Finite Group Databasebeta

← #22600583

Subgroups — #22600583

Order 9088.

460 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 123 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#17986714C211yesmaximal, characteristic, complemented
2D4×D284C211yesmaximal, characteristic, complemented
3#17986346C211yesmaximal, characteristic, complemented
4#17986324C212yesmaximal, complemented
5#17986290C211yesmaximal, characteristic, complemented
6#17986252C212yesmaximal, complemented
7#17986195C211yesmaximal, characteristic, complemented
8#17986135C212yesmaximal, complemented
9C4×D568C211yesmaximal, characteristic, complemented
10#17986005C212yesmaximal, complemented
11#436×C71C211yesmaximal, characteristic, complemented
12C22×D284C2212yes—
13C2×D4×D71—21no—
14C2×D568C2211yescharacteristic
15C2×D568C2212yescomplemented
16#14556128C2211yescharacteristic, complemented
17#14556125C2211yescharacteristic
18#14556114—22no—
19#14556103—21no—
20#14556102C2212yes—
21#14556102C2212yescomplemented
22#14556099—21no—
23#14556098—22no—
24C4×D284C2211yescharacteristic
25C4×D284C2211yescharacteristic, complemented
26#14556087C2212yescomplemented
27#14556086—21no—
28#14556062C2211yescharacteristic, complemented
29#14556060C2212yescomplemented
30#14556050C2211yescharacteristic
31#14556050C2212yes—
32#14556050C2212yescomplemented
33#14556050C2212yescomplemented
34#14556050C2211yescharacteristic, complemented
35#14556050C2212yescomplemented
36C2×M16×C71C2212yescomplemented
37C4×D4×C71C2211yescharacteristic
38C4⋊C8×C71C2211yescharacteristic, complemented
39C22⋊C8×C71C2212yescomplemented
40C4×C8×C71C2211yescharacteristic, complemented
41C23×D71—22no—
42C2×Dic142—21no—
43C2×D284—24no—
44C2×D284—21no—
45C2×D284—22no—
46C2×D284—21no—
47C2×D284—42no—
48C2×D284C2312yes—
49C2×D284C2311yescharacteristic
50C2×D284C2311yescharacteristic
51C2×C22⋊D71—22no—
52C2×C22⋊D71—22no—
53D4×D71—44no—
54C2×C4×D71—21no—
55C2×C4×D71—21no—
56D568—22no—
57D568—42no—
58C4⋊Dic71—21no—
59C4⋊Dic71C2312yes—
60C4⋊Dic71C2311yescharacteristic
61C4⋊Dic71C2311yescharacteristic
62C22⋊Dic71—22no—
63(C2×D71)⋊C4—22no—
64(C2×D71)⋊C4—21no—
65(C2×D71)⋊C4—21no—
66(C2×D71)⋊C4—22no—
67C4×Dic71—22no—
68C2×D4×C71D411yescharacteristic
69C22×C4×C71C2312yes—
70M16×C71—42no—
71C2×C8×C71C2311yescharacteristic
72C2×C8×C71C2311yescharacteristic
73C2×C8×C71C2312yes—
74C4⋊C4×C71D411yescharacteristic
75C22⋊C4×C71D412yes—
76C42×C71C2311yescharacteristic
77C23×C71C2×D412yes—
78C22×D71—21no—
79C22×D71—21no—
80C22×D71—22no—
81C22×D71—44no—
82C22×D71—44no—
83C22×D71—82no—
84D4×C71—22no—
85C2×C4×C71—22no—
86C2×C4×C71—22no—
87C2×C4×C71C2×D411yescharacteristic
88C2×C4×C71C2×D412yes—
89C2×C4×C71C2×D411yescharacteristic
90C2×C4×C71C2411yescharacteristic
91C22⋊D71—44no—
92C22⋊D71—82no—
93C2×Dic71—21no—
94C2×Dic71—22no—
95C2×Dic71—21no—
96D284—21no—
97D284—22no—
98D284—21no—
99D284—21no—
100D284—22no—
101D284—44no—
102D284—42no—
103D284—81no—
104D284#4112yes—
105C4×D71—42no—
106C4×D71—42no—
107Dic142—81no—
108C8×C71—21no—
109C8×C71—21no—
110C8×C71—22no—
111C22×C71—22no—
112C22×C71—22no—
113C22×C71—22no—
114C22×C71C22×D411yescharacteristic
115C2×D71—22no—
116C2×D71—41no—
117C2×D71—41no—
118C2×D71—42no—
119C2×D71—41no—
120C2×D71—41no—
121C2×D71—42no—
122C2×D71—42no—
123C2×D71—41no—
124C2×D71—88no—
125C4×C71—22no—
126C4×C71—21no—
127C4×C71AGL(1,ℤ/8)12yes—
128C4×C71C2×#4111yescharacteristic
129C4×C71H(5,𝔽2)11yescharacteristic
130Dic71—41no—
131Dic71—41no—
132Dic71—42no—
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#476—711no—
147#469—712no—
148#459—711no—
149#447—712no—
150C4×D8—711no—
151#436D7111yescharacteristic, complemented
152#418—712no—
153C22×D4—712no—
154C22×D4—1421no—
155C2×D8—712no—
156C2×D8—711no—
157C2×M16C2×D7112yescomplemented
158C4⋊Q8—711no—
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#106—712no—
173C4⋊C8C2×D7111yescharacteristic, complemented
174#102—711no—
175#102—712no—
176#102—712no—
177#102—712no—
178#102—711no—
179#102—712no—
180C22⋊C8C2×D7112yescomplemented
181C4×C8C2×D7111yescharacteristic, complemented
182C24—1422no—
183C2×Q8—1421no—
184C2×D4—2842no—
185C2×D4—2844no—
186C2×D4—711no—
187C2×D4—712no—
188C2×D4—711no—
189C2×D4—1424no—
190C2×D4—1422no—
191C2×D4—1422no—
192C2×D4—1422no—
193C2×D4—1421no—
194C2×D4—1421no—
195C2×D4D28411yescharacteristic
196C22×C4—1421no—
197C22×C4—1421no—
198C22×C4C22×D7112yes—
199D8—2842no—
200D8—1422no—
201M16—42no—
202C2×C8C22×D7111yescharacteristic
203C2×C8C22×D7111yescharacteristic
204C2×C8C22×D7112yes—
205C4⋊C4—712no—
206C4⋊C4—711no—
207C4⋊C4—711no—
208C4⋊C4—1421no—
209C4⋊C4D28411yescharacteristic
210C22⋊C4—1422no—
211C22⋊C4—1422no—
212C22⋊C4—1421no—
213C22⋊C4—1421no—
214C22⋊C4—1422no—
215C22⋊C4D28412yes—
216C42—1422no—
217C42C22×D7111yescharacteristic
218C23—2844no—
219C23—2844no—
220C23—5682no—
221C23—1421no—
222C23—1422no—
223C23—1421no—
224C23C2×D28412yes—
225Q8—5681no—
226D4—22no—
227D4—2844no—
228D4—2844no—
229D4—2842no—
230D4—5681no—
231D4—5682no—
232D4—712no—
233D4—1421no—
234D4—1422no—
235D4—1421no—
236D4—1422no—
237D4—1421no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—2842no—
241C2×C4—2842no—
242C2×C4—1421no—
243C2×C4—1422no—
244C2×C4—1421no—
245C2×C4C2×D28412yes—
246C2×C4C2×D28411yescharacteristic
247C2×C4C2×D28411yescharacteristic
248C2×C4C23×D7111yescharacteristic
249C8—21no—
250C8—21no—
251C8—22no—
252C22—22no—
253C22—22no—
254C22—22no—
255C22—2841no—
256C22—2841no—
257C22—2842no—
258C22—2841no—
259C22—2841no—
260C22—2842no—
261C22—2841no—
262C22—2842no—
263C22—5688no—
264C22—1422no—
265C22C22×D28411yescharacteristic
266C4—22no—
267C4—21no—
268C4—2841no—
269C4—2841no—
270C4—2842no—
271C4#1455615612yes—
272C4#41×D7111yescharacteristic
273C4#1455617311yescharacteristic
274C2—42no—
275C2—2842no—
276C2—5682no—
277C2—5681no—
278C2#1798664411yescharacteristic
279C2C2×#1455615611yescharacteristic
280C2#1798680411yescharacteristic
subgroupG/H
centerC22C22×D284
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#436×C71C2
socleC22×C71C22×D4
hypercenter#436D71
hypocenterC71#3325
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C4×D4 ◁ #436

Complements

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

NH ≅ G/Ntype
1. #17986714C2semidirect
2. D4×D284C2semidirect
3. #17986346C2semidirect
4. #17986324C2semidirect
5. #17986290C2semidirect
6. #17986252C2semidirect
7. #17986195C2semidirect
8. #17986135C2semidirect
9. C4×D568C2semidirect
10. #17986005C2semidirect
11. #436×C71C2semidirect
15. C2×D568C22semidirect
16. #14556128C22semidirect
21. #14556102C22semidirect
25. C4×D284C22semidirect
26. #14556087C22semidirect
28. #14556062C22semidirect
29. #14556060C22semidirect
32. #14556050C22semidirect
33. #14556050C22semidirect
34. #14556050C22semidirect
35. #14556050C22semidirect
36. C2×M16×C71C22semidirect
38. C4⋊C8×C71C22semidirect
39. C22⋊C8×C71C22semidirect
40. C4×C8×C71C22semidirect
141. C71#3325semidirect
151. #436D71semidirect
157. C2×M16C2×D71semidirect
173. C4⋊C8C2×D71semidirect
180. C22⋊C8C2×D71semidirect
181. C4×C8C2×D71semidirect