Finite Group Databasebeta

← #22600152

Subgroups — #22600152

Order 9088.

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

normal subgroups only

This group has 494 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#17986652C211yesmaximal, characteristic, complemented
2#17986634C211yesmaximal, characteristic, complemented
3#17986311C212yesmaximal, complemented
4#17986276C211yesmaximal, characteristic, complemented
5#17986234C212yesmaximal, complemented
6#17986191C211yesmaximal, characteristic, complemented
7#17986134C212yesmaximal, complemented
8C4×D568C211yesmaximal, characteristic, complemented
9C8×D284C211yesmaximal, characteristic, complemented
10#102×D71C212yesmaximal, complemented
11#486×C71C211yesmaximal, characteristic, complemented
12C2×#11765866—21no—
13C2×D4×D71—21no—
14C2×D4×D71C2212yes—
15C2×D568C2211yescharacteristic
16C2×#11765850C2212yescomplemented
17C2×C8×D71C2212yescomplemented
18#14556113—22no—
19#14556104—21no—
20#14556103C2212yescomplemented
21#14556099—21no—
22#14556099C2212yes—
23#14556096—22no—
24C4×D284C2211yescharacteristic
25C4×D284C2211yescharacteristic
26C4×D284C2211yescharacteristic, complemented
27C4⋊C4×D71C2212yescomplemented
28#14556062C2211yescharacteristic, complemented
29#14556061C2212yescomplemented
30C4⋊(C71⋊C8)C2211yescharacteristic, complemented
31#14556050C2211yescharacteristic
32#14556050C2211yescharacteristic, complemented
33#14556049C2212yescomplemented
34#14556048C2212yes—
35(C2×D71)⋊C8C2212yescomplemented
36C4⋊Q8×C71C2211yescharacteristic, complemented
37#127×C71C2211yescharacteristic
38#102×C71C2212yescomplemented
39#102×C71C2212yescomplemented
40C4×C8×C71C2211yescharacteristic, complemented
41C23×D71—22no—
42#11765866—44no—
43C2×D284—22no—
44C2×D284D411yescharacteristic
45C2×D284C2311yescharacteristic
46C2×D284C2311yescharacteristic
47C2×C22⋊D71—22no—
48C2×C22⋊D71—22no—
49D4×D71—24no—
50D4×D71—42no—
51D4×D71—44no—
52C2×C4×D71—22no—
53C2×C4×D71—21no—
54C2×C4×D71—21no—
55C2×C4×D71C2312yes—
56D568—22no—
57#11765850—42no—
58C2×C71⋊C8C2312yes—
59C8×D71—42no—
60C4⋊Dic71D411yescharacteristic
61C4⋊Dic71C2311yescharacteristic
62C4⋊Dic71C2311yescharacteristic
63Dic71⋊C4—22no—
64C22⋊Dic71—22no—
65(C2×D71)⋊C4—21no—
66(C2×D71)⋊C4—21no—
67(C2×D71)⋊C4—22no—
68(C2×D71)⋊C4D412yes—
69C2×Q8×C71—21no—
70C2×D4×C71—21no—
71C2×D4×C71—21no—
72C2×D4×C71C2312yes—
73C2×C8×C71C2311yescharacteristic
74C2×C8×C71C2311yescharacteristic
75C4⋊C4×C71—21no—
76C4⋊C4×C71C2312yes—
77C42×C71C2311yescharacteristic
78C23×C71—22no—
79C22×D71—21no—
80C22×D71—21no—
81C22×D71—44no—
82C22×D71—44no—
83C22×D71—82no—
84C22×D71C2×D412yes—
85Q8×C71—42no—
86D4×C71—22no—
87D4×C71—22no—
88D4×C71—42no—
89D4×C71—81no—
90C2×C4×C71—22no—
91C2×C4×C71C2×D411yescharacteristic
92C2×C4×C71C2×D411yescharacteristic
93C2×C4×C71C2411yescharacteristic
94C22⋊D71—44no—
95C22⋊D71—44no—
96C2×Dic71—21no—
97C2×Dic71—21no—
98C2×Dic71C2×D412yes—
99D284—21no—
100D284—21no—
101D284—21no—
102D284—44no—
103D284D814yescomplemented
104D284#4112yescomplemented
105C4×D71—22no—
106C4×D71—22no—
107C4×D71—44no—
108C4×D71—42no—
109C4×D71—42no—
110C8×C71—21no—
111C8×C71—21no—
112C71⋊C8—22no—
113C22×C71—42no—
114C22×C71—42no—
115C22×C71—42no—
116C22×C71C22×D411yescharacteristic
117C2×D71—22no—
118C2×D71—22no—
119C2×D71—22no—
120C2×D71—41no—
121C2×D71—41no—
122C2×D71—41no—
123C2×D71—41no—
124C2×D71—41no—
125C2×D71—88no—
126C2×D71C2×D814yes—
127C4×C71—21no—
128C4×C71—42no—
129C4×C71C2×D812yes—
130C4×C71C2×#4111yescharacteristic
131C4×C71H(5,𝔽2)11yescharacteristic
132Dic71—22no—
133Dic71—41no—
134Dic71—41no—
135C2×C71—82no—
136C2×C71#54611yescharacteristic
137C2×C71C22×D811yescharacteristic
138C2×C71#57711yescharacteristic
139D71—24no—
140D71—42no—
141D71—81no—
142#3294—711nomaximal, Sylow
143C71#329411yescharacteristic, complemented, Sylow
144#547—711no—
145D42—711no—
146#486D7111yescharacteristic, complemented
147#480—712no—
148#474—711no—
149#466—712no—
150#459—711no—
151#447—712no—
152C4×D8—711no—
153C8×D4—711no—
154C2×#102—712no—
155C2×#41—1421no—
156C22×D4—712no—
157C22×D4—1421no—
158C2×D8—712no—
159C2×D8—711no—
160C22×C8—712no—
161C4⋊Q8C2×D7111yescharacteristic, complemented
162#127C2×D7111yescharacteristic
163#123—1422no—
164C22⋊Q8—1421no—
165#121—712no—
166#121—712no—
167#121—1421no—
168#120—1422no—
169C4×D4—711no—
170C4×D4—711no—
171C4×D4—711no—
172C2×C4⋊C4—712no—
173#107—711no—
174#107—712no—
175C4⋊C8—711no—
176#102—712no—
177#102—711no—
178#102—711no—
179#102—712no—
180#102C2×D7112yescomplemented
181#102C2×D7112yescomplemented
182C22⋊C8—712no—
183C4×C8C2×D7111yescharacteristic, complemented
184C24—1422no—
185#41—2844no—
186C2×Q8—21no—
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—2844no—
190C2×D4—2842no—
191C2×D4—711no—
192C2×D4—711no—
193C2×D4—711no—
194C2×D4—1422no—
195C2×D4—1422no—
196C2×D4—1424no—
197C2×D4—1422no—
198C2×D4C22×D7112yes—
199C22×C4—712no—
200C22×C4—1422no—
201C22×C4—1421no—
202C22×C4—1421no—
203D8—2842no—
204D8—1422no—
205C2×C8—2842no—
206C2×C8—712no—
207C2×C8C22×D7111yescharacteristic
208C2×C8C22×D7111yescharacteristic
209C4⋊C4—21no—
210C4⋊C4—711no—
211C4⋊C4—711no—
212C4⋊C4—711no—
213C4⋊C4—1422no—
214C4⋊C4C22×D7112yes—
215C22⋊C4—712no—
216C22⋊C4—1422no—
217C22⋊C4—1422no—
218C22⋊C4—1421no—
219C22⋊C4—1421no—
220C42C22×D7111yescharacteristic
221C23—22no—
222C23—2844no—
223C23—2844no—
224C23—5682no—
225C23—712no—
226C23—1421no—
227C23—1421no—
228Q8—42no—
229D4—22no—
230D4—22no—
231D4—42no—
232D4—81no—
233D4—2844no—
234D4—2844no—
235D4—2844no—
236D4—712no—
237D4—714no—
238D4—1421no—
239D4—1421no—
240D4—1421no—
241C2×C4—22no—
242C2×C4—2844no—
243C2×C4—2842no—
244C2×C4—2842no—
245C2×C4—712no—
246C2×C4—1421no—
247C2×C4—1421no—
248C2×C4—1422no—
249C2×C4—1422no—
250C2×C4D4×D7111yescharacteristic
251C2×C4D4×D7111yescharacteristic
252C2×C4C23×D7111yescharacteristic
253C8—21no—
254C8—21no—
255C8—1422no—
256C22—42no—
257C22—42no—
258C22—42no—
259C22—2841no—
260C22—2841no—
261C22—2841no—
262C22—2841no—
263C22—2841no—
264C22—5688no—
265C22—714no—
266C22—1422no—
267C22—1422no—
268C22—1422no—
269C22C2×D4×D7111yescharacteristic
270C4—21no—
271C4—42no—
272C4—2841no—
273C4—2841no—
274C4—1422no—
275C4D8×D7112yes—
276C4#41×D7111yescharacteristic
277C4#1455617411yescharacteristic
278C2—82no—
279C2—2842no—
280C2—5681no—
281C2—1424no—
282C2#1798664211yescharacteristic
283C2C2×D8×D7111yescharacteristic
284C2#1798680511yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#486×C71C2
socleC22×C71C22×D4
hypercenter#486D71
hypocenterC71#3294
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #17986652C2semidirect
2. #17986634C2semidirect
3. #17986311C2semidirect
4. #17986276C2semidirect
5. #17986234C2semidirect
6. #17986191C2semidirect
7. #17986134C2semidirect
8. C4×D568C2semidirect
9. C8×D284C2semidirect
10. #102×D71C2semidirect
11. #486×C71C2semidirect
16. C2×#11765850C22semidirect
17. C2×C8×D71C22semidirect
20. #14556103C22semidirect
26. C4×D284C22semidirect
27. C4⋊C4×D71C22semidirect
28. #14556062C22semidirect
29. #14556061C22semidirect
30. C4⋊(C71⋊C8)C22semidirect
32. #14556050C22semidirect
33. #14556049C22semidirect
35. (C2×D71)⋊C8C22semidirect
36. C4⋊Q8×C71C22semidirect
38. #102×C71C22semidirect
39. #102×C71C22semidirect
40. C4×C8×C71C22semidirect
103. D284D8semidirect
104. D284#41semidirect
143. C71#3294semidirect
146. #486D71semidirect
161. C4⋊Q8C2×D71semidirect
180. #102C2×D71semidirect
181. #102C2×D71semidirect
183. C4×C8C2×D71semidirect