Finite Group Databasebeta

← #22599108

Subgroups — #22599108

Order 9088.

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

normal subgroups only

This group has 472 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#17986512C211yesmaximal, characteristic, complemented
2C2×#14556099C211yesmaximal, characteristic, complemented
3#17986332C211yesmaximal, characteristic, complemented
4#17986322C212yesmaximal, complemented
5#17986318C211yesmaximal, characteristic, complemented
6#17986234C212yesmaximal, complemented
7#17986225C212yesmaximal, complemented
8#17986143C212yesmaximal, complemented
9#17986133C211yesmaximal, characteristic, complemented
10C22⋊C8×D71C211yesmaximal, characteristic, complemented
11#450×C71C211yesmaximal, characteristic, complemented
12C2×#11765866C2211yescharacteristic, complemented
13C22×C22⋊D71—21no—
14C22×C4×D71C2211yescharacteristic
15C2×#11765850C2212yescomplemented
16C2×C8×D71C2212yescomplemented
17#14556113—21no—
18#14556104C2211yescharacteristic, complemented
19#14556103C2211yescharacteristic, complemented
20#14556102C2211yescharacteristic
21#14556099—21no—
22#14556099—21no—
23#14556099C2212yes—
24#14556099C2212yes—
25#14556098—21no—
26#14556090—21no—
27(C22⋊D71)⋊C4—21no—
28#14556086C2211yescharacteristic, complemented
29C2×C4⋊Dic71C2211yescharacteristic
30C2×C22⋊Dic71—21no—
31#14556061C2212yescomplemented
32#14556060C2212yescomplemented
33#14556052C2212yescomplemented
34#14556050C2212yescomplemented
35#14556049C2212yescomplemented
36C22⋊(C71⋊C8)C2211yescharacteristic
37(C2×D71)⋊C8C2212yescomplemented
38C22×D4×C71C2211yescharacteristic
39C2×SD16×C71C2212yescomplemented
40C22⋊Q8×C71C2211yescharacteristic
41#102×C71C2212yescomplemented
42C22⋊C8×C71C2211yescharacteristic
43C23×D71D411yescharacteristic
44#11765866—42no—
45C2×D284—21no—
46C2×D284—21no—
47C2×D284C2311yescharacteristic
48C2×C22⋊D71—21no—
49C2×C22⋊D71—22no—
50C2×C22⋊D71—22no—
51C2×C22⋊D71—44no—
52C22×Dic71—21no—
53C22×Dic71D411yescharacteristic
54C2×C4×D71—21no—
55C2×C4×D71—42no—
56C2×C4×D71D411yescharacteristic
57C2×C4×D71D411yescharacteristic
58C2×C4×D71C2311yescharacteristic
59C2×C4×D71C2311yescharacteristic
60#11765850—42no—
61C2×C71⋊C8C2312yes—
62C8×D71—42no—
63C4⋊Dic71—21no—
64C4⋊Dic71C2311yescharacteristic
65C4⋊Dic71C2312yes—
66Dic71⋊C4—21no—
67C22⋊Dic71—22no—
68C22⋊Dic71—22no—
69(C2×D71)⋊C4—21no—
70(C2×D71)⋊C4—21no—
71(C2×D71)⋊C4—21no—
72(C2×D71)⋊C4—21no—
73C4×Dic71—21no—
74C24×C71—21no—
75C2×Q8×C71C2311yescharacteristic
76C2×D4×C71—21no—
77C2×D4×C71—42no—
78C2×D4×C71C2312yes—
79C22×C4×C71C2311yescharacteristic
80SD16×C71—42no—
81C2×C8×C71C2312yes—
82C4⋊C4×C71—21no—
83C4⋊C4×C71C2311yescharacteristic
84C22⋊C4×C71—21no—
85C23×C71—22no—
86C23×C71—42no—
87C23×C71—42no—
88C23×C71—42no—
89C23×C71C2×D411yescharacteristic
90C22×D71—22no—
91C22×D71—22no—
92C22×D71—21no—
93C22×D71—41no—
94C22×D71C2×D411yescharacteristic
95C22×D71C2×D411yescharacteristic
96Q8×C71—22no—
97D4×C71—22no—
98D4×C71—22no—
99D4×C71—42no—
100C2×C4×C71—22no—
101C2×C4×C71—21no—
102C2×C4×C71—21no—
103C2×C4×C71C2×D411yescharacteristic
104C2×C4×C71C2411yescharacteristic
105C22⋊D71—44no—
106C22⋊D71—42no—
107C22⋊D71—44no—
108C2×Dic71—21no—
109C2×Dic71—22no—
110C2×Dic71—41no—
111C2×Dic71—42no—
112C2×Dic71C2×D411yescharacteristic
113C2×Dic71C2×D411yescharacteristic
114D284—22no—
115D284—42no—
116D284—42no—
117C4×D71—22no—
118C4×D71—22no—
119C4×D71—41no—
120C4×D71—41no—
121C4×D71—42no—
122C8×C71—22no—
123C71⋊C8—22no—
124C22×C71—21no—
125C22×C71—21no—
126C22×C71—42no—
127C22×C71—42no—
128C22×C71—42no—
129C22×C71—84no—
130C22×C71#13512yes—
131C22×C71C22×D411yescharacteristic
132C2×D71—21no—
133C2×D71—21no—
134C2×D71—21no—
135C2×D71—21no—
136C2×D71—21no—
137C2×D71—44no—
138C2×D71—41no—
139C2×D71—41no—
140C2×D71—41no—
141C2×D71AGL(1,ℤ/8)12yes—
142C4×C71—21no—
143C4×C71—41no—
144C4×C71—41no—
145C4×C71H(5,𝔽2)12yes—
146Dic71—21no—
147Dic71—21no—
148Dic71—41no—
149Dic71—42no—
150C2×C71—22no—
151C2×C71—82no—
152C2×C71#53411yescharacteristic
153C2×C71C2×#13511yescharacteristic
154C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
155D71—22no—
156D71—41no—
157D71—81no—
158#3188—711nomaximal, Sylow
159C71#318811yescharacteristic, complemented, Sylow
160#525—711no—
161C2×#121—711no—
162#484—711no—
163#482—712no—
164#481—711no—
165#466—712no—
166#465—712no—
167#450D7111yescharacteristic, complemented
168#449—712no—
169#447—711no—
170C2×C22⋊C8—711no—
171C2×#41—711no—
172C22×D4—1421no—
173C22×D4C2×D7111yescharacteristic
174C23×C4—711no—
175C2×SD16C2×D7112yescomplemented
176C2×D8—712no—
177C22×C8—712no—
178#123—1421no—
179C22⋊Q8—711no—
180C22⋊Q8C2×D7111yescharacteristic
181#121—712no—
182#121—712no—
183#121—711no—
184#121—711no—
185#121—1421no—
186#121—1421no—
187#120—1421no—
188C4×D4—1421no—
189C4×D4—1421no—
190#117—711no—
191C2×C4⋊C4—711no—
192C2×C22⋊C4—1421no—
193#107—712no—
194#106—712no—
195#103—712no—
196#102—712no—
197#102—712no—
198#102C2×D7112yescomplemented
199C22⋊C8—711no—
200C22⋊C8—712no—
201C22⋊C8C2×D7111yescharacteristic
202C24—21no—
203C24—711no—
204#41—2842no—
205C2×Q8C22×D7111yescharacteristic
206C2×D4—21no—
207C2×D4—42no—
208C2×D4—2844no—
209C2×D4—711no—
210C2×D4—1421no—
211C2×D4—1421no—
212C2×D4—1422no—
213C2×D4—1422no—
214C2×D4—1421no—
215C2×D4C22×D7112yes—
216C22×C4—2842no—
217C22×C4—711no—
218C22×C4—711no—
219C22×C4—711no—
220C22×C4—711no—
221C22×C4—711no—
222C22×C4—1421no—
223C22×C4—1421no—
224C22×C4C22×D7111yescharacteristic
225SD16—42no—
226D8—2842no—
227C2×C8—2842no—
228C2×C8—712no—
229C2×C8C22×D7112yes—
230C4⋊C4—21no—
231C4⋊C4—711no—
232C4⋊C4—712no—
233C4⋊C4—1421no—
234C4⋊C4—1421no—
235C4⋊C4C22×D7111yescharacteristic
236C22⋊C4—21no—
237C22⋊C4—1421no—
238C22⋊C4—1421no—
239C22⋊C4—1421no—
240C22⋊C4—1422no—
241C22⋊C4—1422no—
242C22⋊C4—1421no—
243C42—1421no—
244C23—22no—
245C23—42no—
246C23—42no—
247C23—42no—
248C23—2841no—
249C23—711no—
250C23—711no—
251C23—1422no—
252C23—1421no—
253C23—1422no—
254C23D4×D7111yescharacteristic
255Q8—22no—
256D4—22no—
257D4—22no—
258D4—42no—
259D4—2842no—
260D4—2842no—
261D4—2844no—
262D4—2844no—
263D4—2842no—
264D4—1422no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—2841no—
269C2×C4—2841no—
270C2×C4—2842no—
271C2×C4—2842no—
272C2×C4—2841no—
273C2×C4—711no—
274C2×C4—711no—
275C2×C4—1421no—
276C2×C4—1422no—
277C2×C4—1422no—
278C2×C4—1422no—
279C2×C4D4×D7111yescharacteristic
280C2×C4C23×D7111yescharacteristic
281C8—22no—
282C8—1422no—
283C22—21no—
284C22—21no—
285C22—42no—
286C22—42no—
287C22—42no—
288C22—84no—
289C22—2844no—
290C22—2841no—
291C22—2841no—
292C22—2841no—
293C22—712no—
294C22—1421no—
295C22—1421no—
296C22—1421no—
297C22—1421no—
298C22—1421no—
299C22#1455614912yes—
300C22C2×D4×D7111yescharacteristic
301C4—21no—
302C4—41no—
303C4—41no—
304C4—2841no—
305C4—2842no—
306C4—1421no—
307C4—1421no—
308C4#1455617312yes—
309C2—22no—
310C2—82no—
311C2—2841no—
312C2—5681no—
313C2—1422no—
314C2#1798655911yescharacteristic
315C2C2×#1455614911yescharacteristic
316C2AGL(1,ℤ/8)×D7111yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#450×C71C2
socleC22×C71C22×D4
hypercenter#450D71
hypocenterC71#3188
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #17986512C2semidirect
2. C2×#14556099C2semidirect
3. #17986332C2semidirect
4. #17986322C2semidirect
5. #17986318C2semidirect
6. #17986234C2semidirect
7. #17986225C2semidirect
8. #17986143C2semidirect
9. #17986133C2semidirect
10. C22⋊C8×D71C2semidirect
11. #450×C71C2semidirect
12. C2×#11765866C22semidirect
15. C2×#11765850C22semidirect
16. C2×C8×D71C22semidirect
18. #14556104C22semidirect
19. #14556103C22semidirect
28. #14556086C22semidirect
31. #14556061C22semidirect
32. #14556060C22semidirect
33. #14556052C22semidirect
34. #14556050C22semidirect
35. #14556049C22semidirect
37. (C2×D71)⋊C8C22semidirect
39. C2×SD16×C71C22semidirect
41. #102×C71C22semidirect
159. C71#3188semidirect
167. #450D71semidirect
175. C2×SD16C2×D71semidirect
198. #102C2×D71semidirect