Finite Group Databasebeta

← #22600415

Subgroups — #22600415

Order 9088.

410 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17986645C211yesmaximal, characteristic
2#17986361C211yesmaximal, characteristic, complemented
3#17986334C211yesmaximal, characteristic, complemented
4#17986312C211yesmaximal, characteristic, complemented
5#17986276C211yesmaximal, characteristic, complemented
6#17986247C211yesmaximal, characteristic, complemented
7#17986238C211yesmaximal, characteristic, complemented
8#17986221C211yesmaximal, characteristic, complemented
9#17986159C211yesmaximal, characteristic, complemented
10#17986149C211yesmaximal, characteristic, complemented
11#17986097C211yesmaximal, characteristic, complemented
12#17986071C211yesmaximal, characteristic, complemented
13#17985998C211yesmaximal, characteristic, complemented
14C2×#14556048C211yesmaximal, characteristic, complemented
15#554×C71C211yesmaximal, characteristic, complemented
16C2×#11765865C2211yescharacteristic
17C22×D284C2211yescharacteristic
18C2×#11765855C2211yescharacteristic, complemented
19C2×#11765852C2211yescharacteristic
20C2×#11765852C2211yescharacteristic
21C2×#11765849C2211yescharacteristic, complemented
22C22×C71⋊C8C2211yescharacteristic
23#14556116C2211yescharacteristic
24#14556113—21no—
25#14556107C2211yescharacteristic
26#14556102C2211yescharacteristic
27#14556101—21no—
28#14556096—21no—
29C4×D284C2211yescharacteristic
30C22⋊C4×D71—21no—
31#14556061C2211yescharacteristic, complemented
32#14556061C2211yescharacteristic, complemented
33#14556061C2211yescharacteristic, complemented
34C4⋊(C71⋊C8)C2211yescharacteristic, complemented
35#14556052C2211yescharacteristic, complemented
36#14556051C2211yescharacteristic, complemented
37#14556051C2211yescharacteristic, complemented
38#14556051C2211yescharacteristic, complemented
39#14556048C2211yescharacteristic
40#14556048C2211yescharacteristic
41#14556048C2211yescharacteristic
42#14556048C2211yescharacteristic
43#14556048C2211yescharacteristic
44#14556048C2211yescharacteristic
45C22⋊(C71⋊C8)C2211yescharacteristic
46C22⋊(C71⋊C8)C2211yescharacteristic, complemented
47#14556039C2211yescharacteristic, complemented
48C4⋊Q8×C71C2211yescharacteristic, complemented
49#125×C71—21no—
50C22⋊Q8×C71—21no—
51C22⋊Q8×C71—21no—
52C22⋊Q8×C71C2211yescharacteristic
53C22⋊Q8×C71C2211yescharacteristic, complemented
54C4×Q8×C71C2211yescharacteristic, complemented
55C4×D4×C71—21no—
56C4×D4×C71C2211yescharacteristic
57C2×C4⋊C4×C71C2211yescharacteristic
58C2×C4⋊C4×C71C2211yescharacteristic, complemented
59C23×D71—21no—
60#11765865—22no—
61#11765865—41no—
62C2×Dic142C2311yescharacteristic
63C2×D284—22no—
64C2×D284—21no—
65C2×D284—41no—
66C2×D284C2311yescharacteristic
67C2×D284C2311yescharacteristic
68C2×C22⋊D71—21no—
69C2×C22⋊D71—21no—
70C2×C4×D71—21no—
71C2×C4×D71—21no—
72#11765855—41no—
73#11765852—22no—
74#11765852—41no—
75#11765849—41no—
76C2×C71⋊C8—41no—
77C2×C71⋊C8C2311yescharacteristic
78C2×C71⋊C8C2311yescharacteristic
79C2×C71⋊C8C2311yescharacteristic
80C2×C71⋊C8C2311yescharacteristic
81C4⋊Dic71C2311yescharacteristic
82Dic71⋊C4—21no—
83C22⋊Dic71—21no—
84(C2×D71)⋊C4—21no—
85(C2×D71)⋊C4—21no—
86(C2×D71)⋊C4—21no—
87(C2×D71)⋊C4—21no—
88C2×Q8×C71—21no—
89C2×Q8×C71C2311yescharacteristic
90C2×D4×C71D411yescharacteristic
91C22×C4×C71—21no—
92C22×C4×C71—21no—
93C22×C4×C71C2311yescharacteristic
94C22×C4×C71C2311yescharacteristic
95C4⋊C4×C71—21no—
96C4⋊C4×C71—21no—
97C4⋊C4×C71—21no—
98C4⋊C4×C71—21no—
99C4⋊C4×C71—21no—
100C4⋊C4×C71—21no—
101C4⋊C4×C71D411yescharacteristic
102C4⋊C4×C71C2311yescharacteristic
103C4⋊C4×C71C2311yescharacteristic
104C4⋊C4×C71C2311yescharacteristic
105C22⋊C4×C71—21no—
106C22⋊C4×C71—21no—
107C22⋊C4×C71D411yescharacteristic
108C22⋊C4×C71D411yescharacteristic
109C42×C71—21no—
110C42×C71C2311yescharacteristic
111C23×C71C2×D411yescharacteristic
112C23×C71C2×D411yescharacteristic
113C22×D71—21no—
114C22×D71—21no—
115C22×D71—42no—
116C22×D71—42no—
117C22×D71—81no—
118Q8×C71—21no—
119Q8×C71—21no—
120Q8×C71—81no—
121D4×C71—22no—
122C2×C4×C71—21no—
123C2×C4×C71—21no—
124C2×C4×C71—21no—
125C2×C4×C71—21no—
126C2×C4×C71—21no—
127C2×C4×C71—21no—
128C2×C4×C71—21no—
129C2×C4×C71—21no—
130C2×C4×C71—81no—
131C2×C4×C71—81no—
132C2×C4×C71C2×D411yescharacteristic
133C2×C4×C71C2×D411yescharacteristic
134C2×C4×C71C2×D411yescharacteristic
135C2×C4×C71C2×D411yescharacteristic
136C2×C4×C71C2411yescharacteristic
137C22⋊D71—42no—
138C22⋊D71—81no—
139C2×Dic71—21no—
140C2×Dic71—21no—
141D284—21no—
142D284—21no—
143D284—21no—
144D284—42no—
145D284#4112yes—
146C4×D71—42no—
147C4×D71—81no—
148Dic142—21no—
149Dic142—21no—
150C71⋊C8—21no—
151C71⋊C8—21no—
152C71⋊C8—21no—
153C71⋊C8—21no—
154C22×C71—21no—
155C22×C71—21no—
156C22×C71—21no—
157C22×C71—21no—
158C22×C71—21no—
159C22×C71#13712yes—
160C22×C71C22×D411yescharacteristic
161C2×D71—22no—
162C2×D71—41no—
163C2×D71—41no—
164C2×D71—41no—
165C2×D71—41no—
166C2×D71—41no—
167C2×D71—84no—
168C4×C71—21no—
169C4×C71—21no—
170C4×C71—21no—
171C4×C71—21no—
172C4×C71—41no—
173C4×C71—41no—
174C4×C71—41no—
175C4×C71—41no—
176C4×C71C2×#4111yescharacteristic
177C4×C71H(5,𝔽2)11yescharacteristic
178Dic71—41no—
179Dic71—41no—
180C2×C71—22no—
181C2×C71—41no—
182C2×C71#54611yescharacteristic
183C2×C71C2×#13711yescharacteristic
184C2×C71#57611yescharacteristic
185D71—42no—
186D71—81no—
187#3314—711nomaximal, Sylow
188C71#331411yescharacteristic, complemented, Sylow
189#554D7111yescharacteristic, complemented
190#546—711no—
191#489—711no—
192#484—711no—
193#480—711no—
194#474—711no—
195#468—711no—
196#467—711no—
197#464—711no—
198#452—711no—
199#450—711no—
200#440—711no—
201#435—711no—
202#417—711no—
203C2×#102—711no—
204C2×#41—711no—
205C22×D4—711no—
206C2×Q16—711no—
207C2×SD16—711no—
208C2×SD16—711no—
209C2×M16—711no—
210C22×C8—711no—
211C4⋊Q8C2×D7111yescharacteristic, complemented
212#125—21no—
213#124—711no—
214#123—1421no—
215C22⋊Q8—21no—
216C22⋊Q8—21no—
217C22⋊Q8—711no—
218C22⋊Q8C2×D7111yescharacteristic
219C22⋊Q8C2×D7111yescharacteristic, complemented
220#121—711no—
221#121—1421no—
222#120—1421no—
223C4×Q8C2×D7111yescharacteristic, complemented
224C4×D4—21no—
225C4×D4—711no—
226C4×D4C2×D7111yescharacteristic
227C2×C4⋊C4C2×D7111yescharacteristic
228C2×C4⋊C4C2×D7111yescharacteristic, complemented
229C2×C22⋊C4—1421no—
230#107—711no—
231#107—711no—
232#107—711no—
233C4⋊C8—711no—
234#103—711no—
235#103—711no—
236#103—711no—
237#103—711no—
238#102—711no—
239#102—711no—
240#102—711no—
241#102—711no—
242#102—711no—
243#102—711no—
244C22⋊C8—711no—
245C22⋊C8—711no—
246#97—711no—
247C24—1421no—
248#41—2841no—
249#41—1422no—
250C2×Q8—21no—
251C2×Q8—711no—
252C2×Q8C22×D7111yescharacteristic
253C2×D4—2841no—
254C2×D4—711no—
255C2×D4—711no—
256C2×D4—1422no—
257C2×D4—1421no—
258C2×D4—1421no—
259C2×D4—1421no—
260C2×D4C22⋊D7111yescharacteristic
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—1421no—
264C22×C4—1421no—
265C22×C4C22×D7111yescharacteristic
266C22×C4C22×D7111yescharacteristic
267Q16—2841no—
268SD16—2841no—
269SD16—1422no—
270M16—2841no—
271C2×C8—2841no—
272C2×C8—711no—
273C2×C8—711no—
274C2×C8—711no—
275C2×C8—711no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—711no—
283C4⋊C4—1421no—
284C4⋊C4C22⋊D7111yescharacteristic
285C4⋊C4C22×D7111yescharacteristic
286C4⋊C4C22×D7111yescharacteristic
287C4⋊C4C22×D7111yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—1421no—
291C22⋊C4—1421no—
292C22⋊C4—1421no—
293C22⋊C4—1421no—
294C22⋊C4—1421no—
295C22⋊C4C22⋊D7111yescharacteristic
296C22⋊C4C22⋊D7111yescharacteristic
297C42—21no—
298C42C22×D7111yescharacteristic
299C23—2842no—
300C23—2842no—
301C23—5681no—
302C23—1421no—
303C23—1421no—
304C23C2×C22⋊D7111yescharacteristic
305C23C2×C22⋊D7111yescharacteristic
306Q8—21no—
307Q8—21no—
308Q8—81no—
309Q8—1421no—
310Q8—1421no—
311D4—22no—
312D4—2842no—
313D4—2842no—
314D4—5681no—
315D4—712no—
316D4—1421no—
317D4—1421no—
318D4—1421no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—2842no—
330C2×C4—5681no—
331C2×C4—1421no—
332C2×C4—1421no—
333C2×C4C2×C22⋊D7111yescharacteristic
334C2×C4C2×C22⋊D7111yescharacteristic
335C2×C4C2×C22⋊D7111yescharacteristic
336C2×C4C2×C22⋊D7111yescharacteristic
337C2×C4C23×D7111yescharacteristic
338C8—1421no—
339C8—1421no—
340C8—1421no—
341C8—1421no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—2841no—
348C22—2841no—
349C22—2841no—
350C22—2841no—
351C22—2841no—
352C22—5684no—
353C22—1422no—
354C22#1455615912yes—
355C22C22×C22⋊D7111yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—41no—
361C4—41no—
362C4—41no—
363C4—41no—
364C4—2841no—
365C4—2841no—
366C4#41×D7111yescharacteristic
367C4#1455617411yescharacteristic
368C2—22no—
369C2—41no—
370C2—2842no—
371C2—5681no—
372C2#1798663711yescharacteristic
373C2C2×#1455615911yescharacteristic
374C2#1798680211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#554×C71C2
socleC22×C71C22×D4
hypercenter#554D71
hypocenterC71#3314
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C4×D4 ◁ #554

Complements

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

NH ≅ G/Ntype
2. #17986361C2semidirect
3. #17986334C2semidirect
4. #17986312C2semidirect
5. #17986276C2semidirect
6. #17986247C2semidirect
7. #17986238C2semidirect
8. #17986221C2semidirect
9. #17986159C2semidirect
10. #17986149C2semidirect
11. #17986097C2semidirect
12. #17986071C2semidirect
13. #17985998C2semidirect
14. C2×#14556048C2semidirect
15. #554×C71C2semidirect
18. C2×#11765855C22semidirect
21. C2×#11765849C22semidirect
31. #14556061C22semidirect
32. #14556061C22semidirect
33. #14556061C22semidirect
34. C4⋊(C71⋊C8)C22semidirect
35. #14556052C22semidirect
36. #14556051C22semidirect
37. #14556051C22semidirect
38. #14556051C22semidirect
46. C22⋊(C71⋊C8)C22semidirect
47. #14556039C22semidirect
48. C4⋊Q8×C71C22semidirect
53. C22⋊Q8×C71C22semidirect
54. C4×Q8×C71C22semidirect
58. C2×C4⋊C4×C71C22semidirect
188. C71#3314semidirect
189. #554D71semidirect
211. C4⋊Q8C2×D71semidirect
219. C22⋊Q8C2×D71semidirect
223. C4×Q8C2×D71semidirect
228. C2×C4⋊C4C2×D71semidirect