Finite Group Databasebeta

← #22598281

Subgroups — #22598281

Order 9088.

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

normal subgroups only

This group has 462 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
2C2×#14556087C211yesmaximal, characteristic, complemented
3#17986324C214yesmaximal, complemented
4#17986318C212yesmaximal, complemented
5#17986313C212yesmaximal, complemented
6#17986041C212yesmaximal, complemented
7#17986005C212yesmaximal, complemented
8#407×C71C211yesmaximal, characteristic, complemented
9C22×D284C2211yescharacteristic
10C22×C22⋊D71—21no—
11#14556102—21no—
12#14556102—21no—
13#14556102C2212yes—
14#14556102C2212yes—
15#14556087—21no—
16#14556087—21no—
17#14556087C2212yes—
18#14556087C2212yescomplemented
19C2×C4⋊Dic71C2211yescharacteristic
20C2×C4⋊Dic71C2211yescharacteristic, complemented
21C2×C4⋊Dic71C2211yescharacteristic, complemented
22C2×C22⋊Dic71—21no—
23C2×(C2×D71)⋊C4—21no—
24C2×C4×Dic71—21no—
25#14556062C2214yescomplemented
26#14556060C2214yescomplemented
27#14556050C2214yescomplemented
28#14556050C2214yescomplemented
29C23×C4×C71C2211yescharacteristic
30C2×M16×C71C2212yes—
31C22⋊C8×C71C2212yescomplemented
32C22⋊C8×C71C2212yescomplemented
33C23×D71—21no—
34C2×D284—24no—
35C2×D284—21no—
36C2×D284C2312yes—
37C2×C22⋊D71—22no—
38C2×C22⋊D71—22no—
39C2×C22⋊D71—44no—
40C22×Dic71—21no—
41C22×Dic71—21no—
42C22×Dic71—21no—
43C4⋊Dic71—21no—
44C4⋊Dic71—21no—
45C4⋊Dic71—21no—
46C4⋊Dic71C2312yes—
47C4⋊Dic71C2312yes—
48C4⋊Dic71C2312yes—
49C22⋊Dic71—22no—
50C22⋊Dic71—22no—
51(C2×D71)⋊C4—22no—
52(C2×D71)⋊C4—22no—
53C4×Dic71—22no—
54C4×Dic71—22no—
55C24×C71D411yescharacteristic
56C22×C4×C71—24no—
57C22×C4×C71D411yescharacteristic
58C22×C4×C71D411yescharacteristic
59C22×C4×C71D411yescharacteristic
60C22×C4×C71C2311yescharacteristic
61C22×C4×C71C2311yescharacteristic
62C22×C4×C71C2311yescharacteristic
63M16×C71—24no—
64C2×C8×C71C2314yes—
65C23×C71—22no—
66C23×C71—22no—
67C23×C71—41no—
68C23×C71C2×D411yescharacteristic
69C23×C71C2×D411yescharacteristic
70C23×C71C2×D411yescharacteristic
71C22×D71—22no—
72C22×D71—42no—
73C22×D71—42no—
74C22×D71—42no—
75C2×C4×C71—22no—
76C2×C4×C71—22no—
77C2×C4×C71—22no—
78C2×C4×C71—22no—
79C2×C4×C71—22no—
80C2×C4×C71C2×D411yescharacteristic
81C2×C4×C71C2×D411yescharacteristic
82C2×C4×C71C2×D411yescharacteristic
83C2×C4×C71#4114yes—
84C2×C4×C71C2411yescharacteristic
85C22⋊D71—44no—
86C22⋊D71—44no—
87C2×Dic71—22no—
88C2×Dic71—22no—
89C2×Dic71—22no—
90C2×Dic71—42no—
91C2×Dic71—42no—
92C2×Dic71—42no—
93D284—24no—
94D284—42no—
95C8×C71—24no—
96C22×C71—21no—
97C22×C71—21no—
98C22×C71—21no—
99C22×C71—21no—
100C22×C71—21no—
101C22×C71—21no—
102C22×C71—21no—
103C22×C71—44no—
104C22×C71#12312yes—
105C22×C71AGL(1,ℤ/8)12yes—
106C22×C71C22×D411yescharacteristic
107C2×D71—42no—
108C2×D71—42no—
109C2×D71—42no—
110C2×D71—84no—
111C4×C71—21no—
112C4×C71—21no—
113C4×C71#12312yes—
114C4×C71C2×#4112yes—
115Dic71—42no—
116Dic71—42no—
117Dic71—42no—
118C2×C71—22no—
119C2×C71—22no—
120C2×C71—41no—
121C2×C71C2×#12311yescharacteristic
122C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
123C2×C71#57511yescharacteristic
124D71—82no—
125#3094—711nomaximal, Sylow
126C71#309411yescharacteristic, complemented, Sylow
127C2×#121—711no—
128C2×#117—711no—
129#482—714no—
130#481—712no—
131#480—712no—
132#428—712no—
133#418—712no—
134#407D7111yescharacteristic, complemented
135C22×D4—711no—
136C22×D4—1421no—
137C23×C4C2×D7111yescharacteristic
138C2×M16C2×D7112yes—
139#121—712no—
140#121—712no—
141#121—1421no—
142#121—1421no—
143#117—712no—
144#117—712no—
145#117—1421no—
146#117—1421no—
147C2×C4⋊C4—711no—
148C2×C4⋊C4—711no—
149C2×C4⋊C4—711no—
150C2×C22⋊C4—1421no—
151C2×C22⋊C4—1421no—
152C2×C42—1421no—
153#107—714no—
154#106—714no—
155#102—714no—
156#102—714no—
157C22⋊C8C2×D7112yescomplemented
158C22⋊C8C2×D7112yescomplemented
159C24—1421no—
160C24D28411yescharacteristic
161C2×D4—2844no—
162C2×D4—712no—
163C2×D4—1422no—
164C2×D4—1422no—
165C2×D4—1421no—
166C2×D4—1424no—
167C22×C4—24no—
168C22×C4—1421no—
169C22×C4—1421no—
170C22×C4—1421no—
171C22×C4D28411yescharacteristic
172C22×C4D28411yescharacteristic
173C22×C4D28411yescharacteristic
174C22×C4C22×D7111yescharacteristic
175C22×C4C22×D7111yescharacteristic
176C22×C4C22×D7111yescharacteristic
177M16—24no—
178C2×C8C22×D7114yes—
179C4⋊C4—712no—
180C4⋊C4—712no—
181C4⋊C4—712no—
182C4⋊C4—1421no—
183C4⋊C4—1421no—
184C4⋊C4—1421no—
185C22⋊C4—1422no—
186C22⋊C4—1422no—
187C22⋊C4—1422no—
188C22⋊C4—1422no—
189C42—1422no—
190C42—1422no—
191C23—22no—
192C23—22no—
193C23—41no—
194C23—2842no—
195C23—2842no—
196C23—2842no—
197C23—1422no—
198C23C2×D28411yescharacteristic
199C23C2×D28411yescharacteristic
200C23C2×D28411yescharacteristic
201D4—2844no—
202D4—2844no—
203D4—2842no—
204D4—1424no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—2842no—
211C2×C4—2842no—
212C2×C4—2842no—
213C2×C4—1422no—
214C2×C4—1422no—
215C2×C4—1422no—
216C2×C4C2×D28411yescharacteristic
217C2×C4C2×D28411yescharacteristic
218C2×C4C2×D28411yescharacteristic
219C2×C4#1176586414yes—
220C2×C4C23×D7111yescharacteristic
221C8—24no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—44no—
230C22—2842no—
231C22—2842no—
232C22—2842no—
233C22—5684no—
234C22#1455611212yes—
235C22#1455615612yes—
236C22C22×D28411yescharacteristic
237C4—21no—
238C4—21no—
239C4—2842no—
240C4—2842no—
241C4—2842no—
242C4#1455611212yes—
243C4C2×#1176586412yes—
244C2—22no—
245C2—22no—
246C2—41no—
247C2—5682no—
248C2C2×#1455611211yescharacteristic
249C2C2×#1455615611yescharacteristic
250C2#1798679911yescharacteristic
subgroupG/H
centerC22C22×D284
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#407×C71C2
socleC22×C71C22×D4
hypercenter#407D71
hypocenterC71#3094
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×#14556087C2semidirect
3. #17986324C2semidirect
4. #17986318C2semidirect
5. #17986313C2semidirect
6. #17986041C2semidirect
7. #17986005C2semidirect
8. #407×C71C2semidirect
18. #14556087C22semidirect
20. C2×C4⋊Dic71C22semidirect
21. C2×C4⋊Dic71C22semidirect
25. #14556062C22semidirect
26. #14556060C22semidirect
27. #14556050C22semidirect
28. #14556050C22semidirect
31. C22⋊C8×C71C22semidirect
32. C22⋊C8×C71C22semidirect
126. C71#3094semidirect
134. #407D71semidirect
157. C22⋊C8C2×D71semidirect
158. C22⋊C8C2×D71semidirect