Finite Group Databasebeta

← #22599565

Subgroups — #22599565

Order 9088.

362 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C4⋊Q8×D71C211yesmaximal, characteristic, complemented
2#17986536C211yesmaximal, characteristic
3#17986407C212yesmaximal, complemented
4#17986360C212yesmaximal, complemented
5#17986328C212yesmaximal, complemented
6#17986314C212yesmaximal
7#17986294C211yesmaximal, characteristic, complemented
8#17986290C211yesmaximal, characteristic, complemented
9#17986286C211yesmaximal, characteristic, complemented
10#17986055C211yesmaximal, characteristic, complemented
11#477×C71C211yesmaximal, characteristic, complemented
12C2×Q8×D71—21no—
13#14556128C2211yescharacteristic
14#14556127—21no—
15#14556127C2212yes—
16#14556126—21no—
17#14556123—21no—
18#14556119C2211yescharacteristic
19#14556117C2211yescharacteristic
20#14556113—21no—
21#14556104C2211yescharacteristic
22#14556103C2211yescharacteristic
23C4×D284—21no—
24C4×D284C2211yescharacteristic
25#14556085—21no—
26C4⋊C4×D71—21no—
27C4⋊C4×D71C2212yes—
28C42×D71C2211yescharacteristic
29#14556061C2212yes—
30#14556060C2212yes—
31#14556059C2212yes—
32C4⋊(C71⋊C8)C2211yescharacteristic
33#14556052C2212yes—
34#14556050C2212yes—
35#14556048C2212yes—
36(C2×D71)⋊C8C2212yes—
37#14556038C2212yes—
38C4⋊Q8×C71C2211yescharacteristic
39C4×Q8×C71C2211yescharacteristic
40#107×C71C2212yes—
41C4⋊C8×C71C2211yescharacteristic
42#103×C71C2212yes—
43C2×Dic142—21no—
44C2×Dic142—22no—
45Q8×D71—44no—
46C2×D284—21no—
47C2×D284C2311yescharacteristic
48C2×C4×D71—22no—
49C2×C4×D71—21no—
50C2×C4×D71D411yescharacteristic
51C2×C4×D71D411yescharacteristic
52C2×C4×D71C2311yescharacteristic
53C2×C71⋊C8C2312yes—
54C4⋊Dic71—21no—
55C4⋊Dic71—21no—
56C4⋊Dic71C2312yes—
57C4⋊Dic71C2311yescharacteristic
58Dic71⋊C4—22no—
59Dic71⋊C4—22no—
60Dic71⋊C4—21no—
61Dic71⋊C4—21no—
62(C2×D71)⋊C4—21no—
63(C2×D71)⋊C4—21no—
64(C2×D71)⋊C4—21no—
65(C2×D71)⋊C4—21no—
66(C2×D71)⋊C4—21no—
67C4×Dic71D411yescharacteristic
68C4×Dic71D411yescharacteristic
69C4×Dic71C2311yescharacteristic
70C2×Q8×C71—21no—
71C2×Q8×C71C2311yescharacteristic
72C2×C8×C71C2312yes—
73C4⋊C4×C71—21no—
74C4⋊C4×C71—21no—
75C4⋊C4×C71C2312yes—
76C4⋊C4×C71C2311yescharacteristic
77C42×C71—21no—
78C42×C71C2311yescharacteristic
79C22×D71—21no—
80C22×D71C2×D411yescharacteristic
81Q8×C71—21no—
82Q8×C71—21no—
83Q8×C71—42no—
84C2×C4×C71—21no—
85C2×C4×C71—21no—
86C2×C4×C71—22no—
87C2×C4×C71C2×D411yescharacteristic
88C2×C4×C71C2×D411yescharacteristic
89C2×C4×C71C2411yescharacteristic
90C2×Dic71—21no—
91C2×Dic71—22no—
92C2×Dic71C2×D411yescharacteristic
93C2×Dic71C2×D411yescharacteristic
94C2×Dic71C2×D411yescharacteristic
95D284—21no—
96D284—21no—
97D284—42no—
98C4×D71—22no—
99C4×D71—22no—
100C4×D71—21no—
101C4×D71—21no—
102C4×D71—44no—
103C4×D71—42no—
104Dic142—44no—
105Dic142—42no—
106C8×C71—22no—
107C71⋊C8—22no—
108C22×C71C22×D411yescharacteristic
109C2×D71—21no—
110C2×D71—21no—
111C2×D71—41no—
112C2×D71—41no—
113C2×D71—41no—
114C2×D71AGL(1,ℤ/8)12yes—
115C4×C71—21no—
116C4×C71—41no—
117C4×C71—42no—
118C4×C71—41no—
119C4×C71#13712yes—
120C4×C712−1+411yescharacteristic
121C4×C712−1+411yescharacteristic
122Dic71—21no—
123Dic71—21no—
124Dic71—21no—
125Dic71—42no—
126Dic71—41no—
127C2×C71#53611yescharacteristic
128C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
129C2×C71C2×#13711yescharacteristic
130D71—22no—
131D71—81no—
132#3238—711nomaximal, Sylow
133C71#323811yescharacteristic, complemented, Sylow
134C2×C4⋊Q8—711no—
135#529—711no—
136#501—712no—
137#489—712no—
138#483—712no—
139#480—712no—
140#477—711no—
141#477D7111yescharacteristic, complemented
142#476—711no—
143#476—711no—
144#432—711no—
145C22×Q8—1421no—
146C4⋊Q8—712no—
147C4⋊Q8—711no—
148C4⋊Q8—1421no—
149C4⋊Q8—1421no—
150C4⋊Q8C2×D7111yescharacteristic
151#126—1421no—
152#125—711no—
153#124—711no—
154#123—1421no—
155C22⋊Q8—711no—
156#121—711no—
157C4×Q8C2×D7111yescharacteristic
158C4×D4—711no—
159C4×D4—1421no—
160#117—1421no—
161C2×C4⋊C4—712no—
162C2×C4⋊C4—1421no—
163C2×C42—711no—
164#107—712no—
165#107C2×D7112yes—
166#106—712no—
167#106—712no—
168C4⋊C8—711no—
169C4⋊C8C2×D7111yescharacteristic
170#103—712no—
171#103C2×D7112yes—
172#102—712no—
173#102—712no—
174C22⋊C8—712no—
175#97—712no—
176C2×Q8—21no—
177C2×Q8—2844no—
178C2×Q8—1422no—
179C2×Q8—1421no—
180C2×Q8C22×D7111yescharacteristic
181C2×D4—711no—
182C2×D4—1421no—
183C22×C4—711no—
184C22×C4—711no—
185C22×C4—711no—
186C22×C4—1422no—
187C22×C4—1421no—
188C2×C8—712no—
189C2×C8C22×D7112yes—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—712no—
193C4⋊C4—711no—
194C4⋊C4—1422no—
195C4⋊C4—1422no—
196C4⋊C4—1421no—
197C4⋊C4—1421no—
198C4⋊C4—1421no—
199C4⋊C4—1421no—
200C4⋊C4C22×D7112yes—
201C4⋊C4C22×D7111yescharacteristic
202C22⋊C4—1421no—
203C22⋊C4—1421no—
204C22⋊C4—1421no—
205C22⋊C4—1421no—
206C22⋊C4—1421no—
207C42—21no—
208C42—711no—
209C42—711no—
210C42—711no—
211C42C22×D7111yescharacteristic
212C23—711no—
213C23—1421no—
214Q8—21no—
215Q8—21no—
216Q8—42no—
217Q8—2842no—
218Q8—2844no—
219D4—2842no—
220D4—1421no—
221D4—1421no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—2844no—
226C2×C4—2842no—
227C2×C4—711no—
228C2×C4—711no—
229C2×C4—711no—
230C2×C4—1422no—
231C2×C4—1421no—
232C2×C4—1422no—
233C2×C4—1422no—
234C2×C4—1421no—
235C2×C4—1421no—
236C2×C4D4×D7111yescharacteristic
237C2×C4D4×D7111yescharacteristic
238C2×C4C23×D7111yescharacteristic
239C8—22no—
240C8—1422no—
241C22—2841no—
242C22—2841no—
243C22—2841no—
244C22—712no—
245C22—1421no—
246C22—1421no—
247C22C2×D4×D7111yescharacteristic
248C4—21no—
249C4—41no—
250C4—42no—
251C4—41no—
252C4—2842no—
253C4—2841no—
254C4—1421no—
255C4—1421no—
256C4—1421no—
257C4#1455616012yes—
258C4#1455617511yescharacteristic
259C4#1455617611yescharacteristic
260C2—5681no—
261C2—1422no—
262C2#1798657211yescharacteristic
263C2AGL(1,ℤ/8)×D7111yescharacteristic
264C2C2×#1455616011yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#477×C71C2
socleC22×C71C22×D4
hypercenter#477D71
hypocenterC71#3238
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
1. C4⋊Q8×D71C2semidirect
3. #17986407C2semidirect
4. #17986360C2semidirect
5. #17986328C2semidirect
7. #17986294C2semidirect
8. #17986290C2semidirect
9. #17986286C2semidirect
10. #17986055C2semidirect
11. #477×C71C2semidirect
133. C71#3238semidirect
141. #477D71semidirect