Finite Group Databasebeta

← #22599088

Subgroups — #22599088

Order 9088.

520 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 137 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×#14556102C211yesmaximal, characteristic, complemented
2#17986312C212yesmaximal, complemented
3#17986310C212yesmaximal, complemented
4#17986233C214yesmaximal, complemented
5#17986133C212yesmaximal, complemented
6#17986132C212yesmaximal, complemented
7C2×C22⋊(C71⋊C8)C211yesmaximal, characteristic, complemented
8C2×#121×C71C211yesmaximal, characteristic, complemented
9C22×D284C2211yescharacteristic, complemented
10C22×C22⋊D71—21no—
11C2×#11765850C2214yescomplemented
12C22×C71⋊C8C2212yescomplemented
13#14556102—21no—
14#14556102—21no—
15#14556102C2212yes—
16#14556102C2212yescomplemented
17C2×C4⋊Dic71C2211yescharacteristic, complemented
18C2×(C2×D71)⋊C4—21no—
19#14556061C2214yescomplemented
20#14556049C2214yescomplemented
21#14556048C2214yescomplemented
22C22⋊(C71⋊C8)C2212yescomplemented
23C22⋊(C71⋊C8)C2212yescomplemented
24C22×D4×C71—21no—
25C22×D4×C71C2211yescharacteristic, complemented
26C23×C4×C71C2211yescharacteristic
27#121×C71—21no—
28#121×C71—21no—
29#121×C71C2212yes—
30#121×C71C2212yescomplemented
31C2×C4⋊C4×C71C2211yescharacteristic, complemented
32C2×C22⋊C4×C71—21no—
33C23×D71—21no—
34C2×D284—21no—
35C2×D284—42no—
36C2×D284C2312yes—
37C2×C22⋊D71—22no—
38C2×C22⋊D71—22no—
39C2×C22⋊D71—44no—
40C22×Dic71—21no—
41#11765850—44no—
42C2×C71⋊C8—42no—
43C2×C71⋊C8C2314yes—
44C4⋊Dic71—21no—
45C4⋊Dic71C2312yes—
46(C2×D71)⋊C4—22no—
47(C2×D71)⋊C4—22no—
48C24×C71—21no—
49C24×C71D411yescharacteristic
50C2×D4×C71—22no—
51C2×D4×C71—22no—
52C2×D4×C71—21no—
53C2×D4×C71—42no—
54C2×D4×C71—44no—
55C2×D4×C71C2312yes—
56C22×C4×C71—21no—
57C22×C4×C71—42no—
58C22×C4×C71D411yescharacteristic
59C22×C4×C71D411yescharacteristic
60C22×C4×C71D411yescharacteristic
61C22×C4×C71C2311yescharacteristic
62C22×C4×C71C2311yescharacteristic
63C22×C4×C71C2311yescharacteristic
64C4⋊C4×C71—21no—
65C4⋊C4×C71C2312yes—
66C22⋊C4×C71—22no—
67C22⋊C4×C71—22no—
68C23×C71—22no—
69C23×C71—41no—
70C23×C71—41no—
71C23×C71—42no—
72C23×C71—42no—
73C23×C71—42no—
74C23×C71D814yescomplemented
75C23×C71C2×D411yescharacteristic
76C23×C71C2×D411yescharacteristic
77C23×C71C2×D411yescharacteristic
78C22×D71—22no—
79C22×D71—42no—
80C22×D71—42no—
81C22×D71—42no—
82D4×C71—24no—
83D4×C71—42no—
84D4×C71—44no—
85D4×C71—44no—
86C2×C4×C71—22no—
87C2×C4×C71—22no—
88C2×C4×C71—22no—
89C2×C4×C71—22no—
90C2×C4×C71—41no—
91C2×C4×C71—41no—
92C2×C4×C71—41no—
93C2×C4×C71—42no—
94C2×C4×C71C2×D411yescharacteristic
95C2×C4×C71C2×D411yescharacteristic
96C2×C4×C71C2×D411yescharacteristic
97C2×C4×C71C2411yescharacteristic
98C22⋊D71—44no—
99C22⋊D71—44no—
100C2×Dic71—22no—
101C2×Dic71—42no—
102D284—24no—
103D284—42no—
104C71⋊C8—24no—
105C22×C71—21no—
106C22×C71—21no—
107C22×C71—21no—
108C22×C71—21no—
109C22×C71—21no—
110C22×C71—21no—
111C22×C71—44no—
112C22×C71—42no—
113C22×C71—42no—
114C22×C71—42no—
115C22×C71—84no—
116C22×C71C2×D812yes—
117C22×C71C2×D812yes—
118C22×C71C2×D812yes—
119C22×C71C22×D411yescharacteristic
120C2×D71—42no—
121C2×D71—42no—
122C2×D71—42no—
123C2×D71—84no—
124C4×C71—21no—
125C4×C71—21no—
126C4×C71—21no—
127C4×C71—42no—
128C4×C71H(5,𝔽2)12yes—
129Dic71—42no—
130C2×C71—22no—
131C2×C71—22no—
132C2×C71—22no—
133C2×C71—82no—
134C2×C71#53411yescharacteristic
135C2×C71C22×D811yescharacteristic
136C2×C71#57511yescharacteristic
137D71—82no—
138#3186—711nomaximal, Sylow
139C71#318611yescharacteristic, complemented, Sylow
140C2×#121—711no—
141C2×#121D7111yescharacteristic, complemented
142#480—712no—
143#480—712no—
144#466—714no—
145#447—712no—
146#447—712no—
147C2×C22⋊C8—711no—
148C22×D4—21no—
149C22×D4—711no—
150C22×D4—1421no—
151C22×D4C2×D7111yescharacteristic, complemented
152C23×C4C2×D7111yescharacteristic
153C2×D8—714no—
154C22×C8—712no—
155#121—21no—
156#121—21no—
157#121—712no—
158#121—712no—
159#121—1421no—
160#121—1421no—
161#121C2×D7112yes—
162#121C2×D7112yescomplemented
163C2×C4⋊C4—711no—
164C2×C4⋊C4C2×D7111yescharacteristic, complemented
165C2×C22⋊C4—21no—
166C2×C22⋊C4—1421no—
167#107—714no—
168#102—714no—
169#102—714no—
170C22⋊C8—712no—
171C22⋊C8—712no—
172C24—21no—
173C24—1421no—
174C24C22⋊D7111yescharacteristic
175C2×D4—22no—
176C2×D4—22no—
177C2×D4—21no—
178C2×D4—42no—
179C2×D4—44no—
180C2×D4—2842no—
181C2×D4—2844no—
182C2×D4—712no—
183C2×D4—1422no—
184C2×D4—1422no—
185C2×D4—1421no—
186C2×D4C22×D7112yes—
187C22×C4—21no—
188C22×C4—42no—
189C22×C4—1421no—
190C22×C4C22⋊D7111yescharacteristic
191C22×C4C22⋊D7111yescharacteristic
192C22×C4C22⋊D7111yescharacteristic
193C22×C4C22×D7111yescharacteristic
194C22×C4C22×D7111yescharacteristic
195C22×C4C22×D7111yescharacteristic
196D8—2844no—
197C2×C8—2842no—
198C2×C8—714no—
199C4⋊C4—21no—
200C4⋊C4—712no—
201C4⋊C4—1421no—
202C4⋊C4C22×D7112yes—
203C22⋊C4—22no—
204C22⋊C4—22no—
205C22⋊C4—1422no—
206C22⋊C4—1422no—
207C23—22no—
208C23—41no—
209C23—41no—
210C23—42no—
211C23—42no—
212C23—42no—
213C23—2842no—
214C23—2842no—
215C23—2842no—
216C23—1422no—
217C23#1176585014yescomplemented
218C23C2×C22⋊D7111yescharacteristic
219C23C2×C22⋊D7111yescharacteristic
220C23C2×C22⋊D7111yescharacteristic
221D4—24no—
222D4—42no—
223D4—44no—
224D4—44no—
225D4—2842no—
226D4—2844no—
227D4—2844no—
228D4—1424no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—42no—
237C2×C4—2842no—
238C2×C4—1422no—
239C2×C4C2×C22⋊D7111yescharacteristic
240C2×C4C2×C22⋊D7111yescharacteristic
241C2×C4C2×C22⋊D7111yescharacteristic
242C2×C4C23×D7111yescharacteristic
243C8—1424no—
244C22—21no—
245C22—21no—
246C22—21no—
247C22—21no—
248C22—21no—
249C22—21no—
250C22—44no—
251C22—42no—
252C22—42no—
253C22—42no—
254C22—84no—
255C22—2842no—
256C22—2842no—
257C22—2842no—
258C22—5684no—
259C22C2×#1176585012yes—
260C22C2×#1176585012yes—
261C22C2×#1176585012yes—
262C22C22×C22⋊D7111yescharacteristic
263C4—21no—
264C4—21no—
265C4—21no—
266C4—42no—
267C4—2842no—
268C4#1455617312yes—
269C2—22no—
270C2—22no—
271C2—22no—
272C2—82no—
273C2—5682no—
274C2#1798655811yescharacteristic
275C2C22×#1176585011yescharacteristic
276C2#1798679711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
FittingC2×#121×C71C2
socleC22×C71C22×D4
hypercenterC2×#121D71
hypocenterC71#3186
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#14556102C2semidirect
2. #17986312C2semidirect
3. #17986310C2semidirect
4. #17986233C2semidirect
5. #17986133C2semidirect
6. #17986132C2semidirect
7. C2×C22⋊(C71⋊C8)C2semidirect
8. C2×#121×C71C2semidirect
9. C22×D284C22semidirect
11. C2×#11765850C22semidirect
12. C22×C71⋊C8C22semidirect
16. #14556102C22semidirect
17. C2×C4⋊Dic71C22semidirect
19. #14556061C22semidirect
20. #14556049C22semidirect
21. #14556048C22semidirect
22. C22⋊(C71⋊C8)C22semidirect
23. C22⋊(C71⋊C8)C22semidirect
25. C22×D4×C71C22semidirect
30. #121×C71C22semidirect
31. C2×C4⋊C4×C71C22semidirect
74. C23×C71D8semidirect
139. C71#3186semidirect
141. C2×#121D71semidirect
151. C22×D4C2×D71semidirect
162. #121C2×D71semidirect
164. C2×C4⋊C4C2×D71semidirect
217. C23#11765850semidirect