14556117C4semidirect
48. (C2×D71)⋊C4C2×C4semidirect
49. (C2×D71)⋊C4C2×C4semidirect
51. C4×Dic71C2×C4semidirect
54. C2×Q8×C71C2×C4semidirect
124. C71#2379semidirect
127. #391D71semidirect
159. C2×Q8C4×D71semidirect