Finite Group Databasebeta

← #22597598

Subgroups — #22597598

Order 9088.

754 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C22×#11765865C211yesmaximal, characteristic
2C2×#14556159C212yesmaximal, complemented
3#17986159C214yesmaximal, complemented
4#17986154C212yesmaximal, complemented
5#17986149C212yesmaximal, complemented
6#17985998C212yesmaximal, complemented
7#17985963C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8×C71C211yesmaximal, characteristic, complemented
9C2×#11765865—28no—
10C2×#11765865—21no—
11C2×#11765865—21no—
12C2×#11765865C2212yes—
13C2×#11765865C2212yes—
14C22×Dic142C2211yescharacteristic
15C22×D284C2211yescharacteristic
16C22×C22⋊D71—21no—
17C22×C4×D71—21no—
18#14556159—28no—
19C2×#11765855C2214yescomplemented
20C2×#11765852C2214yescomplemented
21C2×#11765849C2212yescomplemented
22#14556051C2214yescomplemented
23#14556048C2214yescomplemented
24C22⋊(C71⋊C8)C2212yes—
25C22⋊(C71⋊C8)C2212yescomplemented
26C22×Q8×C71C2211yescharacteristic, complemented
27C23×C4×C71C2211yescharacteristic
28C22⋊Q8×C71—21no—
29C22⋊Q8×C71—21no—
30C22⋊Q8×C71C2212yes—
31C22⋊Q8×C71C2212yescomplemented
32C2×C4⋊C4×C71—21no—
33C2×C4⋊C4×C71C2211yescharacteristic, complemented
34C2×C22⋊C4×C71—21no—
35C23×D71—21no—
36#11765865—28no—
37#11765865—28no—
38#11765865—24no—
39#11765865—24no—
40#11765865—24no—
41#11765865D418yes—
42C2×Dic142—21no—
43C2×Dic142—24no—
44C2×Dic142C2312yes—
45C2×D284—21no—
46C2×D284—24no—
47C2×D284C2312yes—
48C2×C22⋊D71—28no—
49C2×C22⋊D71—22no—
50C2×C22⋊D71—22no—
51C22×Dic71—21no—
52C2×C4×D71—28no—
53C2×C4×D71—22no—
54C2×C4×D71—22no—
55#11765855—28no—
56#11765852—28no—
57#11765849—24no—
58C2×C71⋊C8C2314yes—
59C24×C71D411yescharacteristic
60C2×Q8×C71—21no—
61C2×Q8×C71—24no—
62C2×Q8×C71C2312yes—
63C22×C4×C71—24no—
64C22×C4×C71—21no—
65C22×C4×C71—21no—
66C22×C4×C71D411yescharacteristic
67C22×C4×C71D411yescharacteristic
68C22×C4×C71D411yescharacteristic
69C22×C4×C71C2311yescharacteristic
70C22×C4×C71C2311yescharacteristic
71C22×C4×C71C2311yescharacteristic
72C4⋊C4×C71—22no—
73C4⋊C4×C71—22no—
74C4⋊C4×C71—21no—
75C4⋊C4×C71C2312yes—
76C22⋊C4×C71—22no—
77C22⋊C4×C71—22no—
78C23×C71—22no—
79C23×C71—22no—
80C23×C71—41no—
81C23×C71C2×D411yescharacteristic
82C23×C71C2×D411yescharacteristic
83C23×C71C2×D411yescharacteristic
84C22×D71—24no—
85C22×D71—22no—
86C22×D71—42no—
87C22×D71—42no—
88Q8×C71—24no—
89Q8×C71—42no—
90C2×C4×C71—22no—
91C2×C4×C71—22no—
92C2×C4×C71—22no—
93C2×C4×C71—22no—
94C2×C4×C71—22no—
95C2×C4×C71—22no—
96C2×C4×C71—22no—
97C2×C4×C71—42no—
98C2×C4×C71—42no—
99C2×C4×C71C2×D414yes—
100C2×C4×C71C2×D411yescharacteristic
101C2×C4×C71C2×D411yescharacteristic
102C2×C4×C71C2×D411yescharacteristic
103C2×C4×C71C2411yescharacteristic
104C22⋊D71—28no—
105C22⋊D71—28no—
106C2×Dic71—24no—
107C2×Dic71—22no—
108D284—24no—
109D284—22no—
110D284C2×D414yes—
111C4×D71—28no—
112C4×D71—28no—
113Dic142—24no—
114Dic142—22no—
115Dic142C2×D414yes—
116C71⋊C8—24no—
117C22×C71—21no—
118C22×C71—21no—
119C22×C71—21no—
120C22×C71—21no—
121C22×C71—21no—
122C22×C71—21no—
123C22×C71—21no—
124C22×C71—44no—
125C22×C71#12012yes—
126C22×C71#13712yescomplemented
127C22×C71C22×D411yescharacteristic
128C2×D71—24no—
129C2×D71—48no—
130C2×D71—42no—
131C2×D71—42no—
132C4×C71—21no—
133C4×C71—21no—
134C4×C71—42no—
135C4×C71—42no—
136C4×C71#12012yes—
137C4×C71C22×D412yes—
138Dic71—24no—
139C2×C71—22no—
140C2×C71—22no—
141C2×C71—41no—
142C2×C71C2×#12011yescharacteristic
143C2×C71C2×#13711yescharacteristic
144C2×C71#57511yescharacteristic
145D71—44no—
146#3007—711nomaximal, Sylow
147C71#300711yescharacteristic, complemented, Sylow
148C22×#41—711no—
149C2×#137—712no—
150C2×C22⋊Q8D7111yescharacteristic, complemented
151#452—714no—
152#451—712no—
153#450—712no—
154#417—712no—
155#407—711no—
156C2×#41—712no—
157C2×#41—712no—
158C2×#41—1428no—
159C2×#41—1421no—
160C2×#41—1421no—
161C22×Q8—711no—
162C22×Q8C2×D7111yescharacteristic, complemented
163C22×D4—711no—
164C22×D4—1421no—
165C23×C4—1421no—
166C23×C4C2×D7111yescharacteristic
167#137—1428no—
168C2×Q16—714no—
169C2×SD16—714no—
170C2×M16—712no—
171C22⋊Q8—21no—
172C22⋊Q8—21no—
173C22⋊Q8C2×D7112yes—
174C22⋊Q8C2×D7112yescomplemented
175C2×C4⋊C4—21no—
176C2×C4⋊C4C2×D7111yescharacteristic, complemented
177C2×C22⋊C4—21no—
178#103—714no—
179#102—714no—
180C22⋊C8—712no—
181C22⋊C8—712no—
182C24—1421no—
183C24C22⋊D7111yescharacteristic
184#41—718no—
185#41—1424no—
186#41—1424no—
187#41—1428no—
188#41—1428no—
189#41—1424no—
190C2×Q8—21no—
191C2×Q8—24no—
192C2×Q8—712no—
193C2×Q8—1424no—
194C2×Q8—1421no—
195C2×Q8C22×D7112yes—
196C2×D4—712no—
197C2×D4—1424no—
198C2×D4—1428no—
199C2×D4—1422no—
200C2×D4—1422no—
201C2×D4—1421no—
202C22×C4—24no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—1428no—
206C22×C4—1421no—
207C22×C4—1422no—
208C22×C4—1422no—
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—22no—
220C4⋊C4—22no—
221C4⋊C4—21no—
222C4⋊C4C22×D7112yes—
223C22⋊C4—22no—
224C22⋊C4—22no—
225C23—22no—
226C23—22no—
227C23—41no—
228C23—2842no—
229C23—2842no—
230C23—1424no—
231C23—1422no—
232C23C2×C22⋊D7111yescharacteristic
233C23C2×C22⋊D7111yescharacteristic
234C23C2×C22⋊D7111yescharacteristic
235Q8—24no—
236Q8—42no—
237Q8—714no—
238Q8—1424no—
239Q8—1422no—
240D4—714no—
241D4—1424no—
242D4—1428no—
243D4—1428no—
244D4—1422no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—1428no—
255C2×C4—1424no—
256C2×C4—1422no—
257C2×C4—1428no—
258C2×C4D4×D7114yes—
259C2×C4C2×C22⋊D7111yescharacteristic
260C2×C4C2×C22⋊D7111yescharacteristic
261C2×C4C2×C22⋊D7111yescharacteristic
262C2×C4C23×D7111yescharacteristic
263C8—1424no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—44no—
272C22—2848no—
273C22—2842no—
274C22—2842no—
275C22—1424no—
276C22#1455609612yes—
277C22#1455615912yescomplemented
278C22C22×C22⋊D7111yescharacteristic
279C4—21no—
280C4—21no—
281C4—42no—
282C4—42no—
283C4—1424no—
284C4#1455609612yes—
285C4C2×D4×D7112yes—
286C2—22no—
287C2—22no—
288C2—41no—
289C2—2844no—
290C2C2×#1455609611yescharacteristic
291C2C2×#1455615911yescharacteristic
292C2#1798679711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
FittingC2×C22⋊Q8×C71C2
socleC22×C71C22×D4
hypercenterC2×C22⋊Q8D71
hypocenterC71#3007
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. C2×#14556159C2semidirect
3. #17986159C2semidirect
4. #17986154C2semidirect
5. #17986149C2semidirect
6. #17985998C2semidirect
7. #17985963C2semidirect
8. C2×C22⋊Q8×C71C2semidirect
19. C2×#11765855C22semidirect
20. C2×#11765852C22semidirect
21. C2×#11765849C22semidirect
22. #14556051C22semidirect
23. #14556048C22semidirect
25. C22⋊(C71⋊C8)C22semidirect
26. C22×Q8×C71C22semidirect
31. C22⋊Q8×C71C22semidirect
33. C2×C4⋊C4×C71C22semidirect
126. C22×C71#137semidirect
147. C71#3007semidirect
150. C2×C22⋊Q8D71semidirect
162. C22×Q8C2×D71semidirect
174. C22⋊Q8C2×D71semidirect
176. C2×C4⋊C4C2×D71semidirect
277. C22#14556159semidirect