Finite Group Databasebeta

← #22600579

Subgroups — #22600579

Order 9088.

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

normal subgroups only

This group has 460 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#17986715C211yesmaximal, characteristic, complemented
2D4×D284C211yesmaximal, characteristic, complemented
3#17986347C211yesmaximal, characteristic, complemented
4#17986325C212yesmaximal, complemented
5#17986253C212yesmaximal, complemented
6#17986195C211yesmaximal, characteristic, complemented
7#17986134C212yesmaximal, complemented
8C4×#11765850C211yesmaximal, characteristic, complemented
9#17986076C211yesmaximal, characteristic, complemented
10#17986002C212yesmaximal, complemented
11#476×C71C211yesmaximal, characteristic, complemented
12C22×D284—21no—
13C2×D4×D71C2212yes—
14C2×D568C2212yescomplemented
15C2×#11765850C2211yescharacteristic
16C2×#11765848C2212yescomplemented
17#14556125C2211yescharacteristic
18#14556117—22no—
19#14556103—21no—
20#14556103C2212yes—
21#14556103C2212yescomplemented
22#14556102—21no—
23#14556098—22no—
24C4×D284C2211yescharacteristic
25C4×D284C2211yescharacteristic, complemented
26#14556086—21no—
27#14556086C2212yescomplemented
28#14556061C2211yescharacteristic, complemented
29#14556050C2212yescomplemented
30#14556049C2211yescharacteristic
31#14556048C2212yescomplemented
32#14556048C2211yescharacteristic, complemented
33#14556048C2212yescomplemented
34(C2×D71)⋊C8C2212yescomplemented
35C4×C71⋊C8C2211yescharacteristic, complemented
36C4⋊Q8×C71C2211yescharacteristic, complemented
37C4×D4×C71C2211yescharacteristic
38#106×C71C2212yescomplemented
39C4⋊C8×C71C2211yescharacteristic, complemented
40#102×C71C2212yes—
41C23×D71—22no—
42C2×D284—22no—
43C2×D284—21no—
44C2×D284—22no—
45C2×D284—21no—
46C2×D284—44no—
47C2×D284D411yescharacteristic
48C2×D284C2312yes—
49C2×D284C2311yescharacteristic
50C2×C22⋊D71—22no—
51D4×D71—24no—
52D4×D71—42no—
53C2×C4×D71—21no—
54C2×C4×D71C2312yes—
55D568—42no—
56#11765850—22no—
57#11765848—42no—
58C2×C71⋊C8C2311yescharacteristic
59C2×C71⋊C8C2311yescharacteristic
60C4⋊Dic71D411yescharacteristic
61C4⋊Dic71C2311yescharacteristic
62(C2×D71)⋊C4—22no—
63(C2×D71)⋊C4—21no—
64(C2×D71)⋊C4—22no—
65(C2×D71)⋊C4—22no—
66(C2×D71)⋊C4D412yes—
67C4×Dic71—22no—
68C2×Q8×C71—21no—
69C2×D4×C71C2311yescharacteristic
70C22×C4×C71—21no—
71C2×C8×C71C2312yes—
72C4⋊C4×C71—21no—
73C4⋊C4×C71C2312yes—
74C4⋊C4×C71C2311yescharacteristic
75C22⋊C4×C71—21no—
76C42×C71C2311yescharacteristic
77C23×C71—21no—
78C22×D71—21no—
79C22×D71—22no—
80C22×D71—44no—
81C22×D71—44no—
82C22×D71—82no—
83C22×D71C2×D412yes—
84Q8×C71—81no—
85D4×C71—21no—
86D4×C71#4112yes—
87C2×C4×C71—21no—
88C2×C4×C71—22no—
89C2×C4×C71—42no—
90C2×C4×C71C2×D411yescharacteristic
91C2×C4×C71C2×D411yescharacteristic
92C2×C4×C71C2411yescharacteristic
93C22⋊D71—44no—
94C2×Dic71—21no—
95C2×Dic71C2×D412yes—
96D284—22no—
97D284—21no—
98D284—22no—
99D284—21no—
100D284—22no—
101D284—44no—
102D284—42no—
103D284—81no—
104D284—82no—
105C4×D71—22no—
106C4×D71—22no—
107C4×D71—42no—
108C8×C71—22no—
109C71⋊C8—21no—
110C71⋊C8—21no—
111C22×C71—22no—
112C22×C71—41no—
113C22×C71—41no—
114C22×C71C22×D411yescharacteristic
115C2×D71—22no—
116C2×D71—22no—
117C2×D71—22no—
118C2×D71—42no—
119C2×D71—41no—
120C2×D71—41no—
121C2×D71—42no—
122C2×D71—41no—
123C2×D71—42no—
124C2×D71—88no—
125C4×C71—21no—
126C4×C71—41no—
127C4×C71—42no—
128C4×C71AGL(1,ℤ/8)12yes—
129C4×C71C2×#4111yescharacteristic
130C4×C71H(5,𝔽2)11yescharacteristic
131Dic71—22no—
132Dic71—41no—
133C2×C71—42no—
134C2×C71#54611yescharacteristic
135C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
136C2×C71#57611yescharacteristic
137D71—42no—
138D71—81no—
139D71—82no—
140#3325—711nomaximal, Sylow
141C71#332511yescharacteristic, complemented, Sylow
142#555—711no—
143D42—711no—
144#486—711no—
145#482—712no—
146#476D7111yescharacteristic, complemented
147#469—712no—
148#459—711no—
149#447—712no—
150C4×D8—711no—
151#436—711no—
152#418—712no—
153C22×D4—712no—
154C22×D4—1421no—
155C2×D8—711no—
156C2×D8—712no—
157C2×M16—712no—
158C4⋊Q8C2×D7111yescharacteristic, complemented
159#127—711no—
160#124—1422no—
161#121—712no—
162#121—712no—
163#121—1421no—
164#121—1421no—
165#120—1422no—
166C4×D4—711no—
167C4×D4—711no—
168C4×D4C2×D7111yescharacteristic
169#117—712no—
170#117—1421no—
171#107—711no—
172#106C2×D7112yescomplemented
173C4⋊C8C2×D7111yescharacteristic, complemented
174#102—711no—
175#102—712no—
176#102—712no—
177#102—711no—
178#102—712no—
179#102C2×D7112yes—
180C22⋊C8—712no—
181C4×C8—711no—
182C24—1422no—
183C2×Q8—21no—
184C2×D4—2844no—
185C2×D4—2842no—
186C2×D4—711no—
187C2×D4—712no—
188C2×D4—711no—
189C2×D4—1422no—
190C2×D4—1422no—
191C2×D4—1422no—
192C2×D4—1424no—
193C2×D4—1421no—
194C2×D4—1421no—
195C2×D4C22×D7111yescharacteristic
196C22×C4—21no—
197C22×C4—712no—
198C22×C4—1421no—
199D8—2842no—
200D8—1422no—
201M16—2842no—
202C2×C8—711no—
203C2×C8—711no—
204C2×C8C22×D7112yes—
205C4⋊C4—21no—
206C4⋊C4—711no—
207C4⋊C4—711no—
208C4⋊C4C22×D7112yes—
209C4⋊C4C22×D7111yescharacteristic
210C22⋊C4—21no—
211C22⋊C4—712no—
212C22⋊C4—1422no—
213C22⋊C4—1422no—
214C22⋊C4—1422no—
215C22⋊C4—1421no—
216C42—1422no—
217C42C22×D7111yescharacteristic
218C23—21no—
219C23—2844no—
220C23—2844no—
221C23—5682no—
222C23—712no—
223C23—1421no—
224C23—1422no—
225Q8—81no—
226D4—21no—
227D4—2844no—
228D4—2844no—
229D4—2842no—
230D4—5681no—
231D4—5682no—
232D4—1421no—
233D4—1422no—
234D4—1421no—
235D4—1422no—
236D4—1422no—
237D4#1176586612yes—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—42no—
241C2×C4—2842no—
242C2×C4—712no—
243C2×C4—1421no—
244C2×C4—1422no—
245C2×C4—1422no—
246C2×C4D4×D7111yescharacteristic
247C2×C4D4×D7111yescharacteristic
248C2×C4C23×D7111yescharacteristic
249C8—22no—
250C8—1421no—
251C8—1421no—
252C22—22no—
253C22—41no—
254C22—41no—
255C22—2842no—
256C22—2841no—
257C22—2841no—
258C22—2842no—
259C22—2841no—
260C22—2842no—
261C22—5688no—
262C22—1422no—
263C22—1422no—
264C22—1422no—
265C22C2×D4×D7111yescharacteristic
266C4—21no—
267C4—41no—
268C4—42no—
269C4—2841no—
270C4—1422no—
271C4#1455615512yes—
272C4C2×#1176586611yescharacteristic
273C4#1455617411yescharacteristic
274C2—42no—
275C2—2842no—
276C2—5682no—
277C2—5681no—
278C2#1798664011yescharacteristic
279C2C2×#1455615511yescharacteristic
280C2#1798680311yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#476×C71C2
socleC22×C71C22×D4
hypercenter#476D71
hypocenterC71#3325
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #17986715C2semidirect
2. D4×D284C2semidirect
3. #17986347C2semidirect
4. #17986325C2semidirect
5. #17986253C2semidirect
6. #17986195C2semidirect
7. #17986134C2semidirect
8. C4×#11765850C2semidirect
9. #17986076C2semidirect
10. #17986002C2semidirect
11. #476×C71C2semidirect
14. C2×D568C22semidirect
16. C2×#11765848C22semidirect
21. #14556103C22semidirect
25. C4×D284C22semidirect
27. #14556086C22semidirect
28. #14556061C22semidirect
29. #14556050C22semidirect
31. #14556048C22semidirect
32. #14556048C22semidirect
33. #14556048C22semidirect
34. (C2×D71)⋊C8C22semidirect
35. C4×C71⋊C8C22semidirect
36. C4⋊Q8×C71C22semidirect
38. #106×C71C22semidirect
39. C4⋊C8×C71C22semidirect
141. C71#3325semidirect
146. #476D71semidirect
158. C4⋊Q8C2×D71semidirect
172. #106C2×D71semidirect
173. C4⋊C8C2×D71semidirect