Finite Group Databasebeta

← #22600419

Subgroups — #22600419

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#17986704C211yesmaximal, characteristic, complemented
2#17986640C211yesmaximal, characteristic
3#17986360C211yesmaximal, characteristic, complemented
4#17986314C211yesmaximal, characteristic, complemented
5#17986278C211yesmaximal, characteristic, complemented
6#17986246C211yesmaximal, characteristic, complemented
7#17986238C211yesmaximal, characteristic, complemented
8#17986224C211yesmaximal, characteristic, complemented
9#17986164C211yesmaximal, characteristic, complemented
10#17986152C211yesmaximal, characteristic, complemented
11#17986098C211yesmaximal, characteristic, complemented
12#17986067C211yesmaximal, characteristic, complemented
13#17985999C211yesmaximal, characteristic, complemented
14C2×#14556048C211yesmaximal, characteristic, complemented
15#484×C71C211yesmaximal, characteristic, complemented
16C2×#11765866C2211yescharacteristic
17C22×D284C2211yescharacteristic
18C2×#11765855C2211yescharacteristic, complemented
19C2×#11765854C2211yescharacteristic
20C2×#11765854C2211yescharacteristic
21C2×#11765848C2211yescharacteristic, complemented
22C22×C71⋊C8C2211yescharacteristic
23#14556127C2211yescharacteristic, complemented
24#14556118—21no—
25#14556117C2211yescharacteristic
26#14556113—21no—
27#14556108C2211yescharacteristic, complemented
28#14556107—21no—
29#14556107C2211yescharacteristic
30#14556106—21no—
31#14556103—21no—
32#14556103C2211yescharacteristic
33#14556098—21no—
34#14556093C2211yescharacteristic, complemented
35#14556090C2211yescharacteristic
36C4×C22⋊D71—21no—
37C4×C22⋊D71C2211yescharacteristic
38C4⋊C4×D71C2211yescharacteristic, complemented
39C22⋊C4×D71—21no—
40#14556061C2211yescharacteristic, complemented
41Dic71⋊C8C2211yescharacteristic, complemented
42#14556052C2211yescharacteristic, complemented
43#14556051C2211yescharacteristic, complemented
44#14556050C2211yescharacteristic
45#14556050C2211yescharacteristic
46#14556048C2211yescharacteristic
47#14556048C2211yescharacteristic
48#14556048C2211yescharacteristic
49#14556048C2211yescharacteristic
50(C2×D71)⋊C8C2211yescharacteristic, complemented
51#14556038C2211yescharacteristic, complemented
52C22⋊Q8×C71C2211yescharacteristic
53C2×C4⋊C4×C71C2211yescharacteristic
54#107×C71C2211yescharacteristic, complemented
55#107×C71C2211yescharacteristic, complemented
56#103×C71C2211yescharacteristic, complemented
57#103×C71C2211yescharacteristic, complemented
58C22⋊C8×C71C2211yescharacteristic
59C23×D71—21no—
60#11765866—22no—
61#11765866—41no—
62C2×Dic142—21no—
63C2×Dic142C2311yescharacteristic
64C2×D284—22no—
65C2×D284—21no—
66C2×D284—21no—
67C2×D284—21no—
68C2×D284—41no—
69C2×D284C2311yescharacteristic
70C2×D284C2311yescharacteristic
71C2×C22⋊D71D411yescharacteristic
72C2×C4×D71—21no—
73C2×C4×D71—21no—
74C2×C4×D71—21no—
75C2×C4×D71C2311yescharacteristic
76#11765855—41no—
77#11765854—22no—
78#11765854—41no—
79#11765848—41no—
80C2×C71⋊C8—41no—
81C2×C71⋊C8C2311yescharacteristic
82C2×C71⋊C8C2311yescharacteristic
83C4⋊Dic71—21no—
84C4⋊Dic71C2311yescharacteristic
85Dic71⋊C4—21no—
86Dic71⋊C4—21no—
87Dic71⋊C4—21no—
88Dic71⋊C4—21no—
89Dic71⋊C4D411yescharacteristic
90C22⋊Dic71—21no—
91C22⋊Dic71D411yescharacteristic
92(C2×D71)⋊C4—21no—
93(C2×D71)⋊C4—21no—
94(C2×D71)⋊C4—21no—
95(C2×D71)⋊C4—21no—
96(C2×D71)⋊C4—21no—
97(C2×D71)⋊C4D411yescharacteristic
98C4×Dic71—21no—
99C4×Dic71C2311yescharacteristic
100C2×Q8×C71C2311yescharacteristic
101C22×C4×C71—21no—
102C22×C4×C71C2311yescharacteristic
103C2×C8×C71C2311yescharacteristic
104C2×C8×C71C2311yescharacteristic
105C4⋊C4×C71—21no—
106C4⋊C4×C71—21no—
107C4⋊C4×C71C2311yescharacteristic
108C4⋊C4×C71C2311yescharacteristic
109C4⋊C4×C71C2311yescharacteristic
110C22⋊C4×C71—21no—
111C23×C71C2×D411yescharacteristic
112C22×D71—21no—
113C22×D71—21no—
114C22×D71—42no—
115C22×D71—42no—
116C22×D71—81no—
117C22×D71C2×D411yescharacteristic
118Q8×C71—21no—
119Q8×C71—21no—
120C2×C4×C71—21no—
121C2×C4×C71—21no—
122C2×C4×C71—21no—
123C2×C4×C71—21no—
124C2×C4×C71—21no—
125C2×C4×C71—21no—
126C2×C4×C71—81no—
127C2×C4×C71C2×D411yescharacteristic
128C2×C4×C71C2411yescharacteristic
129C22⋊D71—22no—
130C2×Dic71—21no—
131C2×Dic71—21no—
132C2×Dic71C2×D411yescharacteristic
133C2×Dic71C2×D411yescharacteristic
134C2×Dic71C2×D411yescharacteristic
135D284—21no—
136D284—21no—
137D284—21no—
138D284—42no—
139D284—42no—
140D284—81no—
141D284#4112yes—
142C4×D71—21no—
143C4×D71—21no—
144C4×D71—42no—
145C4×D71—81no—
146C4×D71—81no—
147Dic142—21no—
148Dic142—21no—
149Dic142—81no—
150C8×C71—21no—
151C8×C71—21no—
152C71⋊C8—21no—
153C71⋊C8—21no—
154C22×C71—21no—
155C22×C71—21no—
156C22×C71#13712yes—
157C22×C71C22×D411yescharacteristic
158C2×D71—21no—
159C2×D71—22no—
160C2×D71—21no—
161C2×D71—21no—
162C2×D71—41no—
163C2×D71—41no—
164C2×D71—41no—
165C2×D71—41no—
166C2×D71—41no—
167C2×D71—84no—
168C4×C71—21no—
169C4×C71—41no—
170C4×C71—41no—
171C4×C71—41no—
172C4×C71—41no—
173C4×C71C2×#4111yescharacteristic
174C4×C71H(5,𝔽2)11yescharacteristic
175Dic71—21no—
176Dic71—21no—
177Dic71—21no—
178Dic71—41no—
179Dic71—41no—
180C2×C71—22no—
181C2×C71#54611yescharacteristic
182C2×C71C2×#13711yescharacteristic
183C2×C71#57611yescharacteristic
184D71—41no—
185D71—42no—
186D71—81no—
187#3314—711nomaximal, Sylow
188C71#331411yescharacteristic, complemented, Sylow
189#554—711no—
190#546—711no—
191#489—711no—
192#484D7111yescharacteristic, complemented
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⋊Q8—711no—
212#125—1421no—
213#124—711no—
214#123—1421no—
215C22⋊Q8—711no—
216C22⋊Q8—711no—
217C22⋊Q8—1421no—
218C22⋊Q8—1421no—
219C22⋊Q8C2×D7111yescharacteristic
220#121—711no—
221#121—1421no—
222#120—1421no—
223C4×Q8—711no—
224C4×D4—711no—
225C4×D4—711no—
226C4×D4—1421no—
227C2×C4⋊C4—711no—
228C2×C4⋊C4C2×D7111yescharacteristic
229C2×C22⋊C4—1421no—
230#107—711no—
231#107C2×D7111yescharacteristic, complemented
232#107C2×D7111yescharacteristic, complemented
233C4⋊C8—711no—
234#103—711no—
235#103—711no—
236#103C2×D7111yescharacteristic, complemented
237#103C2×D7111yescharacteristic, complemented
238#102—711no—
239#102—711no—
240#102—711no—
241#102—711no—
242#102—711no—
243#102—711no—
244C22⋊C8—711no—
245C22⋊C8C2×D7111yescharacteristic
246#97—711no—
247C24—1421no—
248#41—2841no—
249#41—1422no—
250C2×Q8—711no—
251C2×Q8—1421no—
252C2×Q8C22×D7111yescharacteristic
253C2×D4—2841no—
254C2×D4—711no—
255C2×D4—711no—
256C2×D4—711no—
257C2×D4—1421no—
258C2×D4—1422no—
259C2×D4—1421no—
260C2×D4—1421no—
261C22×C4—21no—
262C22×C4—711no—
263C22×C4—1421no—
264C22×C4—1421no—
265C22×C4—1421no—
266C22×C4C22×D7111yescharacteristic
267Q16—2841no—
268SD16—2841no—
269SD16—1422no—
270M16—2841no—
271C2×C8—2841no—
272C2×C8—711no—
273C2×C8—711no—
274C2×C8C22×D7111yescharacteristic
275C2×C8C22×D7111yescharacteristic
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—711no—
279C4⋊C4—711no—
280C4⋊C4—1421no—
281C4⋊C4—1421no—
282C4⋊C4—1421no—
283C4⋊C4—1421no—
284C4⋊C4—1421no—
285C4⋊C4C22×D7111yescharacteristic
286C4⋊C4C22×D7111yescharacteristic
287C4⋊C4C22×D7111yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—711no—
290C22⋊C4—711no—
291C22⋊C4—1421no—
292C22⋊C4—1421no—
293C22⋊C4—1421no—
294C22⋊C4—1421no—
295C22⋊C4—1421no—
296C22⋊C4—1421no—
297C42—711no—
298C42—1421no—
299C23—2842no—
300C23—2842no—
301C23—5681no—
302C23—711no—
303C23—1421no—
304C23—1421no—
305C23D4×D7111yescharacteristic
306Q8—21no—
307Q8—21no—
308Q8—5681no—
309Q8—1421no—
310Q8—1421no—
311D4—2842no—
312D4—2842no—
313D4—5681no—
314D4—712no—
315D4—1422no—
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—81no—
326C2×C4—2842no—
327C2×C4—5681no—
328C2×C4—5681no—
329C2×C4—711no—
330C2×C4—711no—
331C2×C4—711no—
332C2×C4—1421no—
333C2×C4—1421no—
334C2×C4—1421no—
335C2×C4—1421no—
336C2×C4D4×D7111yescharacteristic
337C2×C4C23×D7111yescharacteristic
338C8—21no—
339C8—21no—
340C8—1421no—
341C8—1421no—
342C22—21no—
343C22—21no—
344C22—2841no—
345C22—2841no—
346C22—2841no—
347C22—2841no—
348C22—2841no—
349C22—5684no—
350C22—1422no—
351C22—1421no—
352C22—1421no—
353C22—1421no—
354C22#1455616012yes—
355C22C2×D4×D7111yescharacteristic
356C4—21no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—2841no—
362C4—2841no—
363C4—1421no—
364C4—1421no—
365C4—1421no—
366C4#41×D7111yescharacteristic
367C4#1455617411yescharacteristic
368C2—22no—
369C2—2842no—
370C2—2841no—
371C2—5681no—
372C2#1798664111yescharacteristic
373C2C2×#1455616011yescharacteristic
374C2#1798680311yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#484×C71C2
socleC22×C71C22×D4
hypercenter#484D71
hypocenterC71#3314
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
1. #17986704C2semidirect
3. #17986360C2semidirect
4. #17986314C2semidirect
5. #17986278C2semidirect
6. #17986246C2semidirect
7. #17986238C2semidirect
8. #17986224C2semidirect
9. #17986164C2semidirect
10. #17986152C2semidirect
11. #17986098C2semidirect
12. #17986067C2semidirect
13. #17985999C2semidirect
14. C2×#14556048C2semidirect
15. #484×C71C2semidirect
18. C2×#11765855C22semidirect
21. C2×#11765848C22semidirect
23. #14556127C22semidirect
27. #14556108C22semidirect
34. #14556093C22semidirect
38. C4⋊C4×D71C22semidirect
40. #14556061C22semidirect
41. Dic71⋊C8C22semidirect
42. #14556052C22semidirect
43. #14556051C22semidirect
50. (C2×D71)⋊C8C22semidirect
51. #14556038C22semidirect
54. #107×C71C22semidirect
55. #107×C71C22semidirect
56. #103×C71C22semidirect
57. #103×C71C22semidirect
188. C71#3314semidirect
192. #484D71semidirect
231. #107C2×D71semidirect
232. #107C2×D71semidirect
236. #103C2×D71semidirect
237. #103C2×D71semidirect