Finite Group Databasebeta

← #22597946

Subgroups — #22597946

Order 9088.

500 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 141 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×#14556148C211yesmaximal, characteristic, complemented
2C22×#11765849C211yesmaximal, characteristic, complemented
3#17986510C211yesmaximal, characteristic, complemented
4#17986255C212yesmaximal, complemented
5#17986250C212yesmaximal, complemented
6#17986247C212yesmaximal, complemented
7#17986243C212yesmaximal, complemented
8#17986035C211yesmaximal, characteristic, complemented
9#17985995C211yesmaximal, characteristic, complemented
10#17985993C211yesmaximal, characteristic, complemented
11#525×C71C211yesmaximal, characteristic, complemented
12C2×#11765865C2211yescharacteristic
13#14556148—24no—
14C2×#11765855C2211yescharacteristic, complemented
15C2×#11765853C2211yescharacteristic, complemented
16C2×#11765852C2211yescharacteristic, complemented
17C2×#11765850C2211yescharacteristic, complemented
18C2×#11765849—24no—
19C2×#11765849C2212yes—
20C2×#11765849C2212yes—
21C22×C71⋊C8C2211yescharacteristic
22C22×C71⋊C8C2211yescharacteristic
23#14556111—21no—
24#14556107C2212yes—
25#14556102C2212yes—
26#14556097—21no—
27C4×C22⋊D71—22no—
28#14556087C2211yescharacteristic
29#14556061C2212yescomplemented
30#14556059C2212yescomplemented
31#14556052C2212yescomplemented
32#14556051C2212yescomplemented
33#14556049C2212yescomplemented
34#14556048C2212yescomplemented
35C2×#41×C71C2211yescharacteristic
36C23×C4×C71C2211yescharacteristic
37#123×C71—21no—
38C22⋊Q8×C71C2212yes—
39#121×C71C2212yes—
40#120×C71—21no—
41C4×D4×C71—22no—
42#117×C71C2211yescharacteristic
43#11765865—22no—
44#11765865—22no—
45C2×Dic142C2311yescharacteristic
46C2×D284C2311yescharacteristic
47C2×C22⋊D71—21no—
48C2×C22⋊D71—22no—
49C2×C4×D71—21no—
50#11765855—22no—
51#11765853—22no—
52#11765852—22no—
53#11765850—22no—
54#11765849—24no—
55#11765849—24no—
56C2×C71⋊C8—22no—
57C2×C71⋊C8D414yescomplemented
58C2×C71⋊C8C2312yes—
59C2×C71⋊C8C2311yescharacteristic
60C2×C71⋊C8C2311yescharacteristic
61C4⋊Dic71C2312yes—
62Dic71⋊C4—22no—
63C22⋊Dic71—21no—
64C22⋊Dic71—22no—
65(C2×D71)⋊C4—22no—
66C4×Dic71—21no—
67C24×C71D411yescharacteristic
68#41×C71—22no—
69#41×C71—22no—
70C2×Q8×C71C2311yescharacteristic
71C2×D4×C71—22no—
72C2×D4×C71—21no—
73C2×D4×C71C2311yescharacteristic
74C22×C4×C71—22no—
75C22×C4×C71—22no—
76C22×C4×C71—21no—
77C22×C4×C71D412yes—
78C22×C4×C71D411yescharacteristic
79C22×C4×C71C2312yes—
80C22×C4×C71C2311yescharacteristic
81C4⋊C4×C71—22no—
82C4⋊C4×C71C2312yes—
83C22⋊C4×C71—21no—
84C22⋊C4×C71—22no—
85C22⋊C4×C71—22no—
86C42×C71—21no—
87C23×C71—22no—
88C23×C71—21no—
89C23×C71—42no—
90C23×C71C2×D412yes—
91C23×C71C2×D411yescharacteristic
92C22×D71—21no—
93Q8×C71—21no—
94Q8×C71—21no—
95D4×C71—21no—
96D4×C71—21no—
97D4×C71—44no—
98D4×C71—42no—
99C2×C4×C71—22no—
100C2×C4×C71—24no—
101C2×C4×C71—22no—
102C2×C4×C71—22no—
103C2×C4×C71—22no—
104C2×C4×C71—21no—
105C2×C4×C71—42no—
106C2×C4×C71C2×D412yes—
107C2×C4×C71C2×D412yes—
108C2×C4×C71C2×D411yescharacteristic
109C2×C4×C71#4112yes—
110C2×C4×C71C2411yescharacteristic
111C22⋊D71—44no—
112C22⋊D71—42no—
113C2×Dic71—22no—
114C2×Dic71—21no—
115D284—21no—
116D284—21no—
117C4×D71—42no—
118Dic142—21no—
119Dic142—21no—
120C71⋊C8—22no—
121C71⋊C8C2×D412yes—
122C71⋊C8C2×D412yes—
123C22×C71—22no—
124C22×C71—22no—
125C22×C71—22no—
126C22×C71—22no—
127C22×C71—44no—
128C22×C71—41no—
129C22×C71—42no—
130C22×C71#12112yes—
131C22×C71C22×D411yescharacteristic
132C2×D71—41no—
133C2×D71—42no—
134C4×C71—22no—
135C4×C71—41no—
136C4×C71—42no—
137C4×C71#12112yes—
138C4×C71C22×D411yescharacteristic
139C4×C71C2×#4111yescharacteristic
140Dic71—41no—
141Dic71—42no—
142C2×C71—22no—
143C2×C71—42no—
144C2×C71—81no—
145C2×C71C2×#12111yescharacteristic
146C2×C71#57512yes—
147D71—81no—
148#3054—711nomaximal, Sylow
149C71#305411yescharacteristic, complemented, Sylow
150C2×#135—711no—
151C22×M16—711no—
152#525—711no—
153#525D7111yescharacteristic, complemented
154#470—712no—
155#469—712no—
156#468—712no—
157#468—712no—
158#427—711no—
159#416—711no—
160#416—711no—
161C2×#41—711no—
162C2×#41C2×D7111yescharacteristic
163C23×C4C2×D7111yescharacteristic
164#135—1424no—
165C2×Q16—711no—
166C2×SD16—711no—
167C2×SD16—711no—
168C2×D8—711no—
169C2×M16—712no—
170C2×M16—712no—
171C2×M16—1424no—
172C22×C8—711no—
173C22×C8—711no—
174#123—21no—
175#123—1421no—
176C22⋊Q8—712no—
177C22⋊Q8C2×D7112yes—
178#121—712no—
179#121C2×D7112yes—
180#120—21no—
181#120—1421no—
182C4×D4—22no—
183C4×D4—1422no—
184#117—711no—
185#117C2×D7111yescharacteristic
186#107—712no—
187#106—712no—
188#103—712no—
189#103—712no—
190#102—712no—
191#102—712no—
192C24C22⋊D7111yescharacteristic
193#41—22no—
194#41—22no—
195#41—1422no—
196#41—1422no—
197C2×Q8—711no—
198C2×Q8C22×D7111yescharacteristic
199C2×D4—22no—
200C2×D4—21no—
201C2×D4—711no—
202C2×D4—1422no—
203C2×D4—1421no—
204C2×D4C22×D7111yescharacteristic
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—1421no—
209C22×C4C22⋊D7111yescharacteristic
210C22×C4C22⋊D7112yes—
211C22×C4C22×D7112yes—
212C22×C4C22×D7111yescharacteristic
213Q16—1422no—
214SD16—1422no—
215SD16—1422no—
216D8—1422no—
217M16—1424no—
218M16—1424no—
219C2×C8—714no—
220C2×C8—711no—
221C2×C8—711no—
222C2×C8—712no—
223C2×C8—1422no—
224C4⋊C4—22no—
225C4⋊C4—712no—
226C4⋊C4—1422no—
227C4⋊C4C22×D7112yes—
228C22⋊C4—21no—
229C22⋊C4—22no—
230C22⋊C4—22no—
231C22⋊C4—1421no—
232C22⋊C4—1422no—
233C22⋊C4—1422no—
234C42—21no—
235C42—1421no—
236C23—22no—
237C23—21no—
238C23—42no—
239C23—1421no—
240C23C2×C22⋊D7112yes—
241C23C2×C22⋊D7111yescharacteristic
242Q8—21no—
243Q8—21no—
244Q8—1421no—
245Q8—1421no—
246D4—21no—
247D4—21no—
248D4—44no—
249D4—42no—
250D4—2844no—
251D4—2842no—
252D4—1421no—
253D4—1421no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—24no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—42no—
261C2×C4—2842no—
262C2×C4—1422no—
263C2×C4—1421no—
264C2×C4D4×D7112yes—
265C2×C4C2×C22⋊D7112yes—
266C2×C4C2×C22⋊D7111yescharacteristic
267C2×C4#1176586412yes—
268C2×C4C23×D7111yescharacteristic
269C8—712no—
270C8—712no—
271C8—1422no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—44no—
277C22—41no—
278C22—42no—
279C22—2841no—
280C22—2842no—
281C22#1455610012yes—
282C22C22×C22⋊D7111yescharacteristic
283C4—22no—
284C4—41no—
285C4—42no—
286C4—2841no—
287C4—2842no—
288C4#1455610012yes—
289C4C2×D4×D7111yescharacteristic
290C4C2×#1176586411yescharacteristic
291C2—22no—
292C2—42no—
293C2—81no—
294C2—5681no—
295C2C2×#1455610011yescharacteristic
296C2#1798679712yes—
subgroupG/H
centerC2×C4C2×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#525×C71C2
socleC22×C71C22×D4
hypercenter#525D71
hypocenterC71#3054
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C2×#14556148C2semidirect
2. C22×#11765849C2semidirect
3. #17986510C2semidirect
4. #17986255C2semidirect
5. #17986250C2semidirect
6. #17986247C2semidirect
7. #17986243C2semidirect
8. #17986035C2semidirect
9. #17985995C2semidirect
10. #17985993C2semidirect
11. #525×C71C2semidirect
14. C2×#11765855C22semidirect
15. C2×#11765853C22semidirect
16. C2×#11765852C22semidirect
17. C2×#11765850C22semidirect
29. #14556061C22semidirect
30. #14556059C22semidirect
31. #14556052C22semidirect
32. #14556051C22semidirect
33. #14556049C22semidirect
34. #14556048C22semidirect
57. C2×C71⋊C8D4semidirect
149. C71#3054semidirect
153. #525D71semidirect