Finite Group Databasebeta

← #22599093

Subgroups — #22599093

Order 9088.

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

normal subgroups only

This group has 468 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
2#17986327C212yesmaximal, complemented
3#17986317C212yesmaximal, complemented
4#17986230C214yesmaximal, complemented
5#17986149C212yesmaximal, complemented
6#17986138C212yesmaximal, complemented
7C2×C22⋊(C71⋊C8)C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8×C71C211yesmaximal, characteristic, complemented
9C22×D284C2211yescharacteristic
10C22×C22⋊D71—21no—
11C2×#11765852C2214yescomplemented
12C22×C71⋊C8C2212yescomplemented
13#14556102—21no—
14#14556102—21no—
15#14556102C2212yes—
16#14556102C2212yes—
17C2×C4⋊Dic71C2211yescharacteristic
18C2×(C2×D71)⋊C4—21no—
19#14556059C2214yescomplemented
20#14556052C2214yescomplemented
21#14556048C2214yescomplemented
22C22⋊(C71⋊C8)C2212yes—
23C22⋊(C71⋊C8)C2212yescomplemented
24C22×Q8×C71C2211yescharacteristic, complemented
25C23×C4×C71C2211yescharacteristic
26C22⋊Q8×C71—21no—
27C22⋊Q8×C71—21no—
28C22⋊Q8×C71C2212yes—
29C22⋊Q8×C71C2212yescomplemented
30C2×C4⋊C4×C71—21no—
31C2×C4⋊C4×C71C2211yescharacteristic, complemented
32C2×C22⋊C4×C71—21no—
33C23×D71—21no—
34C2×D284—21no—
35C2×D284—42no—
36C2×D284C2312yes—
37C2×C22⋊D71—22no—
38C2×C22⋊D71—22no—
39C2×C22⋊D71—44no—
40C22×Dic71—21no—
41#11765852—44no—
42C2×C71⋊C8—42no—
43C2×C71⋊C8C2314yes—
44C4⋊Dic71—21no—
45C4⋊Dic71C2312yes—
46(C2×D71)⋊C4—22no—
47(C2×D71)⋊C4—22no—
48C24×C71D411yescharacteristic
49C2×Q8×C71—21no—
50C2×Q8×C71—42no—
51C2×Q8×C71C2312yes—
52C22×C4×C71—21no—
53C22×C4×C71—21no—
54C22×C4×C71—42no—
55C22×C4×C71D411yescharacteristic
56C22×C4×C71D411yescharacteristic
57C22×C4×C71D411yescharacteristic
58C22×C4×C71C2311yescharacteristic
59C22×C4×C71C2311yescharacteristic
60C22×C4×C71C2311yescharacteristic
61C4⋊C4×C71—22no—
62C4⋊C4×C71—22no—
63C4⋊C4×C71—21no—
64C4⋊C4×C71C2312yes—
65C22⋊C4×C71—22no—
66C22⋊C4×C71—22no—
67C23×C71—41no—
68C23×C71—41no—
69C23×C71SD1614yescomplemented
70C23×C71C2×D411yescharacteristic
71C23×C71C2×D411yescharacteristic
72C23×C71C2×D411yescharacteristic
73C22×D71—22no—
74C22×D71—42no—
75C22×D71—42no—
76C22×D71—42no—
77Q8×C71—24no—
78Q8×C71—42no—
79C2×C4×C71—22no—
80C2×C4×C71—22no—
81C2×C4×C71—22no—
82C2×C4×C71—22no—
83C2×C4×C71—22no—
84C2×C4×C71—41no—
85C2×C4×C71—41no—
86C2×C4×C71—41no—
87C2×C4×C71—42no—
88C2×C4×C71—42no—
89C2×C4×C71C2×D411yescharacteristic
90C2×C4×C71C2×D411yescharacteristic
91C2×C4×C71C2×D411yescharacteristic
92C2×C4×C71C2411yescharacteristic
93C22⋊D71—44no—
94C22⋊D71—44no—
95C2×Dic71—22no—
96C2×Dic71—42no—
97D284—24no—
98D284—42no—
99C71⋊C8—24no—
100C22×C71—21no—
101C22×C71—21no—
102C22×C71—21no—
103C22×C71—21no—
104C22×C71—21no—
105C22×C71—21no—
106C22×C71—44no—
107C22×C71C2×SD1612yes—
108C22×C71C2×SD1612yes—
109C22×C71C2×SD1612yes—
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×C71—42no—
119C4×C71—42no—
120C4×C71H(5,𝔽2)12yes—
121Dic71—42no—
122C2×C71—22no—
123C2×C71—22no—
124C2×C71—22no—
125C2×C71#53411yescharacteristic
126C2×C71C22×SD1611yescharacteristic
127C2×C71#57511yescharacteristic
128D71—82no—
129#3187—711nomaximal, Sylow
130C71#318711yescharacteristic, complemented, Sylow
131C2×C22⋊Q8D7111yescharacteristic, complemented
132C2×#121—711no—
133#483—712no—
134#481—712no—
135#465—714no—
136#450—712no—
137#448—712no—
138C2×C22⋊C8—711no—
139C22×Q8C2×D7111yescharacteristic, complemented
140C22×D4—711no—
141C22×D4—1421no—
142C23×C4C2×D7111yescharacteristic
143C2×SD16—714no—
144C22×C8—712no—
145C22⋊Q8—21no—
146C22⋊Q8—21no—
147C22⋊Q8C2×D7112yes—
148C22⋊Q8C2×D7112yescomplemented
149#121—712no—
150#121—712no—
151#121—1421no—
152#121—1421no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—711no—
155C2×C4⋊C4C2×D7111yescharacteristic, complemented
156C2×C22⋊C4—21no—
157C2×C22⋊C4—1421no—
158#106—714no—
159#103—714no—
160#102—714no—
161C22⋊C8—712no—
162C22⋊C8—712no—
163C24—1421no—
164C24C22⋊D7111yescharacteristic
165C2×Q8—21no—
166C2×Q8—42no—
167C2×Q8C22×D7112yes—
168C2×D4—2842no—
169C2×D4—2844no—
170C2×D4—712no—
171C2×D4—1422no—
172C2×D4—1422no—
173C2×D4—1421no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—42no—
177C22×C4—1421no—
178C22×C4C22⋊D7111yescharacteristic
179C22×C4C22⋊D7111yescharacteristic
180C22×C4C22⋊D7111yescharacteristic
181C22×C4C22×D7111yescharacteristic
182C22×C4C22×D7111yescharacteristic
183C22×C4C22×D7111yescharacteristic
184SD16—2844no—
185C2×C8—2842no—
186C2×C8—714no—
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—21no—
190C4⋊C4—712no—
191C4⋊C4—1421no—
192C4⋊C4C22×D7112yes—
193C22⋊C4—22no—
194C22⋊C4—22no—
195C22⋊C4—1422no—
196C22⋊C4—1422no—
197C23—41no—
198C23—41no—
199C23—2842no—
200C23—2842no—
201C23—2842no—
202C23—1422no—
203C23#1176585214yescomplemented
204C23C2×C22⋊D7111yescharacteristic
205C23C2×C22⋊D7111yescharacteristic
206C23C2×C22⋊D7111yescharacteristic
207Q8—24no—
208Q8—42no—
209D4—2842no—
210D4—2844no—
211D4—2844no—
212D4—1424no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—2842no—
224C2×C4—1422no—
225C2×C4C2×C22⋊D7111yescharacteristic
226C2×C4C2×C22⋊D7111yescharacteristic
227C2×C4C2×C22⋊D7111yescharacteristic
228C2×C4C23×D7111yescharacteristic
229C8—1424no—
230C22—21no—
231C22—21no—
232C22—21no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—44no—
237C22—2842no—
238C22—2842no—
239C22—2842no—
240C22—5684no—
241C22C2×#1176585212yes—
242C22C2×#1176585212yes—
243C22C2×#1176585212yes—
244C22C22×C22⋊D7111yescharacteristic
245C4—21no—
246C4—21no—
247C4—21no—
248C4—42no—
249C4—42no—
250C4—2842no—
251C4#1455617312yes—
252C2—22no—
253C2—22no—
254C2—22no—
255C2—5682no—
256C2#1798655811yescharacteristic
257C2C22×#1176585211yescharacteristic
258C2#1798679711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
FittingC2×C22⋊Q8×C71C2
socleC22×C71C22×D4
hypercenterC2×C22⋊Q8D71
hypocenterC71#3187
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #17986327C2semidirect
3. #17986317C2semidirect
4. #17986230C2semidirect
5. #17986149C2semidirect
6. #17986138C2semidirect
7. C2×C22⋊(C71⋊C8)C2semidirect
8. C2×C22⋊Q8×C71C2semidirect
11. C2×#11765852C22semidirect
12. C22×C71⋊C8C22semidirect
19. #14556059C22semidirect
20. #14556052C22semidirect
21. #14556048C22semidirect
23. C22⋊(C71⋊C8)C22semidirect
24. C22×Q8×C71C22semidirect
29. C22⋊Q8×C71C22semidirect
31. C2×C4⋊C4×C71C22semidirect
69. C23×C71SD16semidirect
130. C71#3187semidirect
131. C2×C22⋊Q8D71semidirect
139. C22×Q8C2×D71semidirect
148. C22⋊Q8C2×D71semidirect
155. C2×C4⋊C4C2×D71semidirect
203. C23#11765852semidirect