Finite Group Databasebeta

← #22599136

Subgroups — #22599136

Order 9088.

514 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 113 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#121×D71C212yesmaximal, complemented
2#17986319C212yesmaximal, complemented
3#17986316C212yesmaximal, complemented
4#17986252C211yesmaximal, characteristic, complemented
5#17986250C212yesmaximal, complemented
6#17986134C212yesmaximal, complemented
7#17986131C212yesmaximal, complemented
8#17985962C211yesmaximal, characteristic, complemented
9#469×C71C211yesmaximal, characteristic, complemented
10C2×D4×D71—22no—
11C2×D4×D71C2212yescomplemented
12C22×C4×D71C2211yescharacteristic
13C2×D568C2211yescharacteristic, complemented
14C2×#11765850C2212yescomplemented
15C2×#11765849C2211yescharacteristic, complemented
16#14556103C2212yes—
17#14556102C2212yescomplemented
18#14556101—22no—
19#14556100—22no—
20#14556099C2212yes—
21C4⋊C4×D71C2212yescomplemented
22C22⋊C4×D71—22no—
23#14556060C2211yescharacteristic, complemented
24#14556059C2212yescomplemented
25#14556050C2212yescomplemented
26#14556049C2212yescomplemented
27#14556048C2212yescomplemented
28(C2×D71)⋊C8C2212yescomplemented
29(C2×D71)⋊C8C2212yescomplemented
30C2×D8×C71C2211yescharacteristic, complemented
31C2×M16×C71C2211yescharacteristic, complemented
32#121×C71C2212yescomplemented
33#106×C71C2211yescharacteristic, complemented
34#102×C71C2212yescomplemented
35C23×D71—22no—
36C23×D71D411yescharacteristic
37C2×D284—22no—
38C2×D284C2312yes—
39C2×C22⋊D71—22no—
40C2×C22⋊D71—22no—
41C2×C22⋊D71—22no—
42D4×D71—42no—
43D4×D71—42no—
44D4×D71—48no—
45C22×Dic71D411yescharacteristic
46C2×C4×D71—22no—
47C2×C4×D71—41no—
48C2×C4×D71—41no—
49C2×C4×D71D412yes—
50C2×C4×D71C2312yes—
51D568—41no—
52#11765850—42no—
53#11765849—41no—
54C2×C71⋊C8C2312yes—
55C4⋊Dic71C2312yes—
56Dic71⋊C4—22no—
57C22⋊Dic71—22no—
58(C2×D71)⋊C4—22no—
59(C2×D71)⋊C4—22no—
60C2×D4×C71—22no—
61C2×D4×C71C2312yes—
62C22×C4×C71C2311yescharacteristic
63D8×C71—41no—
64M16×C71—41no—
65C2×C8×C71C2311yescharacteristic
66C2×C8×C71C2311yescharacteristic
67C4⋊C4×C71C2312yes—
68C22⋊C4×C71—22no—
69C23×C71—22no—
70C23×C71C2×D411yescharacteristic
71C22×D71—24no—
72C22×D71—22no—
73C22×D71—41no—
74C22×D71—44no—
75C22×D71—44no—
76C22×D71—44no—
77C22×D71C2×D412yes—
78D4×C71—22no—
79D4×C71—22no—
80D4×C71—44no—
81C2×C4×C71—21no—
82C2×C4×C71—21no—
83C2×C4×C71—22no—
84C2×C4×C71C2×D411yescharacteristic
85C2×C4×C71C2411yescharacteristic
86C22⋊D71—44no—
87C22⋊D71—44no—
88C22⋊D71—44no—
89C2×Dic71—22no—
90C2×Dic71—41no—
91C2×Dic71C2×D412yes—
92D284—22no—
93D284—22no—
94D284—44no—
95C4×D71—22no—
96C4×D71—22no—
97C4×D71—42no—
98C4×D71—44no—
99C8×C71—21no—
100C8×C71—21no—
101C71⋊C8—22no—
102C22×C71—21no—
103C22×C71—22no—
104C22×C71—42no—
105C22×C71—42no—
106C22×C71—42no—
107C22×C71C22×D411yescharacteristic
108C2×D71—22no—
109C2×D71—22no—
110C2×D71—44no—
111C2×D71—42no—
112C2×D71—42no—
113C2×D71—42no—
114C2×D71—88no—
115C2×D71AGL(1,ℤ/8)14yes—
116C4×C71—21no—
117C4×C71—42no—
118C4×C71H(5,𝔽2)11yescharacteristic
119C4×C71H(5,𝔽2)11yescharacteristic
120Dic71—22no—
121Dic71—42no—
122C2×C71—41no—
123C2×C71—82no—
124C2×C71#53411yescharacteristic
125C2×C71C2×AGL(1,ℤ/8)12yes—
126D71—24no—
127D71—82no—
128#3192—711nomaximal, Sylow
129C71#319211yescharacteristic, complemented, Sylow
130C2×#121—712no—
131#481—712no—
132#481—712no—
133#469—711no—
134#469—712no—
135#469D7111yescharacteristic, complemented
136#447—712no—
137#447—712no—
138#407—711no—
139C22×D4—712no—
140C22×D4—1422no—
141C23×C4—711no—
142C2×D8—712no—
143C2×D8—711no—
144C2×D8C2×D7111yescharacteristic, complemented
145C2×M16—711no—
146C2×M16C2×D7111yescharacteristic, complemented
147#121—712no—
148#121—712no—
149#121—712no—
150#121—1422no—
151#121—1422no—
152#121C2×D7112yescomplemented
153C2×C4⋊C4—712no—
154C2×C22⋊C4—1422no—
155#106—711no—
156#106—712no—
157#106C2×D7111yescharacteristic, complemented
158#102—712no—
159#102—712no—
160#102—712no—
161#102C2×D7112yescomplemented
162C22⋊C8—712no—
163C22⋊C8—712no—
164C24—711no—
165C24—1422no—
166C2×D4—22no—
167C2×D4—2848no—
168C2×D4—2842no—
169C2×D4—2842no—
170C2×D4—712no—
171C2×D4—1422no—
172C2×D4—1422no—
173C2×D4—1422no—
174C2×D4—1422no—
175C2×D4C22×D7112yes—
176C22×C4—2841no—
177C22×C4—2841no—
178C22×C4—711no—
179C22×C4—712no—
180C22×C4—712no—
181C22×C4—1422no—
182C22×C4C22×D7111yescharacteristic
183D8—41no—
184D8—2841no—
185D8—2842no—
186M16—41no—
187M16—2841no—
188C2×C8—712no—
189C2×C8C22×D7111yescharacteristic
190C2×C8C22×D7111yescharacteristic
191C4⋊C4—712no—
192C4⋊C4—1422no—
193C4⋊C4C22×D7112yes—
194C22⋊C4—22no—
195C22⋊C4—1422no—
196C22⋊C4—1422no—
197C22⋊C4—1422no—
198C23—22no—
199C23—2841no—
200C23—2844no—
201C23—2844no—
202C23—2844no—
203C23—712no—
204C23—1424no—
205C23—1422no—
206C23D4×D7111yescharacteristic
207D4—22no—
208D4—22no—
209D4—44no—
210D4—2844no—
211D4—2844no—
212D4—2844no—
213D4—2844no—
214D4—1422no—
215D4—1422no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—2842no—
220C2×C4—2844no—
221C2×C4—2841no—
222C2×C4—712no—
223C2×C4—1422no—
224C2×C4—1422no—
225C2×C4—1422no—
226C2×C4D4×D7111yescharacteristic
227C2×C4C23×D7111yescharacteristic
228C8—21no—
229C8—21no—
230C8—1422no—
231C22—22no—
232C22—21no—
233C22—42no—
234C22—42no—
235C22—42no—
236C22—2844no—
237C22—2842no—
238C22—2842no—
239C22—2842no—
240C22—5688no—
241C22—714no—
242C22—1422no—
243C22—1422no—
244C22C2×D4×D7111yescharacteristic
245C4—21no—
246C4—42no—
247C4—2842no—
248C4—1422no—
249C4#1455617311yescharacteristic
250C4#1455617411yescharacteristic
251C2—41no—
252C2—82no—
253C2—5682no—
254C2—1424no—
255C2#1798655711yescharacteristic
256C2AGL(1,ℤ/8)×D7112yes—
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#469×C71C2
socleC22×C71C22×D4
hypercenter#469D71
hypocenterC71#3192
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. #121×D71C2semidirect
2. #17986319C2semidirect
3. #17986316C2semidirect
4. #17986252C2semidirect
5. #17986250C2semidirect
6. #17986134C2semidirect
7. #17986131C2semidirect
8. #17985962C2semidirect
9. #469×C71C2semidirect
11. C2×D4×D71C22semidirect
13. C2×D568C22semidirect
14. C2×#11765850C22semidirect
15. C2×#11765849C22semidirect
17. #14556102C22semidirect
21. C4⋊C4×D71C22semidirect
23. #14556060C22semidirect
24. #14556059C22semidirect
25. #14556050C22semidirect
26. #14556049C22semidirect
27. #14556048C22semidirect
28. (C2×D71)⋊C8C22semidirect
29. (C2×D71)⋊C8C22semidirect
30. C2×D8×C71C22semidirect
31. C2×M16×C71C22semidirect
32. #121×C71C22semidirect
33. #106×C71C22semidirect
34. #102×C71C22semidirect
129. C71#3192semidirect
135. #469D71semidirect
144. C2×D8C2×D71semidirect
146. C2×M16C2×D71semidirect
152. #121C2×D71semidirect
157. #106C2×D71semidirect
161. #102C2×D71semidirect