Finite Group Databasebeta

← #22597600

Subgroups — #22597600

Order 9088.

754 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 169 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
2C2×#14556107C211yesmaximal, characteristic, complemented
3#17986161C214yesmaximal, complemented
4#17986155C212yesmaximal, complemented
5#17986150C212yesmaximal, complemented
6#17986000C212yesmaximal, complemented
7#17985963C211yesmaximal, characteristic, complemented
8C22×#41×C71C211yesmaximal, characteristic
9C22×Dic142C2211yescharacteristic, complemented
10#14556158—28no—
11C2×#11765855C2214yescomplemented
12C2×#11765853C2214yescomplemented
13C2×#11765849C2212yescomplemented
14#14556107—21no—
15#14556107—21no—
16#14556107C2212yes—
17#14556107C2212yescomplemented
18C2×C4⋊Dic71C2211yescharacteristic, complemented
19C2×Dic71⋊C4—21no—
20C2×C22⋊Dic71—21no—
21#14556052C2214yescomplemented
22#14556049C2214yescomplemented
23C22⋊(C71⋊C8)C2212yes—
24C22⋊(C71⋊C8)C2212yescomplemented
25C2×#41×C71—28no—
26C2×#41×C71—21no—
27C2×#41×C71—21no—
28C2×#41×C71C2212yes—
29C2×#41×C71C2212yes—
30C22×Q8×C71C2211yescharacteristic
31C22×D4×C71—21no—
32C22×D4×C71C2211yescharacteristic
33C23×C4×C71—21no—
34C23×C4×C71C2211yescharacteristic
35C2×Dic142—24no—
36C2×Dic142—21no—
37C2×Dic142C2312yes—
38C22×Dic71—21no—
39C22×Dic71—21no—
40#11765855—28no—
41#11765853—28no—
42#11765849—24no—
43C2×C71⋊C8C2314yes—
44C4⋊Dic71—21no—
45C4⋊Dic71C2312yes—
46Dic71⋊C4—22no—
47Dic71⋊C4—22no—
48C22⋊Dic71—22no—
49C22⋊Dic71—22no—
50C24×C71—21no—
51C24×C71D411yescharacteristic
52#41×C71—28no—
53#41×C71—28no—
54#41×C71—24no—
55#41×C71—24no—
56#41×C71—24no—
57#41×C71D418yes—
58C2×Q8×C71—24no—
59C2×Q8×C71—21no—
60C2×Q8×C71C2312yes—
61C2×D4×C71—28no—
62C2×D4×C71—22no—
63C2×D4×C71—22no—
64C2×D4×C71—24no—
65C2×D4×C71—21no—
66C2×D4×C71C2312yes—
67C22×C4×C71—28no—
68C22×C4×C71—21no—
69C22×C4×C71—22no—
70C22×C4×C71—22no—
71C22×C4×C71—24no—
72C22×C4×C71D411yescharacteristic
73C22×C4×C71D411yescharacteristic
74C22×C4×C71D411yescharacteristic
75C22×C4×C71C2311yescharacteristic
76C22×C4×C71C2311yescharacteristic
77C22×C4×C71C2311yescharacteristic
78C23×C71—24no—
79C23×C71—22no—
80C23×C71—22no—
81C23×C71—22no—
82C23×C71—41no—
83C23×C71—42no—
84C23×C71—42no—
85C23×C71C2×D411yescharacteristic
86C23×C71C2×D411yescharacteristic
87C23×C71C2×D411yescharacteristic
88Q8×C71—24no—
89Q8×C71—22no—
90Q8×C71C2×D414yes—
91D4×C71—28no—
92D4×C71—24no—
93D4×C71—28no—
94D4×C71—22no—
95D4×C71C2×D414yes—
96C2×C4×C71—24no—
97C2×C4×C71—28no—
98C2×C4×C71—22no—
99C2×C4×C71—22no—
100C2×C4×C71—22no—
101C2×C4×C71—28no—
102C2×C4×C71—22no—
103C2×C4×C71—22no—
104C2×C4×C71—22no—
105C2×C4×C71C2×D411yescharacteristic
106C2×C4×C71C2×D411yescharacteristic
107C2×C4×C71C2×D411yescharacteristic
108C2×C4×C71C2×D414yes—
109C2×C4×C71C2411yescharacteristic
110C2×Dic71—22no—
111C2×Dic71—22no—
112C2×Dic71—42no—
113C2×Dic71—42no—
114Dic142—24no—
115Dic142—42no—
116C71⋊C8—24no—
117C22×C71—21no—
118C22×C71—21no—
119C22×C71—21no—
120C22×C71—21no—
121C22×C71—21no—
122C22×C71—24no—
123C22×C71—21no—
124C22×C71—21no—
125C22×C71—44no—
126C22×C71—48no—
127C22×C71—42no—
128C22×C71—42no—
129C22×C71#12012yes—
130C22×C71#13712yescomplemented
131C22×C71C22×D411yescharacteristic
132C4×C71—21no—
133C4×C71—24no—
134C4×C71—21no—
135C4×C71#12012yes—
136C4×C71C22×D412yes—
137Dic71—42no—
138Dic71—42no—
139C2×C71—22no—
140C2×C71—22no—
141C2×C71—41no—
142C2×C71—44no—
143C2×C71C2×#12011yescharacteristic
144C2×C71C2×#13711yescharacteristic
145C2×C71#57511yescharacteristic
146#3007—711nomaximal, Sylow
147C71#300711yescharacteristic, complemented, Sylow
148C22×#41D7111yescharacteristic
149C2×#137—712no—
150C2×C22⋊Q8—711no—
151#452—714no—
152#451—712no—
153#450—712no—
154#417—712no—
155#407—711no—
156C2×#41—28no—
157C2×#41—21no—
158C2×#41—21no—
159C2×#41C2×D7112yes—
160C2×#41C2×D7112yes—
161C22×Q8—711no—
162C22×Q8C2×D7111yescharacteristic
163C22×D4—21no—
164C22×D4C2×D7111yescharacteristic
165C23×C4—21no—
166C23×C4C2×D7111yescharacteristic
167#137—1428no—
168C2×Q16—714no—
169C2×SD16—714no—
170C2×M16—712no—
171C22⋊Q8—712no—
172C22⋊Q8—712no—
173C22⋊Q8—1421no—
174C22⋊Q8—1421no—
175C2×C4⋊C4—711no—
176C2×C4⋊C4—1421no—
177C2×C22⋊C4—1421no—
178#103—714no—
179#102—714no—
180C22⋊C8—712no—
181C22⋊C8—712no—
182C24—21no—
183C24C22⋊D7111yescharacteristic
184#41—28no—
185#41—28no—
186#41—24no—
187#41—24no—
188#41—24no—
189#41C22⋊D7118yes—
190C2×Q8—24no—
191C2×Q8—21no—
192C2×Q8—712no—
193C2×Q8—1424no—
194C2×Q8—1421no—
195C2×Q8C22×D7112yes—
196C2×D4—28no—
197C2×D4—22no—
198C2×D4—22no—
199C2×D4—24no—
200C2×D4—21no—
201C2×D4C22×D7112yes—
202C22×C4—28no—
203C22×C4—21no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—24no—
207C22×C4—1421no—
208C22×C4—1421no—
209C22×C4C22⋊D7111yescharacteristic
210C22×C4C22⋊D7111yescharacteristic
211C22×C4C22⋊D7111yescharacteristic
212C22×C4C22×D7111yescharacteristic
213C22×C4C22×D7111yescharacteristic
214C22×C4C22×D7111yescharacteristic
215Q16—1428no—
216SD16—1428no—
217M16—1424no—
218C2×C8—714no—
219C4⋊C4—712no—
220C4⋊C4—1422no—
221C4⋊C4—1422no—
222C4⋊C4—1421no—
223C22⋊C4—1422no—
224C22⋊C4—1422no—
225C23—24no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—41no—
230C23—42no—
231C23—42no—
232C23C2×C22⋊D7111yescharacteristic
233C23C2×C22⋊D7111yescharacteristic
234C23C2×C22⋊D7111yescharacteristic
235Q8—24no—
236Q8—22no—
237Q8—2842no—
238Q8—1424no—
239Q8C2×C22⋊D7114yes—
240D4—28no—
241D4—24no—
242D4—28no—
243D4—22no—
244D4C2×C22⋊D7114yes—
245C2×C4—24no—
246C2×C4—28no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—28no—
253C2×C4—22no—
254C2×C4—2842no—
255C2×C4—2842no—
256C2×C4—1422no—
257C2×C4—1422no—
258C2×C4C2×C22⋊D7111yescharacteristic
259C2×C4C2×C22⋊D7111yescharacteristic
260C2×C4C2×C22⋊D7114yes—
261C2×C4C2×C22⋊D7111yescharacteristic
262C2×C4C23×D7111yescharacteristic
263C8—1424no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—24no—
270C22—21no—
271C22—21no—
272C22—44no—
273C22—48no—
274C22—42no—
275C22—42no—
276C22#1455609712yes—
277C22#1455615812yescomplemented
278C22C22×C22⋊D7111yescharacteristic
279C4—21no—
280C4—24no—
281C4—21no—
282C4—2842no—
283C4—2842no—
284C4#1455609712yes—
285C4C22×C22⋊D7112yes—
286C2—22no—
287C2—22no—
288C2—41no—
289C2—44no—
290C2C2×#1455609711yescharacteristic
291C2C2×#1455615811yescharacteristic
292C2#1798679711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
FittingC22×#41×C71C2
socleC22×C71C22×D4
hypercenterC22×#41D71
hypocenterC71#3007
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#14556158C2semidirect
2. C2×#14556107C2semidirect
3. #17986161C2semidirect
4. #17986155C2semidirect
5. #17986150C2semidirect
6. #17986000C2semidirect
7. #17985963C2semidirect
9. C22×Dic142C22semidirect
11. C2×#11765855C22semidirect
12. C2×#11765853C22semidirect
13. C2×#11765849C22semidirect
17. #14556107C22semidirect
18. C2×C4⋊Dic71C22semidirect
21. #14556052C22semidirect
22. #14556049C22semidirect
24. C22⋊(C71⋊C8)C22semidirect
130. C22×C71#137semidirect
147. C71#3007semidirect
277. C22#14556158semidirect