Finite Group Databasebeta

← #22598711

Subgroups — #22598711

Order 9088.

484 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 125 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#17986554C211yesmaximal, characteristic, complemented
2#127×D71C211yesmaximal, characteristic
3#17986362C214yesmaximal, complemented
4#17986347C211yesmaximal, characteristic, complemented
5#17986346C211yesmaximal, characteristic, complemented
6#17986345C211yesmaximal, characteristic, complemented
7#17986002C214yesmaximal, complemented
8C4×#11765848C211yesmaximal, characteristic, complemented
9#486×C71C211yesmaximal, characteristic, complemented
10C2×D4×D71—21no—
11C2×D4×D71—21no—
12C2×D4×D71C2212yes—
13C2×#11765848C2212yes—
14#14556128C2211yescharacteristic, complemented
15#14556125C2211yescharacteristic
16#14556124—21no—
17#14556124—21no—
18#14556124C2212yes—
19#14556119C2212yescomplemented
20#14556108—21no—
21#14556104—21no—
22Q8×Dic71—21no—
23#14556093—21no—
24#14556086C2212yes—
25C42×D71C2211yescharacteristic
26#14556050C2214yescomplemented
27#14556049C2214yescomplemented
28#14556048C2214yescomplemented
29#14556038C2212yescomplemented
30C4×C71⋊C8C2211yescharacteristic, complemented
31C4⋊Q8×C71C2211yescharacteristic, complemented
32#127×C71C2211yescharacteristic
33#102×C71C2214yescomplemented
34C4×C8×C71C2211yescharacteristic, complemented
35C23×D71—22no—
36C2×Dic142—21no—
37C2×D284—21no—
38C2×D284—21no—
39C2×D284C2312yes—
40C2×C22⋊D71—22no—
41C2×C22⋊D71—22no—
42C2×C22⋊D71—22no—
43D4×D71—28no—
44D4×D71—44no—
45D4×D71—44no—
46C2×C4×D71D411yescharacteristic
47C2×C4×D71D411yescharacteristic
48C2×C4×D71C2311yescharacteristic
49#11765848—24no—
50C2×C71⋊C8C2312yes—
51C4⋊Dic71—21no—
52C4⋊Dic71—22no—
53C4⋊Dic71C2312yes—
54Dic71⋊C4—22no—
55(C2×D71)⋊C4—22no—
56C4×Dic71—22no—
57C4×Dic71D411yescharacteristic
58C4×Dic71D411yescharacteristic
59C4×Dic71C2311yescharacteristic
60C2×Q8×C71—21no—
61C2×D4×C71—21no—
62C2×D4×C71—21no—
63C2×D4×C71C2312yes—
64C2×C8×C71C2312yes—
65C4⋊C4×C71—21no—
66C4⋊C4×C71C2312yes—
67C42×C71C2311yescharacteristic
68C23×C71—22no—
69C22×D71—22no—
70C22×D71—44no—
71C22×D71—44no—
72C22×D71—44no—
73C22×D71C2×D411yescharacteristic
74Q8×C71—42no—
75D4×C71—24no—
76D4×C71—42no—
77D4×C71—42no—
78C2×C4×C71—22no—
79C2×C4×C71C2×D411yescharacteristic
80C2×C4×C71C2×D411yescharacteristic
81C2×C4×C71C2411yescharacteristic
82C22⋊D71—44no—
83C22⋊D71—44no—
84C22⋊D71—44no—
85C2×Dic71—22no—
86C2×Dic71C2×D411yescharacteristic
87C2×Dic71C2×D411yescharacteristic
88C2×Dic71C2×D411yescharacteristic
89D284—24no—
90D284—42no—
91D284—42no—
92C4×D71—22no—
93C4×D71—22no—
94C4×D71#4114yes—
95Dic142—42no—
96C8×C71—22no—
97C71⋊C8—22no—
98C22×C71—42no—
99C22×C71—42no—
100C22×C71—42no—
101C22×C71C22×D411yescharacteristic
102C2×D71—21no—
103C2×D71—21no—
104C2×D71—42no—
105C2×D71—42no—
106C2×D71—42no—
107C2×D71—88no—
108C2×D71#12412yes—
109C4×C71—42no—
110C4×C71AGL(1,ℤ/8)12yes—
111C4×C71AGL(1,ℤ/8)12yes—
112C4×C71C2×#4112yes—
113Dic71—21no—
114Dic71—21no—
115Dic71—42no—
116Dic71#12412yes—
117C2×C71—82no—
118C2×C71C2×#12411yescharacteristic
119C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
120C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
121D71—22no—
122D71—82no—
123#3138—711nomaximal, Sylow
124C71#313811yescharacteristic, complemented, Sylow
125#533—711no—
126C2×#127—711no—
127#490—714no—
128#486—711no—
129#486—711no—
130#486—711no—
131#486D7111yescharacteristic, complemented
132#418—714no—
133C4×M16—711no—
134C22×D4—712no—
135C22×D4—1421no—
136C22×D4—1421no—
137C2×M16—712no—
138C4⋊Q8—711no—
139C4⋊Q8C2×D7111yescharacteristic, complemented
140#127—712no—
141#127—711no—
142#127—1421no—
143#127—1421no—
144#127C2×D7111yescharacteristic
145#125—712no—
146C22⋊Q8—1421no—
147C22⋊Q8—1421no—
148C4×Q8—1421no—
149C4×Q8—1421no—
150#117—712no—
151C2×C42—711no—
152#102—714no—
153#102—714no—
154#102—714no—
155#102C2×D7114yescomplemented
156#97—712no—
157C4×C8—711no—
158C4×C8C2×D7111yescharacteristic, complemented
159C24—1422no—
160C2×Q8—21no—
161C2×Q8—1421no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—2844no—
165C2×D4—2844no—
166C2×D4—712no—
167C2×D4—1428no—
168C2×D4—1422no—
169C2×D4—1422no—
170C2×D4—1422no—
171C2×D4—1421no—
172C2×D4—1421no—
173C2×D4C22×D7112yes—
174C22×C4—711no—
175C22×C4—711no—
176C22×C4—711no—
177M16—1424no—
178C2×C8—712no—
179C2×C8C22×D7112yes—
180C4⋊C4—21no—
181C4⋊C4—712no—
182C4⋊C4—1422no—
183C4⋊C4—1422no—
184C4⋊C4—1421no—
185C4⋊C4C22×D7112yes—
186C22⋊C4—1422no—
187C42—711no—
188C42—711no—
189C42—711no—
190C42—1422no—
191C42C22×D7111yescharacteristic
192C23—22no—
193C23—2844no—
194C23—2844no—
195C23—2844no—
196C23—711no—
197C23—1422no—
198Q8—42no—
199Q8—2842no—
200D4—24no—
201D4—42no—
202D4—42no—
203D4—2844no—
204D4—2844no—
205D4—2844no—
206D4—2842no—
207D4—2842no—
208D4—1424no—
209C2×C4—22no—
210C2×C4—711no—
211C2×C4—714no—
212C2×C4—711no—
213C2×C4—711no—
214C2×C4—1422no—
215C2×C4—1422no—
216C2×C4—1422no—
217C2×C4D4×D7111yescharacteristic
218C2×C4D4×D7111yescharacteristic
219C2×C4C23×D7111yescharacteristic
220C8—22no—
221C8—1422no—
222C22—42no—
223C22—42no—
224C22—42no—
225C22—2842no—
226C22—2842no—
227C22—2842no—
228C22—5688no—
229C22—712no—
230C22—1421no—
231C22—1421no—
232C22C2×D4×D7111yescharacteristic
233C4—42no—
234C4—2842no—
235C4—712no—
236C4—1421no—
237C4—1421no—
238C4#1455615212yes—
239C4#1455615512yes—
240C4#41×D7112yes—
241C2—82no—
242C2—5682no—
243C2—1422no—
244C2#124×D7111yescharacteristic
245C2C2×#1455615211yescharacteristic
246C2C2×#1455615511yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#486×C71C2
socleC22×C71C22×D4
hypercenter#486D71
hypocenterC71#3138
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. #17986554C2semidirect
3. #17986362C2semidirect
4. #17986347C2semidirect
5. #17986346C2semidirect
6. #17986345C2semidirect
7. #17986002C2semidirect
8. C4×#11765848C2semidirect
9. #486×C71C2semidirect
14. #14556128C22semidirect
19. #14556119C22semidirect
26. #14556050C22semidirect
27. #14556049C22semidirect
28. #14556048C22semidirect
29. #14556038C22semidirect
30. C4×C71⋊C8C22semidirect
31. C4⋊Q8×C71C22semidirect
33. #102×C71C22semidirect
34. C4×C8×C71C22semidirect
124. C71#3138semidirect
131. #486D71semidirect
139. C4⋊Q8C2×D71semidirect
155. #102C2×D71semidirect
158. C4×C8C2×D71semidirect