Finite Group Databasebeta

← #22597815

Subgroups — #22597815

Order 9088.

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

normal subgroups only

This group has 584 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×#14556152C212yesmaximal, complemented
2#127×D71C211yesmaximal, characteristic, complemented
3C4×#11765864C211yesmaximal, characteristic, complemented
4#17986200C211yesmaximal, characteristic, complemented
5#17986198C212yesmaximal, complemented
6#17986196C211yesmaximal, characteristic, complemented
7#17986194C211yesmaximal, characteristic, complemented
8#17986192C212yesmaximal, complemented
9#17986021C211yesmaximal, characteristic, complemented
10#17986002C212yesmaximal, complemented
11#459×C71C211yesmaximal, characteristic, complemented
12C2×#11765864C2211yescharacteristic
13C2×D4×D71—21no—
14C2×D4×D71—21no—
15C2×D4×D71C2212yes—
16#14556152—28no—
17C2×#11765854C2212yescomplemented
18C2×#11765853C2212yescomplemented
19C2×#11765850C2212yescomplemented
20C2×#11765848C2212yes—
21#14556125C2211yescharacteristic
22#14556124—21no—
23#14556124—21no—
24#14556124C2212yes—
25C4×Dic142C2211yescharacteristic, complemented
26#14556093C2211yescharacteristic, complemented
27D4×Dic71C2211yescharacteristic, complemented
28C4×C22⋊D71—21no—
29(C22⋊D71)⋊C4—21no—
30#14556086C2211yescharacteristic
31#14556084—21no—
32C2×C4×Dic71—21no—
33C42×D71C2211yescharacteristic
34C4⋊(C71⋊C8)C2211yescharacteristic, complemented
35Dic71⋊C8C2212yescomplemented
36#14556050C2212yescomplemented
37#14556049C2212yescomplemented
38#14556048C2212yescomplemented
39C2×D8×C71C2212yescomplemented
40#127×C71C2211yescharacteristic
41C4×D4×C71C2211yescharacteristic, complemented
42C4⋊C8×C71C2211yescharacteristic, complemented
43#102×C71C2212yescomplemented
44C23×D71—22no—
45#11765864—22no—
46#11765864D414yescomplemented
47C2×Dic142C2311yescharacteristic
48C2×D284—21no—
49C2×D284—21no—
50C2×D284C2312yes—
51C2×C22⋊D71—21no—
52C2×C22⋊D71—22no—
53C2×C22⋊D71—22no—
54C2×C22⋊D71—22no—
55D4×D71—24no—
56D4×D71—24no—
57D4×D71—44no—
58D4×D71—44no—
59C22×Dic71—21no—
60C22×Dic71—21no—
61C2×C4×D71D411yescharacteristic
62C2×C4×D71D411yescharacteristic
63C2×C4×D71C2311yescharacteristic
64#11765854—24no—
65#11765853—24no—
66#11765850—24no—
67#11765848—24no—
68C2×C71⋊C8C2312yes—
69C4⋊Dic71C2311yescharacteristic
70Dic71⋊C4—21no—
71Dic71⋊C4—21no—
72C22⋊Dic71—21no—
73(C2×D71)⋊C4—21no—
74C4×Dic71—21no—
75C4×Dic71—21no—
76C4×Dic71—21no—
77C4×Dic71D411yescharacteristic
78C4×Dic71D411yescharacteristic
79C4×Dic71C2311yescharacteristic
80C2×D4×C71—21no—
81C2×D4×C71—21no—
82C2×D4×C71C2312yes—
83C2×D4×C71C2311yescharacteristic
84C22×C4×C71—21no—
85D8×C71—24no—
86C2×C8×C71C2312yes—
87C4⋊C4×C71C2311yescharacteristic
88C22⋊C4×C71—21no—
89C42×C71C2311yescharacteristic
90C23×C71—21no—
91C23×C71—22no—
92C22×D71—22no—
93C22×D71—44no—
94C22×D71—44no—
95C22×D71—44no—
96C22×D71C2×D411yescharacteristic
97D4×C71—22no—
98D4×C71—21no—
99D4×C71—22no—
100D4×C71—42no—
101D4×C71—42no—
102D4×C71C2×D412yescomplemented
103C2×C4×C71—21no—
104C2×C4×C71—42no—
105C2×C4×C71C2×D411yescharacteristic
106C2×C4×C71C2×D411yescharacteristic
107C2×C4×C71C2411yescharacteristic
108C22⋊D71—24no—
109C22⋊D71—44no—
110C22⋊D71—44no—
111C22⋊D71—44no—
112C2×Dic71—21no—
113C2×Dic71—24no—
114C2×Dic71—21no—
115C2×Dic71—42no—
116C2×Dic71C2×D411yescharacteristic
117C2×Dic71C2×D411yescharacteristic
118C2×Dic71C2×D411yescharacteristic
119D284—22no—
120D284—22no—
121D284—42no—
122D284—42no—
123C4×D71—22no—
124C4×D71—22no—
125C4×D71C2×D412yes—
126C4×D71#4112yes—
127Dic142—21no—
128Dic142C2×D412yescomplemented
129C8×C71—22no—
130C71⋊C8—22no—
131C22×C71—22no—
132C22×C71—41no—
133C22×C71—41no—
134C22×C71—42no—
135C22×C71—42no—
136C22×C71—42no—
137C22×C71C22×D411yescharacteristic
138C2×D71—21no—
139C2×D71—21no—
140C2×D71—42no—
141C2×D71—42no—
142C2×D71—42no—
143C2×D71—88no—
144C2×D71#12112yes—
145C4×C71—21no—
146C4×C71—41no—
147C4×C71AGL(1,ℤ/8)12yescomplemented
148C4×C71C22×D411yescharacteristic
149C4×C71C2×#4111yescharacteristic
150Dic71—22no—
151Dic71—21no—
152Dic71—41no—
153Dic71#12112yes—
154Dic71AGL(1,ℤ/8)12yescomplemented
155C2×C71—42no—
156C2×C71—82no—
157C2×C71C2×#12111yescharacteristic
158C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
159C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
160D71—22no—
161D71—82no—
162#3038—711nomaximal, Sylow
163C71#303811yescharacteristic, complemented, Sylow
164C2×AGL(1,ℤ/8)—712no—
165C2×#127—711no—
166C4×#41—711no—
167#460—711no—
168#460—711no—
169#460—712no—
170#459—712no—
171#459—711no—
172#459D7111yescharacteristic, complemented
173#423—711no—
174#418—712no—
175C2×#41—711no—
176C22×D4—712no—
177C22×D4—1421no—
178C22×D4—1421no—
179AGL(1,ℤ/8)—1428no—
180C2×SD16—712no—
181C2×SD16—712no—
182C2×D8—712no—
183C2×D8C2×D7112yescomplemented
184C2×M16—712no—
185#127—712no—
186#127—711no—
187#127—1421no—
188#127—1421no—
189#127C2×D7111yescharacteristic
190C4×Q8—711no—
191C4×Q8—711no—
192C4×D4—711no—
193C4×D4—1421no—
194C4×D4—1421no—
195C4×D4C2×D7111yescharacteristic, complemented
196#117—711no—
197#117—1421no—
198C2×C42—711no—
199C2×C42—1421no—
200C4⋊C8—712no—
201C4⋊C8—711no—
202C4⋊C8C2×D7111yescharacteristic, complemented
203#102—712no—
204#102—712no—
205#102—712no—
206#102C2×D7112yescomplemented
207C24—1422no—
208#41—714no—
209#41—1422no—
210C2×Q8—711no—
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—2844no—
214C2×D4—2844no—
215C2×D4—712no—
216C2×D4—1424no—
217C2×D4—1424no—
218C2×D4—1422no—
219C2×D4—1422no—
220C2×D4—1422no—
221C2×D4—1421no—
222C2×D4—1421no—
223C2×D4—1421no—
224C2×D4C22×D7112yes—
225C2×D4C22×D7111yescharacteristic
226C22×C4—21no—
227C22×C4—711no—
228C22×C4—711no—
229C22×C4—711no—
230C22×C4—1421no—
231C22×C4—1421no—
232SD16—1424no—
233SD16—1424no—
234D8—24no—
235D8—1424no—
236M16—1424no—
237C2×C8—712no—
238C2×C8C22×D7112yes—
239C4⋊C4—711no—
240C4⋊C4—1421no—
241C4⋊C4—1421no—
242C4⋊C4C22×D7111yescharacteristic
243C22⋊C4—21no—
244C22⋊C4—1421no—
245C22⋊C4—1421no—
246C42—711no—
247C42—711no—
248C42—711no—
249C42—1421no—
250C42—1421no—
251C42—1421no—
252C42C22×D7111yescharacteristic
253C23—21no—
254C23—22no—
255C23—2844no—
256C23—2844no—
257C23—2844no—
258C23—711no—
259C23—1422no—
260Q8—712no—
261Q8—1421no—
262D4—22no—
263D4—21no—
264D4—22no—
265D4—42no—
266D4—42no—
267D4—2842no—
268D4—2844no—
269D4—2844no—
270D4—2844no—
271D4—2842no—
272D4—1424no—
273D4—1422no—
274D4—1422no—
275D4D4×D7112yescomplemented
276C2×C4—21no—
277C2×C4—42no—
278C2×C4—2842no—
279C2×C4—711no—
280C2×C4—711no—
281C2×C4—712no—
282C2×C4—712no—
283C2×C4—711no—
284C2×C4—1422no—
285C2×C4—1421no—
286C2×C4—1424no—
287C2×C4—1421no—
288C2×C4—1422no—
289C2×C4D4×D7111yescharacteristic
290C2×C4D4×D7111yescharacteristic
291C2×C4C23×D7111yescharacteristic
292C8—22no—
293C8—1422no—
294C22—22no—
295C22—41no—
296C22—41no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—2842no—
301C22—2842no—
302C22—2842no—
303C22—5688no—
304C22—712no—
305C22—1421no—
306C22—1421no—
307C22C2×D4×D7111yescharacteristic
308C4—21no—
309C4—41no—
310C4—2841no—
311C4—712no—
312C4—712no—
313C4—1422no—
314C4—1421no—
315C4#1455615212yescomplemented
316C4C2×D4×D7111yescharacteristic
317C4#41×D7111yescharacteristic
318C2—42no—
319C2—82no—
320C2—5682no—
321C2—1422no—
322C2#121×D7111yescharacteristic
323C2AGL(1,ℤ/8)×D7111yescharacteristic
324C2C2×#1455615211yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#459×C71C2
socleC22×C71C22×D4
hypercenter#459D71
hypocenterC71#3038
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#14556152C2semidirect
2. #127×D71C2semidirect
3. C4×#11765864C2semidirect
4. #17986200C2semidirect
5. #17986198C2semidirect
6. #17986196C2semidirect
7. #17986194C2semidirect
8. #17986192C2semidirect
9. #17986021C2semidirect
10. #17986002C2semidirect
11. #459×C71C2semidirect
17. C2×#11765854C22semidirect
18. C2×#11765853C22semidirect
19. C2×#11765850C22semidirect
25. C4×Dic142C22semidirect
26. #14556093C22semidirect
27. D4×Dic71C22semidirect
34. C4⋊(C71⋊C8)C22semidirect
35. Dic71⋊C8C22semidirect
36. #14556050C22semidirect
37. #14556049C22semidirect
38. #14556048C22semidirect
39. C2×D8×C71C22semidirect
41. C4×D4×C71C22semidirect
42. C4⋊C8×C71C22semidirect
43. #102×C71C22semidirect
46. #11765864D4semidirect
102. D4×C71C2×D4semidirect
128. Dic142C2×D4semidirect
147. C4×C71AGL(1,ℤ/8)semidirect
154. Dic71AGL(1,ℤ/8)semidirect
163. C71#3038semidirect
172. #459D71semidirect
183. C2×D8C2×D71semidirect
195. C4×D4C2×D71semidirect
202. C4⋊C8C2×D71semidirect
206. #102C2×D71semidirect
275. D4D4×D71semidirect
315. C4#14556152semidirect