Finite Group Databasebeta

← #22599357

Subgroups — #22599357

Order 9088.

490 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 115 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
1#127×D71C211yesmaximal, characteristic, complemented
2#17986531C211yesmaximal, characteristic, complemented
3#17986387C212yesmaximal, complemented
4#17986362C212yesmaximal, complemented
5#17986200C211yesmaximal, characteristic, complemented
6#17986196C211yesmaximal, characteristic, complemented
7#17986194C211yesmaximal, characteristic, complemented
8#17986151C212yesmaximal, complemented
9#17986131C212yesmaximal, complemented
10#17986055C211yesmaximal, characteristic, complemented
11#459×C71C211yesmaximal, characteristic, complemented
12C2×D4×D71—21no—
13C2×D4×D71—21no—
14C2×D4×D71C2212yes—
15C2×#11765854C2212yescomplemented
16C2×#11765853C2212yescomplemented
17C2×#11765850C2212yescomplemented
18#14556125C2211yescharacteristic
19#14556124—21no—
20#14556124—21no—
21#14556124C2212yes—
22#14556121—21no—
23#14556119C2211yescharacteristic, complemented
24#14556115C2211yescharacteristic, complemented
25#14556110—21no—
26#14556108C2211yescharacteristic
27#14556099C2211yescharacteristic
28C4×Dic142C2211yescharacteristic, complemented
29C4×C22⋊D71—21no—
30#14556087—21no—
31C42×D71C2211yescharacteristic
32C4⋊(C71⋊C8)C2211yescharacteristic, complemented
33#14556050C2212yescomplemented
34#14556049C2212yescomplemented
35#14556048C2212yescomplemented
36(C2×D71)⋊C8C2212yes—
37#14556038C2212yescomplemented
38C2×D8×C71C2212yescomplemented
39#127×C71C2211yescharacteristic
40C4×D4×C71C2211yescharacteristic, complemented
41C4⋊C8×C71C2211yescharacteristic, complemented
42#102×C71C2212yescomplemented
43C23×D71—22no—
44C2×Dic142C2311yescharacteristic
45C2×D284—21no—
46C2×D284—21no—
47C2×D284C2312yes—
48C2×C22⋊D71—21no—
49C2×C22⋊D71—22no—
50C2×C22⋊D71—22no—
51C2×C22⋊D71—22no—
52D4×D71—42no—
53D4×D71—42no—
54D4×D71—44no—
55D4×D71—44no—
56C2×C4×D71D411yescharacteristic
57C2×C4×D71D411yescharacteristic
58C2×C4×D71C2311yescharacteristic
59#11765854—42no—
60#11765853—42no—
61#11765850—42no—
62C2×C71⋊C8C2312yes—
63C4⋊Dic71—21no—
64C4⋊Dic71—21no—
65C4⋊Dic71C2311yescharacteristic
66Dic71⋊C4—21no—
67Dic71⋊C4—21no—
68C22⋊Dic71—21no—
69C22⋊Dic71—21no—
70C22⋊Dic71—21no—
71(C2×D71)⋊C4—21no—
72C4×Dic71—21no—
73C4×Dic71D411yescharacteristic
74C4×Dic71D411yescharacteristic
75C4×Dic71C2311yescharacteristic
76C2×D4×C71—21no—
77C2×D4×C71—21no—
78C2×D4×C71C2312yes—
79C2×D4×C71C2311yescharacteristic
80C22×C4×C71—21no—
81D8×C71—42no—
82C2×C8×C71C2312yes—
83C4⋊C4×C71C2311yescharacteristic
84C22⋊C4×C71—21no—
85C42×C71C2311yescharacteristic
86C23×C71—21no—
87C23×C71—22no—
88C22×D71—22no—
89C22×D71—44no—
90C22×D71—44no—
91C22×D71—44no—
92C22×D71C2×D411yescharacteristic
93D4×C71—21no—
94D4×C71—22no—
95D4×C71—21no—
96D4×C71—22no—
97D4×C71—42no—
98D4×C71—42no—
99C2×C4×C71—21no—
100C2×C4×C71—42no—
101C2×C4×C71C2×D411yescharacteristic
102C2×C4×C71C2×D411yescharacteristic
103C2×C4×C71C2411yescharacteristic
104C22⋊D71—44no—
105C22⋊D71—44no—
106C22⋊D71—44no—
107C22⋊D71—42no—
108C2×Dic71—21no—
109C2×Dic71—21no—
110C2×Dic71C2×D411yescharacteristic
111C2×Dic71C2×D411yescharacteristic
112C2×Dic71C2×D411yescharacteristic
113D284—22no—
114D284—22no—
115D284—42no—
116D284—42no—
117C4×D71—22no—
118C4×D71—22no—
119C4×D71—21no—
120C4×D71—21no—
121Dic142—21no—
122Dic142—21no—
123C8×C71—22no—
124C71⋊C8—22no—
125C22×C71—42no—
126C22×C71—41no—
127C22×C71—42no—
128C22×C71—42no—
129C22×C71—41no—
130C22×C71—41no—
131C22×C71C22×D411yescharacteristic
132C2×D71—21no—
133C2×D71—21no—
134C2×D71—42no—
135C2×D71—42no—
136C2×D71—42no—
137C2×D71—88no—
138C2×D71AGL(1,ℤ/8)12yes—
139C4×C71—21no—
140C4×C71—41no—
141C4×C71AGL(1,ℤ/8)12yes—
142C4×C71H(5,𝔽2)11yescharacteristic
143C4×C71H(5,𝔽2)11yescharacteristic
144Dic71—21no—
145Dic71—21no—
146Dic71—21no—
147Dic71—41no—
148Dic71—41no—
149C2×C71—82no—
150C2×C71—81no—
151C2×C71#53511yescharacteristic
152C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
153C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
154D71—22no—
155D71—82no—
156#3217—711nomaximal, Sylow
157C71#321711yescharacteristic, complemented, Sylow
158C2×#127—711no—
159#529—711no—
160#496—712no—
161#490—712no—
162#460—711no—
163#460—711no—
164#459—711no—
165#459D7111yescharacteristic, complemented
166#450—712no—
167#447—712no—
168#432—711no—
169C22×D4—712no—
170C22×D4—1421no—
171C22×D4—1421no—
172C2×SD16—712no—
173C2×SD16—712no—
174C2×D8—712no—
175C2×D8C2×D7112yescomplemented
176#127—712no—
177#127—711no—
178#127—1421no—
179#127—1421no—
180#127C2×D7111yescharacteristic
181#126—1421no—
182#125—711no—
183#124—711no—
184#123—1421no—
185C22⋊Q8—711no—
186#121—711no—
187C4×Q8—711no—
188C4×D4—1421no—
189C4×D4C2×D7111yescharacteristic, complemented
190#117—1421no—
191C2×C42—711no—
192C4⋊C8—711no—
193C4⋊C8C2×D7111yescharacteristic, complemented
194#102—712no—
195#102—712no—
196#102—712no—
197#102C2×D7112yescomplemented
198C22⋊C8—712no—
199#97—712no—
200C24—1422no—
201C2×Q8—711no—
202C2×D4—21no—
203C2×D4—21no—
204C2×D4—2842no—
205C2×D4—2842no—
206C2×D4—2844no—
207C2×D4—2844no—
208C2×D4—712no—
209C2×D4—1422no—
210C2×D4—1422no—
211C2×D4—1422no—
212C2×D4—1421no—
213C2×D4—1421no—
214C2×D4—1421no—
215C2×D4C22×D7112yes—
216C2×D4C22×D7111yescharacteristic
217C22×C4—21no—
218C22×C4—711no—
219C22×C4—711no—
220C22×C4—711no—
221SD16—2842no—
222SD16—2842no—
223D8—42no—
224D8—2842no—
225C2×C8—712no—
226C2×C8C22×D7112yes—
227C4⋊C4—711no—
228C4⋊C4—1421no—
229C4⋊C4—1421no—
230C4⋊C4—1421no—
231C4⋊C4—1421no—
232C4⋊C4C22×D7111yescharacteristic
233C22⋊C4—21no—
234C22⋊C4—1421no—
235C22⋊C4—1421no—
236C22⋊C4—1421no—
237C22⋊C4—1421no—
238C42—711no—
239C42—711no—
240C42—711no—
241C42—1421no—
242C42C22×D7111yescharacteristic
243C23—21no—
244C23—22no—
245C23—2844no—
246C23—2844no—
247C23—2844no—
248C23—711no—
249C23—1422no—
250Q8—1421no—
251Q8—1421no—
252D4—21no—
253D4—22no—
254D4—21no—
255D4—22no—
256D4—42no—
257D4—42no—
258D4—2842no—
259D4—2842no—
260D4—2844no—
261D4—2844no—
262D4—2844no—
263D4—2842no—
264D4—1422no—
265D4—1422no—
266C2×C4—21no—
267C2×C4—42no—
268C2×C4—711no—
269C2×C4—711no—
270C2×C4—711no—
271C2×C4—1422no—
272C2×C4—1421no—
273C2×C4—1421no—
274C2×C4—1421no—
275C2×C4—1421no—
276C2×C4—1422no—
277C2×C4D4×D7111yescharacteristic
278C2×C4D4×D7111yescharacteristic
279C2×C4C23×D7111yescharacteristic
280C8—22no—
281C8—1422no—
282C22—42no—
283C22—41no—
284C22—42no—
285C22—41no—
286C22—41no—
287C22—42no—
288C22—2842no—
289C22—2842no—
290C22—2842no—
291C22—5688no—
292C22—712no—
293C22—1421no—
294C22—1421no—
295C22C2×D4×D7111yescharacteristic
296C4—21no—
297C4—41no—
298C4—2841no—
299C4—2841no—
300C4—1421no—
301C4—1421no—
302C4—1421no—
303C4#1455615212yes—
304C4#1455617311yescharacteristic
305C4#1455617411yescharacteristic
306C2—82no—
307C2—81no—
308C2—5682no—
309C2—1422no—
310C2#1798656511yescharacteristic
311C2AGL(1,ℤ/8)×D7111yescharacteristic
312C2C2×#1455615211yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#459×C71C2
socleC22×C71C22×D4
hypercenter#459D71
hypocenterC71#3217
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. #127×D71C2semidirect
2. #17986531C2semidirect
3. #17986387C2semidirect
4. #17986362C2semidirect
5. #17986200C2semidirect
6. #17986196C2semidirect
7. #17986194C2semidirect
8. #17986151C2semidirect
9. #17986131C2semidirect
10. #17986055C2semidirect
11. #459×C71C2semidirect
15. C2×#11765854C22semidirect
16. C2×#11765853C22semidirect
17. C2×#11765850C22semidirect
23. #14556119C22semidirect
24. #14556115C22semidirect
28. C4×Dic142C22semidirect
32. C4⋊(C71⋊C8)C22semidirect
33. #14556050C22semidirect
34. #14556049C22semidirect
35. #14556048C22semidirect
37. #14556038C22semidirect
38. C2×D8×C71C22semidirect
40. C4×D4×C71C22semidirect
41. C4⋊C8×C71C22semidirect
42. #102×C71C22semidirect
157. C71#3217semidirect
165. #459D71semidirect
175. C2×D8C2×D71semidirect
189. C4×D4C2×D71semidirect
193. C4⋊C8C2×D71semidirect
197. #102C2×D71semidirect