Finite Group Databasebeta

← #22600421

Subgroups — #22600421

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#17986706C211yesmaximal, characteristic, complemented
2#17986358C211yesmaximal, characteristic, complemented
3#17986332C211yesmaximal, characteristic, complemented
4#17986310C211yesmaximal, characteristic, complemented
5#17986279C211yesmaximal, characteristic, complemented
6#17986243C211yesmaximal, characteristic, complemented
7#17986238C211yesmaximal, characteristic, complemented
8#17986218C211yesmaximal, characteristic, complemented
9#17986161C211yesmaximal, characteristic, complemented
10#17986150C211yesmaximal, characteristic, complemented
11#17986099C211yesmaximal, characteristic, complemented
12#17986071C211yesmaximal, characteristic, complemented
13#17986000C211yesmaximal, characteristic, complemented
14C2×#14556049C211yesmaximal, characteristic, complemented
15#546×C71C211yesmaximal, characteristic
16C2×#11765855C2211yescharacteristic, complemented
17C2×#11765853C2211yescharacteristic
18C2×#11765853C2211yescharacteristic
19C2×#11765849C2211yescharacteristic, complemented
20C22×C71⋊C8C2211yescharacteristic
21#14556128C2211yescharacteristic, complemented
22#14556119—21no—
23#14556107C2211yescharacteristic
24#14556107C2211yescharacteristic, complemented
25#14556105—21no—
26#14556105—21no—
27C4×Dic142C2211yescharacteristic, complemented
28D4×Dic71—21no—
29C2×C4⋊Dic71C2211yescharacteristic
30C2×C4⋊Dic71C2211yescharacteristic, complemented
31#14556061C2211yescharacteristic, complemented
32#14556061C2211yescharacteristic, complemented
33#14556061C2211yescharacteristic, complemented
34C4⋊(C71⋊C8)C2211yescharacteristic, complemented
35#14556052C2211yescharacteristic, complemented
36#14556052C2211yescharacteristic, complemented
37#14556052C2211yescharacteristic, complemented
38#14556051C2211yescharacteristic, complemented
39#14556049C2211yescharacteristic
40#14556049C2211yescharacteristic
41#14556049C2211yescharacteristic
42#14556049C2211yescharacteristic
43#14556049C2211yescharacteristic
44#14556049C2211yescharacteristic
45C22⋊(C71⋊C8)C2211yescharacteristic
46C22⋊(C71⋊C8)C2211yescharacteristic, complemented
47#14556039C2211yescharacteristic, complemented
48C2×#41×C71C2211yescharacteristic
49C22×D4×C71C2211yescharacteristic
50#124×C71C2211yescharacteristic
51#123×C71—21no—
52C22⋊Q8×C71C2211yescharacteristic
53#121×C71—21no—
54#121×C71C2211yescharacteristic
55#120×C71—21no—
56C4×D4×C71C2211yescharacteristic
57C4×D4×C71C2211yescharacteristic
58C2×C22⋊C4×C71—21no—
59C2×Dic142—21no—
60C2×Dic142C2311yescharacteristic
61C22×Dic71—21no—
62C22×Dic71—21no—
63#11765855—41no—
64#11765853—22no—
65#11765853—41no—
66#11765849—41no—
67C2×C71⋊C8—41no—
68C2×C71⋊C8C2311yescharacteristic
69C2×C71⋊C8C2311yescharacteristic
70C2×C71⋊C8C2311yescharacteristic
71C2×C71⋊C8C2311yescharacteristic
72C4⋊Dic71—21no—
73C4⋊Dic71—21no—
74C4⋊Dic71—21no—
75C4⋊Dic71C2311yescharacteristic
76C4⋊Dic71C2311yescharacteristic
77C4⋊Dic71C2311yescharacteristic
78Dic71⋊C4—21no—
79Dic71⋊C4—21no—
80Dic71⋊C4—21no—
81C22⋊Dic71—21no—
82C22⋊Dic71—21no—
83C4×Dic71—21no—
84C24×C71—21no—
85#41×C71—22no—
86#41×C71—41no—
87C2×Q8×C71C2311yescharacteristic
88C2×D4×C71—22no—
89C2×D4×C71—21no—
90C2×D4×C71—21no—
91C2×D4×C71—21no—
92C2×D4×C71—41no—
93C2×D4×C71D411yescharacteristic
94C2×D4×C71C2311yescharacteristic
95C2×D4×C71C2311yescharacteristic
96C22×C4×C71—21no—
97C22×C4×C71—21no—
98C22×C4×C71C2311yescharacteristic
99C22×C4×C71C2311yescharacteristic
100C4⋊C4×C71—21no—
101C4⋊C4×C71D411yescharacteristic
102C4⋊C4×C71C2311yescharacteristic
103C22⋊C4×C71—21no—
104C22⋊C4×C71—21no—
105C22⋊C4×C71—21no—
106C22⋊C4×C71—21no—
107C22⋊C4×C71—21no—
108C22⋊C4×C71D411yescharacteristic
109C22⋊C4×C71D411yescharacteristic
110C42×C71C2311yescharacteristic
111C23×C71—21no—
112C23×C71—21no—
113C23×C71—42no—
114C23×C71—42no—
115C23×C71—81no—
116C23×C71C2×D411yescharacteristic
117C23×C71C2×D411yescharacteristic
118Q8×C71—21no—
119Q8×C71—21no—
120D4×C71—22no—
121D4×C71—21no—
122D4×C71—21no—
123D4×C71—21no—
124D4×C71—42no—
125D4×C71—42no—
126D4×C71—81no—
127D4×C71#4112yes—
128C2×C4×C71—21no—
129C2×C4×C71—21no—
130C2×C4×C71—21no—
131C2×C4×C71—21no—
132C2×C4×C71—21no—
133C2×C4×C71—21no—
134C2×C4×C71—42no—
135C2×C4×C71—81no—
136C2×C4×C71C2×D411yescharacteristic
137C2×C4×C71C2×D411yescharacteristic
138C2×C4×C71C2×D411yescharacteristic
139C2×C4×C71C2×D411yescharacteristic
140C2×C4×C71C2411yescharacteristic
141C2×Dic71—21no—
142C2×Dic71—21no—
143C2×Dic71—21no—
144C2×Dic71—21no—
145C2×Dic71—81no—
146C2×Dic71—81no—
147Dic142—21no—
148Dic142—21no—
149Dic142—81no—
150C71⋊C8—21no—
151C71⋊C8—21no—
152C71⋊C8—21no—
153C71⋊C8—21no—
154C22×C71—21no—
155C22×C71—21no—
156C22×C71—21no—
157C22×C71—22no—
158C22×C71—21no—
159C22×C71—21no—
160C22×C71—41no—
161C22×C71—41no—
162C22×C71—41no—
163C22×C71—41no—
164C22×C71—41no—
165C22×C71—84no—
166C22×C71#13712yes—
167C22×C71C22×D411yescharacteristic
168C4×C71—21no—
169C4×C71—21no—
170C4×C71—21no—
171C4×C71—21no—
172C4×C71—41no—
173C4×C71—41no—
174C4×C71C2×#4111yescharacteristic
175C4×C71H(5,𝔽2)11yescharacteristic
176Dic71—41no—
177Dic71—41no—
178Dic71—41no—
179Dic71—41no—
180C2×C71—22no—
181C2×C71—41no—
182C2×C71—42no—
183C2×C71—81no—
184C2×C71#54611yescharacteristic
185C2×C71C2×#13711yescharacteristic
186C2×C71#57611yescharacteristic
187#3314—711nomaximal, Sylow
188C71#331411yescharacteristic, complemented, Sylow
189#554—711no—
190#546D7111yescharacteristic
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×#41C2×D7111yescharacteristic
205C22×D4C2×D7111yescharacteristic
206C2×Q16—711no—
207C2×SD16—711no—
208C2×SD16—711no—
209C2×M16—711no—
210C22×C8—711no—
211C4⋊Q8—711no—
212#125—1421no—
213#124C2×D7111yescharacteristic
214#123—21no—
215C22⋊Q8—711no—
216C22⋊Q8—711no—
217C22⋊Q8—1421no—
218C22⋊Q8—1421no—
219C22⋊Q8C2×D7111yescharacteristic
220#121—21no—
221#121C2×D7111yescharacteristic
222#120—21no—
223C4×Q8—711no—
224C4×D4—1421no—
225C4×D4C2×D7111yescharacteristic
226C4×D4C2×D7111yescharacteristic
227C2×C4⋊C4—711no—
228C2×C4⋊C4—711no—
229C2×C22⋊C4—21no—
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—21no—
248#41—22no—
249#41—41no—
250C2×Q8—711no—
251C2×Q8—1421no—
252C2×Q8C22×D7111yescharacteristic
253C2×D4—22no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—41no—
258C2×D4C22⋊D7111yescharacteristic
259C2×D4C22×D7111yescharacteristic
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—711no—
278C4⋊C4—711no—
279C4⋊C4—711no—
280C4⋊C4—1421no—
281C4⋊C4—1421no—
282C4⋊C4—1421no—
283C4⋊C4—1421no—
284C4⋊C4—1421no—
285C4⋊C4—1421no—
286C4⋊C4C22⋊D7111yescharacteristic
287C4⋊C4C22×D7111yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—1421no—
294C22⋊C4—1421no—
295C22⋊C4C22⋊D7111yescharacteristic
296C22⋊C4C22⋊D7111yescharacteristic
297C42—1421no—
298C42C22×D7111yescharacteristic
299C23—21no—
300C23—21no—
301C23—42no—
302C23—42no—
303C23—81no—
304C23C2×C22⋊D7111yescharacteristic
305C23C2×C22⋊D7111yescharacteristic
306Q8—21no—
307Q8—21no—
308Q8—5681no—
309Q8—1421no—
310Q8—1421no—
311D4—22no—
312D4—21no—
313D4—21no—
314D4—21no—
315D4—42no—
316D4—42no—
317D4—81no—
318D4#1176586412yes—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—42no—
326C2×C4—81no—
327C2×C4—5681no—
328C2×C4—5681no—
329C2×C4—1421no—
330C2×C4—1421no—
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—22no—
346C22—21no—
347C22—21no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—84no—
354C22#1455615812yes—
355C22C22×C22⋊D7111yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—41no—
361C4—41no—
362C4—2841no—
363C4—2841no—
364C4—2841no—
365C4—2841no—
366C4C2×#1176586411yescharacteristic
367C4#1455617311yescharacteristic
368C2—22no—
369C2—41no—
370C2—42no—
371C2—81no—
372C2#1798664311yescharacteristic
373C2C2×#1455615811yescharacteristic
374C2#1798680211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#546×C71C2
socleC22×C71C22×D4
hypercenter#546D71
hypocenterC71#3314
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. #17986706C2semidirect
2. #17986358C2semidirect
3. #17986332C2semidirect
4. #17986310C2semidirect
5. #17986279C2semidirect
6. #17986243C2semidirect
7. #17986238C2semidirect
8. #17986218C2semidirect
9. #17986161C2semidirect
10. #17986150C2semidirect
11. #17986099C2semidirect
12. #17986071C2semidirect
13. #17986000C2semidirect
14. C2×#14556049C2semidirect
16. C2×#11765855C22semidirect
19. C2×#11765849C22semidirect
21. #14556128C22semidirect
24. #14556107C22semidirect
27. C4×Dic142C22semidirect
30. C2×C4⋊Dic71C22semidirect
31. #14556061C22semidirect
32. #14556061C22semidirect
33. #14556061C22semidirect
34. C4⋊(C71⋊C8)C22semidirect
35. #14556052C22semidirect
36. #14556052C22semidirect
37. #14556052C22semidirect
38. #14556051C22semidirect
46. C22⋊(C71⋊C8)C22semidirect
47. #14556039C22semidirect
188. C71#3314semidirect