Finite Group Databasebeta

← #22600520

Subgroups — #22600520

Order 9088.

392 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17986658C211yesmaximal, characteristic, complemented
2#17986361C211yesmaximal, characteristic, complemented
3#17986340C211yesmaximal, characteristic, complemented
4#17986323C211yesmaximal, characteristic, complemented
5#17986281C211yesmaximal, characteristic, complemented
6#17986250C211yesmaximal, characteristic, complemented
7#17986226C211yesmaximal, characteristic, complemented
8#17986214C211yesmaximal, characteristic, complemented
9#17986142C211yesmaximal, characteristic, complemented
10#17986137C211yesmaximal, characteristic, complemented
11#17986099C211yesmaximal, characteristic, complemented
12#17986071C211yesmaximal, characteristic, complemented
13#17986007C211yesmaximal, characteristic, complemented
14#17985993C211yesmaximal, characteristic, complemented
15#555×C71C211yesmaximal, characteristic, complemented
16C2×#11765865C2211yescharacteristic
17C22×Dic142C2211yescharacteristic
18C2×#11765853C2211yescharacteristic
19C2×#11765853C2211yescharacteristic
20C2×#11765850C2211yescharacteristic
21C2×#11765849C2211yescharacteristic
22C22×C71⋊C8C2211yescharacteristic
23#14556116C2211yescharacteristic
24#14556114—21no—
25#14556107C2211yescharacteristic
26#14556106—21no—
27#14556102C2211yescharacteristic
28#14556100—21no—
29C4×Dic142C2211yescharacteristic
30(C22⋊D71)⋊C4—21no—
31#14556061C2211yescharacteristic
32#14556059C2211yescharacteristic
33#14556059C2211yescharacteristic
34C4⋊(C71⋊C8)C2211yescharacteristic
35#14556051C2211yescharacteristic
36#14556051C2211yescharacteristic
37#14556051C2211yescharacteristic
38#14556051C2211yescharacteristic
39#14556051C2211yescharacteristic
40#14556051C2211yescharacteristic
41#14556049C2211yescharacteristic
42#14556048C2211yescharacteristic
43#14556048C2211yescharacteristic
44#14556048C2211yescharacteristic
45C22⋊(C71⋊C8)C2211yescharacteristic
46C22⋊(C71⋊C8)C2211yescharacteristic
47#14556039C2211yescharacteristic
48C4⋊Q8×C71C2211yescharacteristic
49#124×C71—21no—
50#124×C71—21no—
51#121×C71—21no—
52#121×C71C2211yescharacteristic
53#121×C71C2211yescharacteristic
54C4×D4×C71C2211yescharacteristic
55C4×D4×C71C2211yescharacteristic
56#117×C71—21no—
57#117×C71C2211yescharacteristic
58#117×C71C2211yescharacteristic
59#11765865—22no—
60#11765865—41no—
61C2×Dic142—22no—
62C2×Dic142—21no—
63C2×Dic142—41no—
64C2×Dic142C2311yescharacteristic
65C2×Dic142C2311yescharacteristic
66C2×D284C2311yescharacteristic
67C2×C22⋊D71—21no—
68C2×C22⋊D71—21no—
69C22×Dic71—21no—
70C2×C4×D71—21no—
71#11765853—22no—
72#11765853—41no—
73#11765850—41no—
74#11765849—41no—
75C2×C71⋊C8—41no—
76C2×C71⋊C8C2311yescharacteristic
77C2×C71⋊C8C2311yescharacteristic
78C2×C71⋊C8C2311yescharacteristic
79C2×C71⋊C8C2311yescharacteristic
80C4⋊Dic71C2311yescharacteristic
81Dic71⋊C4—21no—
82Dic71⋊C4—21no—
83C22⋊Dic71—21no—
84(C2×D71)⋊C4—21no—
85(C2×D71)⋊C4—21no—
86(C2×D71)⋊C4—21no—
87C4×Dic71—21no—
88C2×Q8×C71—21no—
89C2×D4×C71—21no—
90C2×D4×C71—21no—
91C2×D4×C71D411yescharacteristic
92C2×D4×C71C2311yescharacteristic
93C22×C4×C71—21no—
94C22×C4×C71C2311yescharacteristic
95C22×C4×C71C2311yescharacteristic
96C4⋊C4×C71—21no—
97C4⋊C4×C71D411yescharacteristic
98C4⋊C4×C71C2311yescharacteristic
99C4⋊C4×C71C2311yescharacteristic
100C4⋊C4×C71C2311yescharacteristic
101C22⋊C4×C71—21no—
102C22⋊C4×C71—21no—
103C22⋊C4×C71—21no—
104C22⋊C4×C71—21no—
105C22⋊C4×C71—21no—
106C22⋊C4×C71D411yescharacteristic
107C22⋊C4×C71D411yescharacteristic
108C42×C71—21no—
109C42×C71—21no—
110C42×C71C2311yescharacteristic
111C23×C71—21no—
112C23×C71C2×D411yescharacteristic
113C23×C71C2×D411yescharacteristic
114C22×D71—21no—
115Q8×C71—81no—
116D4×C71—21no—
117D4×C71—21no—
118D4×C71—41no—
119D4×C71—81no—
120D4×C71—81no—
121C2×C4×C71—21no—
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—42no—
129C2×C4×C71C2×D411yescharacteristic
130C2×C4×C71C2×D411yescharacteristic
131C2×C4×C71C2×D411yescharacteristic
132C2×C4×C71C2×D411yescharacteristic
133C2×C4×C71C2411yescharacteristic
134C22⋊D71—42no—
135C22⋊D71—42no—
136C2×Dic71—21no—
137C2×Dic71—21no—
138C2×Dic71—21no—
139C2×Dic71—42no—
140D284—21no—
141D284—21no—
142C4×D71—42no—
143Dic142—21no—
144Dic142—21no—
145Dic142—21no—
146Dic142—42no—
147Dic142#4112yes—
148C71⋊C8—21no—
149C71⋊C8—21no—
150C71⋊C8—21no—
151C71⋊C8—21no—
152C22×C71—21no—
153C22×C71—21no—
154C22×C71—21no—
155C22×C71—21no—
156C22×C71—21no—
157C22×C71—21no—
158C22×C71—41no—
159C22×C71—41no—
160C22×C71—41no—
161C22×C71C22×D411yescharacteristic
162C2×D71—41no—
163C2×D71—41no—
164C2×D71—41no—
165C4×C71—21no—
166C4×C71—21no—
167C4×C71—21no—
168C4×C71—21no—
169C4×C71—41no—
170C4×C71—41no—
171C4×C71—41no—
172C4×C71C2×#4111yescharacteristic
173C4×C71H(5,𝔽2)11yescharacteristic
174Dic71—22no—
175Dic71—41no—
176Dic71—41no—
177C2×C71—41no—
178C2×C71—41no—
179C2×C71—81no—
180C2×C71#54611yescharacteristic
181C2×C71#57511yescharacteristic
182C2×C71#57711yescharacteristic
183D71—81no—
184#3321—711nomaximal, Sylow
185C71#332111yescharacteristic, complemented, Sylow
186#555D7111yescharacteristic, complemented
187#548—711no—
188#489—711no—
189#485—711no—
190#482—711no—
191#475—711no—
192#469—711no—
193#465—711no—
194#463—711no—
195#449—711no—
196#448—711no—
197#440—711no—
198#435—711no—
199#419—711no—
200#416—711no—
201C2×#41—711no—
202C22×Q8—711no—
203C2×SD16—711no—
204C2×SD16—711no—
205C2×D8—711no—
206C2×M16—711no—
207C22×C8—711no—
208C4⋊Q8C2×D7111yescharacteristic
209#124—21no—
210#124—21no—
211#124—711no—
212#124—1421no—
213C22⋊Q8—711no—
214C22⋊Q8—1421no—
215#121—21no—
216#121—711no—
217#121—1421no—
218#121C2×D7111yescharacteristic
219#121C2×D7111yescharacteristic
220C4×Q8—711no—
221C4×D4—1421no—
222C4×D4C2×D7111yescharacteristic
223C4×D4C2×D7111yescharacteristic
224#117—21no—
225#117C2×D7111yescharacteristic
226#117C2×D7111yescharacteristic
227#107—711no—
228#106—711no—
229#106—711no—
230C4⋊C8—711no—
231#103—711no—
232#103—711no—
233#103—711no—
234#103—711no—
235#103—711no—
236#103—711no—
237#102—711no—
238#102—711no—
239#102—711no—
240#102—711no—
241C22⋊C8—711no—
242C22⋊C8—711no—
243#97—711no—
244#41—2841no—
245#41—1422no—
246C2×Q8—21no—
247C2×Q8—2841no—
248C2×Q8—711no—
249C2×Q8—711no—
250C2×Q8—1422no—
251C2×Q8—1421no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—711no—
255C2×D4—1421no—
256C2×D4—1421no—
257C2×D4C22⋊D7111yescharacteristic
258C2×D4C22×D7111yescharacteristic
259C22×C4—21no—
260C22×C4—1421no—
261C22×C4—1421no—
262C22×C4C22×D7111yescharacteristic
263C22×C4C22×D7111yescharacteristic
264SD16—2841no—
265SD16—1422no—
266D8—2841no—
267M16—2841no—
268C2×C8—2841no—
269C2×C8—711no—
270C2×C8—711no—
271C2×C8—711no—
272C2×C8—711no—
273C4⋊C4—21no—
274C4⋊C4—711no—
275C4⋊C4—1421no—
276C4⋊C4—1421no—
277C4⋊C4C22⋊D7111yescharacteristic
278C4⋊C4C22×D7111yescharacteristic
279C4⋊C4C22×D7111yescharacteristic
280C4⋊C4C22×D7111yescharacteristic
281C22⋊C4—21no—
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—1421no—
287C22⋊C4—1421no—
288C22⋊C4—1421no—
289C22⋊C4—1421no—
290C22⋊C4C22⋊D7111yescharacteristic
291C22⋊C4C22⋊D7111yescharacteristic
292C42—21no—
293C42—21no—
294C42—1421no—
295C42C22×D7111yescharacteristic
296C23—21no—
297C23—1421no—
298C23C2×C22⋊D7111yescharacteristic
299C23C2×C22⋊D7111yescharacteristic
300Q8—81no—
301Q8—2842no—
302Q8—712no—
303Q8—1421no—
304Q8—1421no—
305Q8—1421no—
306D4—21no—
307D4—21no—
308D4—41no—
309D4—81no—
310D4—81no—
311D4—2842no—
312D4—2842no—
313D4—1421no—
314D4—1421no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—42no—
323C2×C4—2842no—
324C2×C4—2842no—
325C2×C4—1421no—
326C2×C4—1421no—
327C2×C4—1421no—
328C2×C4C2×C22⋊D7111yescharacteristic
329C2×C4C2×C22⋊D7111yescharacteristic
330C2×C4C2×C22⋊D7111yescharacteristic
331C2×C4C2×C22⋊D7111yescharacteristic
332C2×C4C23×D7111yescharacteristic
333C8—1421no—
334C8—1421no—
335C8—1421no—
336C8—1421no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—2841no—
347C22—2841no—
348C22—2841no—
349C22C22×C22⋊D7111yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—2841no—
358C4—2841no—
359C4—1422no—
360C4#41×D7111yescharacteristic
361C4#1455617411yescharacteristic
362C2—41no—
363C2—41no—
364C2—81no—
365C2—5681no—
366C2#1798663711yescharacteristic
367C2#1798679711yescharacteristic
368C2#1798681011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#555×C71C2
socleC22×C71C22×D4
hypercenter#555D71
hypocenterC71#3321
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C4×D4 ◁ #555

Complements

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

NH ≅ G/Ntype
1. #17986658C2semidirect
2. #17986361C2semidirect
3. #17986340C2semidirect
4. #17986323C2semidirect
5. #17986281C2semidirect
6. #17986250C2semidirect
7. #17986226C2semidirect
8. #17986214C2semidirect
9. #17986142C2semidirect
10. #17986137C2semidirect
11. #17986099C2semidirect
12. #17986071C2semidirect
13. #17986007C2semidirect
14. #17985993C2semidirect
15. #555×C71C2semidirect
185. C71#3321semidirect
186. #555D71semidirect