Finite Group Databasebeta

← #22600331

Subgroups — #22600331

Order 9088.

420 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#17986653C211yesmaximal, characteristic, complemented
2#17986358C211yesmaximal, characteristic, complemented
3#17986332C211yesmaximal, characteristic, complemented
4#17986310C211yesmaximal, characteristic, complemented
5#17986294C211yesmaximal, characteristic, complemented
6#17986247C211yesmaximal, characteristic, complemented
7#17986233C211yesmaximal, characteristic, complemented
8#17986211C211yesmaximal, characteristic, complemented
9#17986144C211yesmaximal, characteristic, complemented
10#17986138C211yesmaximal, characteristic, complemented
11#17986097C211yesmaximal, characteristic, complemented
12#17986071C211yesmaximal, characteristic, complemented
13#17986000C211yesmaximal, characteristic, complemented
14C2×#14556052C211yesmaximal, characteristic, complemented
15#548×C71C211yesmaximal, characteristic, complemented
16C2×#11765864—21no—
17C2×#11765852C2211yescharacteristic
18C2×#11765852C2211yescharacteristic
19C2×#11765850C2211yescharacteristic
20C2×#11765849C2211yescharacteristic
21C22×C71⋊C8C2211yescharacteristic
22#14556128C2211yescharacteristic
23#14556112—21no—
24#14556112—21no—
25#14556108—21no—
26#14556102C2211yescharacteristic
27#14556102C2211yescharacteristic
28C4×D284C2211yescharacteristic
29C2×C4⋊Dic71C2211yescharacteristic
30C2×C4⋊Dic71C2211yescharacteristic
31#14556061C2211yescharacteristic
32#14556061C2211yescharacteristic
33#14556061C2211yescharacteristic
34C4⋊(C71⋊C8)C2211yescharacteristic
35#14556052C2211yescharacteristic
36#14556052C2211yescharacteristic
37#14556052C2211yescharacteristic
38#14556052C2211yescharacteristic
39#14556052C2211yescharacteristic
40#14556052C2211yescharacteristic
41#14556049C2211yescharacteristic
42#14556049C2211yescharacteristic
43#14556049C2211yescharacteristic
44#14556048C2211yescharacteristic
45C22⋊(C71⋊C8)C2211yescharacteristic
46C22⋊(C71⋊C8)C2211yescharacteristic
47#14556039C2211yescharacteristic
48C2×#41×C71C2211yescharacteristic
49C22×Q8×C71C2211yescharacteristic
50#124×C71—21no—
51#124×C71C2211yescharacteristic
52C22⋊Q8×C71—21no—
53C22⋊Q8×C71C2211yescharacteristic
54#121×C71—21no—
55#121×C71C2211yescharacteristic
56C4×Q8×C71C2211yescharacteristic
57C4×D4×C71—21no—
58C4×D4×C71C2211yescharacteristic
59#11765864—44no—
60C2×Dic142—21no—
61C2×D284C2311yescharacteristic
62C2×C22⋊D71—21no—
63C2×C22⋊D71—21no—
64C22×Dic71—21no—
65C22×Dic71—21no—
66C2×C4×D71—21no—
67#11765852—22no—
68#11765852—41no—
69#11765850—41no—
70#11765849—41no—
71C2×C71⋊C8—41no—
72C2×C71⋊C8C2311yescharacteristic
73C2×C71⋊C8C2311yescharacteristic
74C2×C71⋊C8C2311yescharacteristic
75C2×C71⋊C8C2311yescharacteristic
76C4⋊Dic71—21no—
77C4⋊Dic71—21no—
78C4⋊Dic71—21no—
79C4⋊Dic71C2311yescharacteristic
80C4⋊Dic71C2311yescharacteristic
81C4⋊Dic71C2311yescharacteristic
82(C2×D71)⋊C4—21no—
83(C2×D71)⋊C4—21no—
84(C2×D71)⋊C4—21no—
85#41×C71—22no—
86#41×C71—41no—
87C2×Q8×C71—21no—
88C2×Q8×C71—22no—
89C2×Q8×C71—41no—
90C2×Q8×C71C2311yescharacteristic
91C2×Q8×C71C2311yescharacteristic
92C2×D4×C71—21no—
93C2×D4×C71—21no—
94C2×D4×C71D411yescharacteristic
95C2×D4×C71C2311yescharacteristic
96C22×C4×C71—21no—
97C22×C4×C71—21no—
98C22×C4×C71C2311yescharacteristic
99C22×C4×C71C2311yescharacteristic
100C4⋊C4×C71—21no—
101C4⋊C4×C71—21no—
102C4⋊C4×C71D411yescharacteristic
103C4⋊C4×C71C2311yescharacteristic
104C22⋊C4×C71—21no—
105C22⋊C4×C71—21no—
106C22⋊C4×C71—21no—
107C22⋊C4×C71—21no—
108C22⋊C4×C71D411yescharacteristic
109C22⋊C4×C71D411yescharacteristic
110C42×C71—21no—
111C42×C71C2311yescharacteristic
112C23×C71—21no—
113C23×C71C2×D411yescharacteristic
114C23×C71C2×D411yescharacteristic
115C22×D71—21no—
116Q8×C71—21no—
117Q8×C71—21no—
118Q8×C71—21no—
119Q8×C71—42no—
120Q8×C71#4112yescomplemented
121D4×C71—22no—
122D4×C71—21no—
123D4×C71—21no—
124D4×C71—42no—
125D4×C71—42no—
126C2×C4×C71—21no—
127C2×C4×C71—21no—
128C2×C4×C71—21no—
129C2×C4×C71—21no—
130C2×C4×C71—21no—
131C2×C4×C71—21no—
132C2×C4×C71—21no—
133C2×C4×C71—42no—
134C2×C4×C71—42no—
135C2×C4×C71C2×D411yescharacteristic
136C2×C4×C71C2×D411yescharacteristic
137C2×C4×C71C2×D411yescharacteristic
138C2×C4×C71C2×D411yescharacteristic
139C2×C4×C71C2411yescharacteristic
140C22⋊D71—42no—
141C22⋊D71—42no—
142C2×Dic71—21no—
143C2×Dic71—21no—
144C2×Dic71—21no—
145C2×Dic71—42no—
146C2×Dic71—42no—
147D284—21no—
148D284—21no—
149C4×D71—42no—
150Dic142—42no—
151C71⋊C8—21no—
152C71⋊C8—21no—
153C71⋊C8—21no—
154C71⋊C8—21no—
155C22×C71—21no—
156C22×C71—21no—
157C22×C71—21no—
158C22×C71—21no—
159C22×C71—21no—
160C22×C71—41no—
161C22×C71—41no—
162C22×C71—41no—
163C22×C71AGL(1,ℤ/8)12yes—
164C22×C71C22×D411yescharacteristic
165C2×D71—41no—
166C2×D71—41no—
167C2×D71—41no—
168C4×C71—21no—
169C4×C71—21no—
170C4×C71—22no—
171C4×C71—21no—
172C4×C71—21no—
173C4×C71—41no—
174C4×C71—41no—
175C4×C71C2×#4111yescharacteristic
176C4×C71H(5,𝔽2)11yescharacteristic
177Dic71—41no—
178Dic71—41no—
179Dic71—41no—
180C2×C71—22no—
181C2×C71—41no—
182C2×C71—81no—
183C2×C71#54611yescharacteristic
184C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
185C2×C71#57811yescharacteristic
186D71—81no—
187#3308—711nomaximal, Sylow
188C71#330811yescharacteristic, complemented, Sylow
189#548D7111yescharacteristic, complemented
190#547—711no—
191#489—711no—
192#484—711no—
193#480—711no—
194#477—711no—
195#468—711no—
196#466—711no—
197#463—711no—
198#449—711no—
199#448—711no—
200#440—711no—
201#435—711no—
202#417—711no—
203C2×#103—711no—
204C2×#41—1421no—
205C2×#41C2×D7111yescharacteristic
206C22×Q8C2×D7111yescharacteristic
207C2×SD16—711no—
208C2×SD16—711no—
209C2×D8—711no—
210C2×M16—711no—
211C22×C8—711no—
212C4⋊Q8—711no—
213#124—21no—
214#124C2×D7111yescharacteristic
215#123—1421no—
216#123—1421no—
217C22⋊Q8—21no—
218C22⋊Q8—1421no—
219C22⋊Q8C2×D7111yescharacteristic
220#121—21no—
221#121—711no—
222#121—711no—
223#121C2×D7111yescharacteristic
224C4×Q8C2×D7111yescharacteristic
225C4×D4—21no—
226C4×D4—711no—
227C4×D4C2×D7111yescharacteristic
228C2×C4⋊C4—711no—
229C2×C4⋊C4—711no—
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#103—711no—
239#103—711no—
240#102—711no—
241#102—711no—
242#102—711no—
243#102—711no—
244C22⋊C8—711no—
245C22⋊C8—711no—
246#97—711no—
247#41—22no—
248#41—41no—
249#41—2844no—
250C2×Q8—21no—
251C2×Q8—22no—
252C2×Q8—41no—
253C2×Q8—1421no—
254C2×Q8C22×D7111yescharacteristic
255C2×Q8C22×D7111yescharacteristic
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—711no—
259C2×D4—1421no—
260C2×D4—1421no—
261C2×D4C22⋊D7111yescharacteristic
262C2×D4C22×D7111yescharacteristic
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—1421no—
266C22×C4—1421no—
267C22×C4—1421no—
268C22×C4C22×D7111yescharacteristic
269C22×C4C22×D7111yescharacteristic
270SD16—2841no—
271SD16—1422no—
272D8—2841no—
273M16—2841no—
274C2×C8—2841no—
275C2×C8—711no—
276C2×C8—711no—
277C2×C8—711no—
278C2×C8—711no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—711no—
282C4⋊C4—711no—
283C4⋊C4—711no—
284C4⋊C4—1421no—
285C4⋊C4—1421no—
286C4⋊C4—1421no—
287C4⋊C4C22⋊D7111yescharacteristic
288C4⋊C4C22×D7111yescharacteristic
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—1421no—
294C22⋊C4—1421no—
295C22⋊C4—1421no—
296C22⋊C4C22⋊D7111yescharacteristic
297C22⋊C4C22⋊D7111yescharacteristic
298C42—21no—
299C42C22×D7111yescharacteristic
300C23—21no—
301C23—1421no—
302C23C2×C22⋊D7111yescharacteristic
303C23C2×C22⋊D7111yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—42no—
308Q8—2842no—
309Q8#1176586412yescomplemented
310D4—22no—
311D4—21no—
312D4—21no—
313D4—42no—
314D4—42no—
315D4—2842no—
316D4—2842no—
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—42no—
327C2×C4—42no—
328C2×C4—2842no—
329C2×C4—2842no—
330C2×C4—2842no—
331C2×C4—1421no—
332C2×C4—1421no—
333C2×C4—1421no—
334C2×C4C2×C22⋊D7111yescharacteristic
335C2×C4C2×C22⋊D7111yescharacteristic
336C2×C4C2×C22⋊D7111yescharacteristic
337C2×C4C2×C22⋊D7111yescharacteristic
338C2×C4C23×D7111yescharacteristic
339C8—1421no—
340C8—1421no—
341C8—1421no—
342C8—1421no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—2841no—
352C22—2841no—
353C22—2841no—
354C22#1455615312yes—
355C22C22×C22⋊D7111yescharacteristic
356C4—21no—
357C4—21no—
358C4—22no—
359C4—21no—
360C4—21no—
361C4—41no—
362C4—41no—
363C4—2841no—
364C4—2841no—
365C4—2841no—
366C4C2×#1176586411yescharacteristic
367C4#1455617311yescharacteristic
368C2—22no—
369C2—41no—
370C2—81no—
371C2—5681no—
372C2#1798664311yescharacteristic
373C2C2×#1455615311yescharacteristic
374C2#1798681411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#548×C71C2
socleC22×C71C22×D4
hypercenter#548D71
hypocenterC71#3308
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #17986653C2semidirect
2. #17986358C2semidirect
3. #17986332C2semidirect
4. #17986310C2semidirect
5. #17986294C2semidirect
6. #17986247C2semidirect
7. #17986233C2semidirect
8. #17986211C2semidirect
9. #17986144C2semidirect
10. #17986138C2semidirect
11. #17986097C2semidirect
12. #17986071C2semidirect
13. #17986000C2semidirect
14. C2×#14556052C2semidirect
15. #548×C71C2semidirect
120. Q8×C71#41semidirect
188. C71#3308semidirect
189. #548D71semidirect
309. Q8#11765864semidirect