Finite Group Databasebeta

← #22599147

Subgroups — #22599147

Order 9088.

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

normal subgroups only

This group has 462 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
1C2×#14556107C211yesmaximal, characteristic, complemented
2C2×#14556102C211yesmaximal, characteristic
3#17986335C212yesmaximal, complemented
4#17986313C212yesmaximal, complemented
5#17986245C214yesmaximal, complemented
6#17986152C212yesmaximal, complemented
7#17986140C212yesmaximal, complemented
8#407×C71C211yesmaximal, characteristic, complemented
9C22×Dic142C2211yescharacteristic, complemented
10C22×D284C2211yescharacteristic
11C22×C22⋊D71—21no—
12C2×#11765854C2214yescomplemented
13#14556107—21no—
14#14556107—21no—
15#14556107C2212yes—
16#14556107C2212yescomplemented
17#14556102—21no—
18#14556102—21no—
19#14556102C2212yes—
20#14556102C2212yes—
21C2×C4⋊Dic71C2211yescharacteristic
22C2×C4⋊Dic71C2211yescharacteristic, complemented
23C2×Dic71⋊C4—21no—
24C2×C22⋊Dic71—21no—
25C2×(C2×D71)⋊C4—21no—
26#14556062C2214yescomplemented
27#14556053C2214yescomplemented
28#14556050C2214yescomplemented
29C23×C4×C71C2211yescharacteristic
30C2×M16×C71C2212yescomplemented
31C22⋊C8×C71C2212yes—
32C22⋊C8×C71C2212yescomplemented
33C23×D71—21no—
34C2×Dic142—21no—
35C2×Dic142—42no—
36C2×Dic142C2312yes—
37C2×D284—21no—
38C2×D284—42no—
39C2×D284C2312yes—
40C2×C22⋊D71—22no—
41C2×C22⋊D71—22no—
42C2×C22⋊D71—44no—
43C22×Dic71—21no—
44C22×Dic71—21no—
45C22×Dic71—21no—
46#11765854—44no—
47C4⋊Dic71—21no—
48C4⋊Dic71—21no—
49C4⋊Dic71C2312yes—
50C4⋊Dic71C2312yes—
51Dic71⋊C4—22no—
52Dic71⋊C4—22no—
53C22⋊Dic71—22no—
54C22⋊Dic71—22no—
55(C2×D71)⋊C4—22no—
56(C2×D71)⋊C4—22no—
57C24×C71D411yescharacteristic
58C22×C4×C71—42no—
59C22×C4×C71D411yescharacteristic
60C22×C4×C71D411yescharacteristic
61C22×C4×C71D411yescharacteristic
62C22×C4×C71C2311yescharacteristic
63C22×C4×C71C2311yescharacteristic
64C22×C4×C71C2311yescharacteristic
65M16×C71—42no—
66C2×C8×C71C2314yes—
67C23×C71—22no—
68C23×C71—22no—
69C23×C71—41no—
70C23×C71C2×D411yescharacteristic
71C23×C71C2×D411yescharacteristic
72C23×C71C2×D411yescharacteristic
73C22×D71—22no—
74C22×D71—42no—
75C22×D71—42no—
76C22×D71—42no—
77C2×C4×C71—22no—
78C2×C4×C71—22no—
79C2×C4×C71—22no—
80C2×C4×C71—41no—
81C2×C4×C71—41no—
82C2×C4×C71—41no—
83C2×C4×C71C2×D411yescharacteristic
84C2×C4×C71C2×D411yescharacteristic
85C2×C4×C71C2×D411yescharacteristic
86C2×C4×C71C2411yescharacteristic
87C22⋊D71—44no—
88C22⋊D71—44no—
89C2×Dic71—22no—
90C2×Dic71—22no—
91C2×Dic71—22no—
92C2×Dic71—42no—
93C2×Dic71—42no—
94C2×Dic71—42no—
95D284—24no—
96D284—42no—
97Dic142—24no—
98Dic142—42no—
99C8×C71—24no—
100C22×C71—21no—
101C22×C71—21no—
102C22×C71—21no—
103C22×C71—21no—
104C22×C71—21no—
105C22×C71—21no—
106C22×C71—21no—
107C22×C71—44no—
108C22×C71AGL(1,ℤ/8)12yes—
109C22×C71#13712yes—
110C22×C71C22×D411yescharacteristic
111C2×D71—42no—
112C2×D71—42no—
113C2×D71—42no—
114C2×D71—84no—
115C4×C71—21no—
116C4×C71—21no—
117C4×C71—21no—
118C4×C71H(5,𝔽2)12yes—
119Dic71—42no—
120Dic71—42no—
121Dic71—42no—
122C2×C71—22no—
123C2×C71—22no—
124C2×C71—41no—
125C2×C71#53411yescharacteristic
126C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
127C2×C71C2×#13711yescharacteristic
128D71—82no—
129#3193—711nomaximal, Sylow
130C71#319311yescharacteristic, complemented, Sylow
131C2×C22⋊Q8—711no—
132C2×#121—711no—
133#484—712no—
134#480—712no—
135#468—714no—
136#450—712no—
137#448—712no—
138#407D7111yescharacteristic, complemented
139C22×Q8—711no—
140C22×D4—711no—
141C22×D4—1421no—
142C23×C4C2×D7111yescharacteristic
143C2×SD16—714no—
144C2×M16C2×D7112yescomplemented
145C22⋊Q8—712no—
146C22⋊Q8—712no—
147C22⋊Q8—1421no—
148C22⋊Q8—1421no—
149#121—712no—
150#121—712no—
151#121—1421no—
152#121—1421no—
153C2×C4⋊C4—711no—
154C2×C4⋊C4—711no—
155C2×C4⋊C4—1421no—
156C2×C22⋊C4—1421no—
157C2×C22⋊C4—1421no—
158#107—714no—
159#103—714no—
160#102—714no—
161C22⋊C8C2×D7112yes—
162C22⋊C8C2×D7112yescomplemented
163C24—1421no—
164C24D28411yescharacteristic
165C2×Q8—2842no—
166C2×Q8—712no—
167C2×Q8—1421no—
168C2×D4—2842no—
169C2×D4—2844no—
170C2×D4—712no—
171C2×D4—1422no—
172C2×D4—1422no—
173C2×D4—1421no—
174C22×C4—42no—
175C22×C4—1421no—
176C22×C4—1421no—
177C22×C4—1421no—
178C22×C4D28411yescharacteristic
179C22×C4D28411yescharacteristic
180C22×C4D28411yescharacteristic
181C22×C4C22×D7111yescharacteristic
182C22×C4C22×D7111yescharacteristic
183C22×C4C22×D7111yescharacteristic
184SD16—2844no—
185M16—42no—
186C2×C8C22×D7114yes—
187C4⋊C4—712no—
188C4⋊C4—712no—
189C4⋊C4—1422no—
190C4⋊C4—1422no—
191C4⋊C4—1421no—
192C4⋊C4—1421no—
193C22⋊C4—1422no—
194C22⋊C4—1422no—
195C22⋊C4—1422no—
196C22⋊C4—1422no—
197C23—22no—
198C23—22no—
199C23—41no—
200C23—2842no—
201C23—2842no—
202C23—2842no—
203C23—1422no—
204C23C2×D28411yescharacteristic
205C23C2×D28411yescharacteristic
206C23C2×D28411yescharacteristic
207Q8—2842no—
208Q8—1424no—
209D4—2842no—
210D4—2844no—
211D4—2844no—
212D4—1424no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—2842no—
220C2×C4—2842no—
221C2×C4—2842no—
222C2×C4—1422no—
223C2×C4—1422no—
224C2×C4—1422no—
225C2×C4C2×D28411yescharacteristic
226C2×C4C2×D28411yescharacteristic
227C2×C4C2×D28411yescharacteristic
228C2×C4C23×D7111yescharacteristic
229C8—24no—
230C22—21no—
231C22—21no—
232C22—21no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—44no—
238C22—2842no—
239C22—2842no—
240C22—2842no—
241C22—5684no—
242C22#1455615612yes—
243C22#1455616112yes—
244C22C22×D28411yescharacteristic
245C4—21no—
246C4—21no—
247C4—21no—
248C4—2842no—
249C4—2842no—
250C4—2842no—
251C4#1455617312yes—
252C2—22no—
253C2—22no—
254C2—41no—
255C2—5682no—
256C2#1798656111yescharacteristic
257C2C2×#1455615611yescharacteristic
258C2C2×#1455616111yescharacteristic
subgroupG/H
centerC22C22×D284
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#407×C71C2
socleC22×C71C22×D4
hypercenter#407D71
hypocenterC71#3193
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
1. C2×#14556107C2semidirect
3. #17986335C2semidirect
4. #17986313C2semidirect
5. #17986245C2semidirect
6. #17986152C2semidirect
7. #17986140C2semidirect
8. #407×C71C2semidirect
9. C22×Dic142C22semidirect
12. C2×#11765854C22semidirect
16. #14556107C22semidirect
22. C2×C4⋊Dic71C22semidirect
26. #14556062C22semidirect
27. #14556053C22semidirect
28. #14556050C22semidirect
30. C2×M16×C71C22semidirect
32. C22⋊C8×C71C22semidirect
130. C71#3193semidirect
138. #407D71semidirect
144. C2×M16C2×D71semidirect
162. C22⋊C8C2×D71semidirect