Finite Group Databasebeta

← #22600425

Subgroups — #22600425

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#17986707C211yesmaximal, characteristic, complemented
2#17986643C211yesmaximal, characteristic
3#17986357C211yesmaximal, characteristic, complemented
4#17986335C211yesmaximal, characteristic, complemented
5#17986311C211yesmaximal, characteristic, complemented
6#17986277C211yesmaximal, characteristic, complemented
7#17986242C211yesmaximal, characteristic, complemented
8#17986238C211yesmaximal, characteristic, complemented
9#17986219C211yesmaximal, characteristic, complemented
10#17986158C211yesmaximal, characteristic, complemented
11#17986100C211yesmaximal, characteristic, complemented
12#17986067C211yesmaximal, characteristic, complemented
13#17985997C211yesmaximal, characteristic, complemented
14C2×#14556049C211yesmaximal, characteristic, complemented
15#450×C71C211yesmaximal, characteristic, complemented
16C2×#11765864C2211yescharacteristic
17C2×#11765855C2211yescharacteristic, complemented
18C2×#11765848C2211yescharacteristic, complemented
19C22×C71⋊C8C2211yescharacteristic
20#14556127C2211yescharacteristic, complemented
21#14556118—21no—
22#14556115C2211yescharacteristic
23#14556109—21no—
24#14556108—21no—
25#14556107C2211yescharacteristic
26#14556105—21no—
27#14556104C2211yescharacteristic, complemented
28#14556100—21no—
29#14556099C2211yescharacteristic
30#14556097—21no—
31Q8×Dic71C2211yescharacteristic, complemented
32D4×Dic71C2211yescharacteristic
33C4×C22⋊D71C2211yescharacteristic
34(C22⋊D71)⋊C4—21no—
35C2×C4⋊Dic71C2211yescharacteristic
36C4⋊C4×D71C2211yescharacteristic, complemented
37C2×C22⋊Dic71—21no—
38#14556062C2211yescharacteristic, complemented
39#14556062C2211yescharacteristic, complemented
40#14556061C2211yescharacteristic, complemented
41Dic71⋊C8C2211yescharacteristic, complemented
42#14556053C2211yescharacteristic, complemented
43#14556053C2211yescharacteristic, complemented
44#14556052C2211yescharacteristic, complemented
45#14556051C2211yescharacteristic, complemented
46#14556049C2211yescharacteristic
47#14556049C2211yescharacteristic
48#14556049C2211yescharacteristic
49#14556049C2211yescharacteristic
50(C2×D71)⋊C8C2211yescharacteristic, complemented
51#14556038C2211yescharacteristic, complemented
52C22×D4×C71C2211yescharacteristic
53C2×SD16×C71C2211yescharacteristic
54C2×SD16×C71C2211yescharacteristic
55C22⋊Q8×C71C2211yescharacteristic
56#102×C71C2211yescharacteristic
57#102×C71C2211yescharacteristic
58C22⋊C8×C71C2211yescharacteristic
59#11765864—22no—
60#11765864—41no—
61C2×Dic142—21no—
62C2×Dic142C2311yescharacteristic
63C2×C22⋊D71—21no—
64C2×C22⋊D71—21no—
65C2×C22⋊D71D411yescharacteristic
66C22×Dic71—21no—
67C22×Dic71—21no—
68C22×Dic71—21no—
69C2×C4×D71—21no—
70C2×C4×D71C2311yescharacteristic
71#11765855—41no—
72#11765848—41no—
73C2×C71⋊C8—41no—
74C2×C71⋊C8C2311yescharacteristic
75C2×C71⋊C8C2311yescharacteristic
76C4⋊Dic71—21no—
77C4⋊Dic71—21no—
78C4⋊Dic71C2311yescharacteristic
79C4⋊Dic71C2311yescharacteristic
80C4⋊Dic71C2311yescharacteristic
81Dic71⋊C4—21no—
82Dic71⋊C4—21no—
83Dic71⋊C4—21no—
84Dic71⋊C4—21no—
85Dic71⋊C4D411yescharacteristic
86C22⋊Dic71—21no—
87C22⋊Dic71—21no—
88C22⋊Dic71—21no—
89C22⋊Dic71—21no—
90C22⋊Dic71—21no—
91C22⋊Dic71D411yescharacteristic
92(C2×D71)⋊C4—21no—
93(C2×D71)⋊C4D411yescharacteristic
94C4×Dic71—21no—
95C4×Dic71C2311yescharacteristic
96C24×C71—21no—
97C2×Q8×C71C2311yescharacteristic
98C2×D4×C71—22no—
99C2×D4×C71—21no—
100C2×D4×C71—41no—
101C2×D4×C71C2311yescharacteristic
102C2×D4×C71C2311yescharacteristic
103C22×C4×C71C2311yescharacteristic
104SD16×C71—22no—
105SD16×C71—41no—
106C2×C8×C71C2311yescharacteristic
107C2×C8×C71C2311yescharacteristic
108C4⋊C4×C71—21no—
109C4⋊C4×C71C2311yescharacteristic
110C22⋊C4×C71—21no—
111C23×C71—21no—
112C23×C71—21no—
113C23×C71—42no—
114C23×C71—42no—
115C23×C71—81no—
116C23×C71C2×D411yescharacteristic
117C22×D71C2×D411yescharacteristic
118Q8×C71—21no—
119Q8×C71—21no—
120D4×C71—21no—
121D4×C71—21no—
122D4×C71—21no—
123D4×C71—42no—
124D4×C71#4112yes—
125C2×C4×C71—21no—
126C2×C4×C71—21no—
127C2×C4×C71—21no—
128C2×C4×C71—21no—
129C2×C4×C71C2×D411yescharacteristic
130C2×C4×C71C2411yescharacteristic
131C22⋊D71—22no—
132C22⋊D71—42no—
133C22⋊D71—81no—
134C2×Dic71—21no—
135C2×Dic71—21no—
136C2×Dic71—21no—
137C2×Dic71—21no—
138C2×Dic71—42no—
139C2×Dic71—81no—
140C2×Dic71—81no—
141C2×Dic71C2×D411yescharacteristic
142C2×Dic71C2×D411yescharacteristic
143C2×Dic71C2×D411yescharacteristic
144C4×D71—21no—
145C4×D71—21no—
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—22no—
157C22×C71—41no—
158C22×C71—41no—
159C22×C71—41no—
160C22×C71—41no—
161C22×C71—41no—
162C22×C71—84no—
163C22×C71#13712yes—
164C22×C71C22×D411yescharacteristic
165C2×D71—21no—
166C2×D71—21no—
167C2×D71—21no—
168C4×C71—21no—
169C4×C71—41no—
170C4×C71—41no—
171C4×C71C2×#4111yescharacteristic
172C4×C71H(5,𝔽2)11yescharacteristic
173Dic71—21no—
174Dic71—21no—
175Dic71—21no—
176Dic71—41no—
177Dic71—41no—
178Dic71—41no—
179Dic71—41no—
180C2×C71—22no—
181C2×C71—42no—
182C2×C71—81no—
183C2×C71#54611yescharacteristic
184C2×C71C2×#13711yescharacteristic
185C2×C71#57611yescharacteristic
186D71—41no—
187#3314—711nomaximal, Sylow
188C71#331411yescharacteristic, complemented, Sylow
189#554—711no—
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#450D7111yescharacteristic, complemented
200#440—711no—
201#435—711no—
202#417—711no—
203C2×#102—711no—
204C2×#41—711no—
205C22×D4C2×D7111yescharacteristic
206C2×Q16—711no—
207C2×SD16C2×D7111yescharacteristic
208C2×SD16C2×D7111yescharacteristic
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⋊C4—711no—
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#102C2×D7111yescharacteristic
243#102C2×D7111yescharacteristic
244C22⋊C8—711no—
245C22⋊C8C2×D7111yescharacteristic
246#97—711no—
247C24—21no—
248#41—2841no—
249#41—1422no—
250C2×Q8—711no—
251C2×Q8—1421no—
252C2×Q8C22×D7111yescharacteristic
253C2×D4—22no—
254C2×D4—21no—
255C2×D4—41no—
256C2×D4—711no—
257C2×D4—1421no—
258C2×D4—1421no—
259C2×D4C22×D7111yescharacteristic
260C2×D4C22×D7111yescharacteristic
261C22×C4—711no—
262C22×C4—1421no—
263C22×C4—1421no—
264C22×C4—1421no—
265C22×C4—1421no—
266C22×C4C22×D7111yescharacteristic
267Q16—2841no—
268SD16—22no—
269SD16—41no—
270M16—2841no—
271C2×C8—2841no—
272C2×C8—711no—
273C2×C8—711no—
274C2×C8C22×D7111yescharacteristic
275C2×C8C22×D7111yescharacteristic
276C4⋊C4—21no—
277C4⋊C4—711no—
278C4⋊C4—711no—
279C4⋊C4—711no—
280C4⋊C4—711no—
281C4⋊C4—1421no—
282C4⋊C4—1421no—
283C4⋊C4—1421no—
284C4⋊C4—1421no—
285C4⋊C4—1421no—
286C4⋊C4—1421no—
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—21no—
300C23—21no—
301C23—42no—
302C23—42no—
303C23—81no—
304C23—711no—
305C23D4×D7111yescharacteristic
306Q8—21no—
307Q8—21no—
308Q8—5681no—
309Q8—1421no—
310Q8—1421no—
311D4—21no—
312D4—21no—
313D4—21no—
314D4—42no—
315D4—2842no—
316D4—5681no—
317D4—1422no—
318D4#1176586412yes—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—2842no—
324C2×C4—5681no—
325C2×C4—5681no—
326C2×C4—5681no—
327C2×C4—711no—
328C2×C4—711no—
329C2×C4—711no—
330C2×C4—1421no—
331C2×C4—1421no—
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—22no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—84no—
351C22—1421no—
352C22—1421no—
353C22—1421no—
354C22#1455615712yes—
355C22C2×D4×D7111yescharacteristic
356C4—21no—
357C4—41no—
358C4—41no—
359C4—2841no—
360C4—2841no—
361C4—2841no—
362C4—2841no—
363C4—1421no—
364C4—1421no—
365C4—1421no—
366C4C2×#1176586411yescharacteristic
367C4#1455617311yescharacteristic
368C2—22no—
369C2—42no—
370C2—81no—
371C2—2841no—
372C2#1798663611yescharacteristic
373C2C2×#1455615711yescharacteristic
374C2#1798680311yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#450×C71C2
socleC22×C71C22×D4
hypercenter#450D71
hypocenterC71#3314
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #17986707C2semidirect
3. #17986357C2semidirect
4. #17986335C2semidirect
5. #17986311C2semidirect
6. #17986277C2semidirect
7. #17986242C2semidirect
8. #17986238C2semidirect
9. #17986219C2semidirect
10. #17986158C2semidirect
11. #17986100C2semidirect
12. #17986067C2semidirect
13. #17985997C2semidirect
14. C2×#14556049C2semidirect
15. #450×C71C2semidirect
17. C2×#11765855C22semidirect
18. C2×#11765848C22semidirect
20. #14556127C22semidirect
27. #14556104C22semidirect
31. Q8×Dic71C22semidirect
36. C4⋊C4×D71C22semidirect
38. #14556062C22semidirect
39. #14556062C22semidirect
40. #14556061C22semidirect
41. Dic71⋊C8C22semidirect
42. #14556053C22semidirect
43. #14556053C22semidirect
44. #14556052C22semidirect
45. #14556051C22semidirect
50. (C2×D71)⋊C8C22semidirect
51. #14556038C22semidirect
188. C71#3314semidirect
199. #450D71semidirect