Finite Group Databasebeta

← #22600006

Subgroups — #22600006

Order 9088.

482 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 133 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×#14556158C212yesmaximal, complemented
2D4×Dic142C211yesmaximal, characteristic, complemented
3#17986371C211yesmaximal, characteristic, complemented
4#17986243C212yesmaximal, complemented
5#17986209C211yesmaximal, characteristic, complemented
6#17986196C211yesmaximal, characteristic, complemented
7#17986161C212yesmaximal, complemented
8#17986137C212yesmaximal, complemented
9C4×#11765855C211yesmaximal, characteristic, complemented
10#17986077C211yesmaximal, characteristic, complemented
11#550×C71C211yesmaximal, characteristic
12C22×Dic142C2212yes—
13#14556158—28no—
14C2×#11765855C2212yes—
15C2×#11765855C2211yescharacteristic
16C2×#11765853C2212yes—
17C2×#11765853C2212yes—
18C2×#11765853C2212yes—
19C2×#11765849C2212yes—
20#14556128C2211yescharacteristic
21#14556127—21no—
22#14556107C2212yes—
23#14556105—22no—
24C4×Dic142C2211yescharacteristic
25D4×Dic71—21no—
26#14556061C2211yescharacteristic
27C4⋊(C71⋊C8)C2211yescharacteristic
28#14556052C2211yescharacteristic
29#14556051C2211yescharacteristic
30#14556051C2212yes—
31#14556049C2212yes—
32C22⋊(C71⋊C8)C2212yes—
33C4×C71⋊C8C2211yescharacteristic
34C2×#41×C71C2212yes—
35#127×C71—21no—
36#127×C71C2211yescharacteristic
37#121×C71—22no—
38#121×C71C2212yes—
39C4×Q8×C71C2211yescharacteristic
40C4×D4×C71—21no—
41C4×D4×C71C2211yescharacteristic
42C2×Dic142—24no—
43C2×Dic142—22no—
44C2×Dic142—42no—
45C2×Dic142C2312yes—
46C2×Dic142C2311yescharacteristic
47C22×Dic71—22no—
48#11765855—24no—
49#11765855D414yescomplemented
50#11765853—24no—
51#11765853—24no—
52#11765853—42no—
53#11765849—24no—
54C2×C71⋊C8C2311yescharacteristic
55C2×C71⋊C8C2312yes—
56C2×C71⋊C8C2311yescharacteristic
57C4⋊Dic71—21no—
58C4⋊Dic71—21no—
59C4⋊Dic71C2311yescharacteristic
60Dic71⋊C4—21no—
61Dic71⋊C4—22no—
62C22⋊Dic71—22no—
63C4×Dic71—21no—
64#41×C71—24no—
65#41×C71—42no—
66C2×Q8×C71C2311yescharacteristic
67C2×D4×C71—22no—
68C2×D4×C71—22no—
69C2×D4×C71—21no—
70C2×D4×C71—21no—
71C2×D4×C71D411yescharacteristic
72C2×D4×C71C2312yes—
73C22×C4×C71—22no—
74C22×C4×C71C2312yes—
75C4⋊C4×C71—21no—
76C4⋊C4×C71D411yescharacteristic
77C4⋊C4×C71C2311yescharacteristic
78C22⋊C4×C71—22no—
79C22⋊C4×C71D412yes—
80C42×C71—21no—
81C42×C71C2311yescharacteristic
82C23×C71—22no—
83C23×C71C2×D412yes—
84Q8×C71—21no—
85Q8×C71C2×D412yes—
86D4×C71—22no—
87D4×C71—22no—
88D4×C71—22no—
89D4×C71—44no—
90D4×C71—42no—
91D4×C71—81no—
92D4×C71—82no—
93C2×C4×C71—21no—
94C2×C4×C71—22no—
95C2×C4×C71—22no—
96C2×C4×C71—21no—
97C2×C4×C71—44no—
98C2×C4×C71C2×D411yescharacteristic
99C2×C4×C71C2×D412yes—
100C2×C4×C71C2×D411yescharacteristic
101C2×C4×C71C2411yescharacteristic
102C2×Dic71—21no—
103C2×Dic71—21no—
104C2×Dic71—22no—
105C2×Dic71—44no—
106Dic142—21no—
107Dic142—22no—
108Dic142—22no—
109Dic142—44no—
110Dic142C2×D412yes—
111C71⋊C8—22no—
112C71⋊C8—21no—
113C71⋊C8C2×D412yes—
114C22×C71—22no—
115C22×C71—22no—
116C22×C71—22no—
117C22×C71—42no—
118C22×C71—42no—
119C22×C71—42no—
120C22×C71C22×D411yescharacteristic
121C4×C71—22no—
122C4×C71—22no—
123C4×C71—21no—
124C4×C71—41no—
125C4×C71#13712yescomplemented
126C4×C71C22×D411yescharacteristic
127C4×C71H(5,𝔽2)11yescharacteristic
128Dic71—22no—
129Dic71—42no—
130Dic71—41no—
131C2×C71—42no—
132C2×C71—82no—
133C2×C71D4211yescharacteristic
134C2×C71C2×#13711yescharacteristic
135C2×C71#57711yescharacteristic
136#3278—711nomaximal, Sylow
137C71#327811yescharacteristic, complemented, Sylow
138C2×#137—712no—
139#550D7111yescharacteristic
140D4×Q8—711no—
141#492—711no—
142#468—712no—
143#462—711no—
144#460—711no—
145#452—712no—
146#448—712no—
147C4×Q16—711no—
148#436—711no—
149C2×#41C2×D7112yes—
150C22×Q8—712no—
151#137—1428no—
152C2×Q16—711no—
153C2×Q16—712no—
154C2×SD16—712no—
155C2×SD16—712no—
156C2×SD16—712no—
157C2×M16—712no—
158C4⋊Q8—711no—
159C4⋊Q8—1421no—
160#127—21no—
161#127C2×D7111yescharacteristic
162C22⋊Q8—712no—
163C22⋊Q8—1422no—
164#121—22no—
165#121C2×D7112yes—
166C4×Q8—711no—
167C4×Q8C2×D7111yescharacteristic
168C4×D4—21no—
169C4×D4—1421no—
170C4×D4C2×D7111yescharacteristic
171#107—711no—
172C4⋊C8—711no—
173#103—711no—
174#103—711no—
175#103—712no—
176#102—712no—
177C22⋊C8—712no—
178C4×C8—711no—
179#41—24no—
180#41—42no—
181C2×Q8—2842no—
182C2×Q8—711no—
183C2×Q8—712no—
184C2×Q8—1424no—
185C2×Q8—1422no—
186C2×Q8C22×D7111yescharacteristic
187C2×D4—22no—
188C2×D4—22no—
189C2×D4—21no—
190C2×D4—21no—
191C2×D4C22⋊D7111yescharacteristic
192C2×D4C22×D7112yes—
193C22×C4—22no—
194C22×C4—1422no—
195C22×C4C22×D7112yes—
196Q16—714no—
197Q16—1424no—
198SD16—2842no—
199SD16—1424no—
200SD16—1424no—
201M16—1424no—
202C2×C8—711no—
203C2×C8—711no—
204C2×C8—712no—
205C4⋊C4—21no—
206C4⋊C4—711no—
207C4⋊C4—1421no—
208C4⋊C4—1422no—
209C4⋊C4—1421no—
210C4⋊C4—1421no—
211C4⋊C4C22⋊D7111yescharacteristic
212C4⋊C4C22×D7111yescharacteristic
213C22⋊C4—22no—
214C22⋊C4—1422no—
215C22⋊C4C22⋊D7112yes—
216C42—21no—
217C42—1421no—
218C42C22×D7111yescharacteristic
219C23—22no—
220C23C2×C22⋊D7112yes—
221Q8—21no—
222Q8—2844no—
223Q8—712no—
224Q8—1422no—
225Q8—1422no—
226Q8—1421no—
227Q8D4×D7112yes—
228D4—22no—
229D4—22no—
230D4—22no—
231D4—44no—
232D4—42no—
233D4—81no—
234D4—82no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—44no—
240C2×C4—2844no—
241C2×C4—1421no—
242C2×C4—1422no—
243C2×C4—1421no—
244C2×C4C2×C22⋊D7111yescharacteristic
245C2×C4C2×C22⋊D7112yes—
246C2×C4C2×C22⋊D7111yescharacteristic
247C2×C4C23×D7111yescharacteristic
248C8—712no—
249C8—1422no—
250C8—1421no—
251C22—22no—
252C22—22no—
253C22—22no—
254C22—42no—
255C22—42no—
256C22—42no—
257C22C22×C22⋊D7111yescharacteristic
258C4—22no—
259C4—22no—
260C4—21no—
261C4—41no—
262C4—2842no—
263C4—2841no—
264C4—1422no—
265C4#1455615812yescomplemented
266C4C2×D4×D7111yescharacteristic
267C4#1455617311yescharacteristic
268C2—42no—
269C2—82no—
270C2D4×C22⋊D7111yescharacteristic
271C2C2×#1455615811yescharacteristic
272C2#1798681011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#550×C71C2
socleC22×C71C22×D4
hypercenter#550D71
hypocenterC71#3278
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C4×D4 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C2×#14556158C2semidirect
2. D4×Dic142C2semidirect
3. #17986371C2semidirect
4. #17986243C2semidirect
5. #17986209C2semidirect
6. #17986196C2semidirect
7. #17986161C2semidirect
8. #17986137C2semidirect
9. C4×#11765855C2semidirect
10. #17986077C2semidirect
49. #11765855D4semidirect
125. C4×C71#137semidirect
137. C71#3278semidirect
265. C4#14556158semidirect