Finite Group Databasebeta

← #22600563

Subgroups — #22600563

Order 9088.

402 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17986690C211yesmaximal, characteristic, complemented
2#17986642C211yesmaximal, characteristic
3#17986353C211yesmaximal, characteristic, complemented
4#17986339C211yesmaximal, characteristic, complemented
5#17986316C211yesmaximal, characteristic, complemented
6#17986300C211yesmaximal, characteristic, complemented
7#17986254C211yesmaximal, characteristic, complemented
8#17986242C211yesmaximal, characteristic, complemented
9#17986160C211yesmaximal, characteristic, complemented
10#17986148C211yesmaximal, characteristic, complemented
11C4×#11765852C211yesmaximal, characteristic, complemented
12#17986076C211yesmaximal, characteristic, complemented
13#17986002C211yesmaximal, characteristic, complemented
14#17985999C211yesmaximal, characteristic, complemented
15#464×C71C211yesmaximal, characteristic, complemented
16C2×#11765866C2211yescharacteristic
17C2×D4×D71C2211yescharacteristic
18C2×#11765852C2211yescharacteristic
19C2×#11765848C2211yescharacteristic
20C2×#11765848C2211yescharacteristic, complemented
21#14556122—21no—
22#14556120C2211yescharacteristic, complemented
23#14556118—21no—
24#14556113—21no—
25#14556110—21no—
26#14556104C2211yescharacteristic
27#14556104C2211yescharacteristic
28#14556104C2211yescharacteristic, complemented
29#14556101—21no—
30#14556099C2211yescharacteristic
31#14556098—21no—
32C4×D284C2211yescharacteristic
33C4×D284C2211yescharacteristic
34#14556086C2211yescharacteristic
35#14556085—21no—
36C4⋊C4×D71C2211yescharacteristic, complemented
37C22⋊C4×D71—21no—
38#14556062C2211yescharacteristic, complemented
39#14556060C2211yescharacteristic, complemented
40#14556059C2211yescharacteristic, complemented
41#14556052C2211yescharacteristic, complemented
42#14556052C2211yescharacteristic, complemented
43#14556052C2211yescharacteristic, complemented
44#14556050C2211yescharacteristic
45#14556050C2211yescharacteristic
46#14556049C2211yescharacteristic
47#14556049C2211yescharacteristic
48#14556048C2211yescharacteristic
49(C2×D71)⋊C8C2211yescharacteristic
50(C2×D71)⋊C8C2211yescharacteristic, complemented
51C4×C71⋊C8C2211yescharacteristic, complemented
52C2×Q16×C71C2211yescharacteristic, complemented
53C2×SD16×C71C2211yescharacteristic
54#124×C71C2211yescharacteristic
55C4×Q8×C71C2211yescharacteristic, complemented
56C4⋊C8×C71C2211yescharacteristic, complemented
57#103×C71C2211yescharacteristic, complemented
58#102×C71C2211yescharacteristic
59C23×D71—21no—
60#11765866—22no—
61#11765866—41no—
62C2×D284—21no—
63C2×D284D411yescharacteristic
64C2×D284C2311yescharacteristic
65C2×C22⋊D71—21no—
66C2×C22⋊D71—21no—
67D4×D71—22no—
68D4×D71—41no—
69C2×C4×D71—21no—
70C2×C4×D71—21no—
71C2×C4×D71—21no—
72C2×C4×D71C2311yescharacteristic
73C2×C4×D71C2311yescharacteristic
74#11765852—22no—
75#11765848—41no—
76#11765848—41no—
77C2×C71⋊C8C2311yescharacteristic
78C2×C71⋊C8C2311yescharacteristic
79C4⋊Dic71D411yescharacteristic
80C4⋊Dic71C2311yescharacteristic
81C4⋊Dic71C2311yescharacteristic
82C4⋊Dic71C2311yescharacteristic
83Dic71⋊C4—21no—
84Dic71⋊C4—21no—
85Dic71⋊C4—21no—
86Dic71⋊C4—21no—
87Dic71⋊C4—21no—
88Dic71⋊C4—21no—
89C22⋊Dic71—21no—
90(C2×D71)⋊C4—21no—
91(C2×D71)⋊C4—21no—
92(C2×D71)⋊C4—21no—
93(C2×D71)⋊C4—21no—
94(C2×D71)⋊C4—21no—
95(C2×D71)⋊C4D411yescharacteristic
96(C2×D71)⋊C4D411yescharacteristic
97C4×Dic71—21no—
98C2×Q8×C71C2311yescharacteristic
99C2×Q8×C71C2311yescharacteristic
100C2×D4×C71C2311yescharacteristic
101Q16×C71—41no—
102SD16×C71—41no—
103C2×C8×C71C2311yescharacteristic
104C2×C8×C71C2311yescharacteristic
105C4⋊C4×C71—21no—
106C4⋊C4×C71C2311yescharacteristic
107C22⋊C4×C71—21no—
108C22⋊C4×C71—21no—
109C42×C71—21no—
110C42×C71C2311yescharacteristic
111C23×C71—21no—
112C22×D71—21no—
113C22×D71—42no—
114C22×D71—42no—
115C22×D71—81no—
116C22×D71C2×D411yescharacteristic
117C22×D71C2×D411yescharacteristic
118Q8×C71—21no—
119Q8×C71—21no—
120Q8×C71—21no—
121Q8×C71—21no—
122D4×C71—21no—
123D4×C71—21no—
124C2×C4×C71—21no—
125C2×C4×C71—21no—
126C2×C4×C71—21no—
127C2×C4×C71C2×D411yescharacteristic
128C2×C4×C71C2×D411yescharacteristic
129C2×C4×C71C2411yescharacteristic
130C22⋊D71—42no—
131C22⋊D71—81no—
132C2×Dic71—21no—
133C2×Dic71—21no—
134C2×Dic71—21no—
135C2×Dic71C2×D411yescharacteristic
136C2×Dic71C2×D411yescharacteristic
137D284—21no—
138D284—41no—
139D284—42no—
140D284#4112yes—
141C4×D71—21no—
142C4×D71—21no—
143C4×D71—21no—
144C4×D71—21no—
145C4×D71—42no—
146C4×D71—42no—
147C4×D71—81no—
148C8×C71—21no—
149C8×C71—21no—
150C71⋊C8—21no—
151C71⋊C8—21no—
152C22×C71—41no—
153C22×C71—41no—
154C22×C71—41no—
155C22×C71C22×D411yescharacteristic
156C2×D71—21no—
157C2×D71—21no—
158C2×D71—22no—
159C2×D71—21no—
160C2×D71—21no—
161C2×D71—21no—
162C2×D71—21no—
163C2×D71—41no—
164C2×D71—41no—
165C2×D71—84no—
166C4×C71—21no—
167C4×C71—21no—
168C4×C71—41no—
169C4×C71—41no—
170C4×C71—41no—
171C4×C71C2×#4111yescharacteristic
172C4×C71H(5,𝔽2)11yescharacteristic
173Dic71—21no—
174Dic71—21no—
175Dic71—41no—
176Dic71—41no—
177Dic71—41no—
178C2×C71—81no—
179C2×C71#54611yescharacteristic
180C2×C71#57511yescharacteristic
181C2×C71#57711yescharacteristic
182D71—41no—
183D71—41no—
184D71—42no—
185#3323—711nomaximal, Sylow
186C71#332311yescharacteristic, complemented, Sylow
187#552—711no—
188#546—711no—
189#488—711no—
190#485—711no—
191#481—711no—
192#478—711no—
193#470—711no—
194#468—711no—
195#464D7111yescharacteristic, complemented
196#452—711no—
197#450—711no—
198C4×SD16—711no—
199#436—711no—
200#418—711no—
201#417—711no—
202C2×#41—711no—
203C22×D4—711no—
204C2×Q16C2×D7111yescharacteristic, complemented
205C2×SD16—711no—
206C2×SD16C2×D7111yescharacteristic
207C2×M16—711no—
208C2×M16—711no—
209#126—1421no—
210#125—711no—
211#125—1421no—
212#124C2×D7111yescharacteristic
213#123—1421no—
214#123—1421no—
215C22⋊Q8—711no—
216C22⋊Q8—711no—
217C22⋊Q8—711no—
218#121—711no—
219#121—1421no—
220#120—1421no—
221C4×Q8C2×D7111yescharacteristic, complemented
222C4×D4—711no—
223C4×D4—711no—
224#117—711no—
225#117—1421no—
226C2×C4⋊C4—711no—
227C2×C22⋊C4—1421no—
228#107—711no—
229#106—711no—
230#106—711no—
231C4⋊C8C2×D7111yescharacteristic, complemented
232#103—711no—
233#103—711no—
234#103—711no—
235#103C2×D7111yescharacteristic, complemented
236#102—711no—
237#102—711no—
238#102—711no—
239#102—711no—
240#102—711no—
241#102C2×D7111yescharacteristic
242C22⋊C8—711no—
243C22⋊C8—711no—
244C4×C8—711no—
245C24—1421no—
246#41—2841no—
247#41—1422no—
248C2×Q8C22×D7111yescharacteristic
249C2×Q8C22×D7111yescharacteristic
250C2×D4—2841no—
251C2×D4—711no—
252C2×D4—711no—
253C2×D4—1421no—
254C2×D4—1421no—
255C2×D4—1422no—
256C2×D4—1421no—
257C2×D4C22×D7111yescharacteristic
258C22×C4—711no—
259C22×C4—711no—
260C22×C4—1421no—
261C22×C4—1421no—
262C22×C4—1421no—
263Q16—41no—
264SD16—41no—
265SD16—1422no—
266M16—2841no—
267M16—2841no—
268C2×C8—711no—
269C2×C8—711no—
270C2×C8C22×D7111yescharacteristic
271C2×C8C22×D7111yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—711no—
274C4⋊C4—711no—
275C4⋊C4—711no—
276C4⋊C4—711no—
277C4⋊C4—1421no—
278C4⋊C4—1421no—
279C4⋊C4—1421no—
280C4⋊C4—1421no—
281C4⋊C4—1421no—
282C4⋊C4—1421no—
283C4⋊C4C22×D7111yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—711no—
287C22⋊C4—711no—
288C22⋊C4—1421no—
289C22⋊C4—1421no—
290C22⋊C4—1421no—
291C22⋊C4—1421no—
292C22⋊C4—1421no—
293C22⋊C4—1421no—
294C42—21no—
295C42—1421no—
296C42C22×D7111yescharacteristic
297C23—21no—
298C23—2842no—
299C23—2842no—
300C23—5681no—
301C23—711no—
302C23—711no—
303C23—1421no—
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—21no—
308D4—21no—
309D4—21no—
310D4—2841no—
311D4—2842no—
312D4—2842no—
313D4—5681no—
314D4—712no—
315D4—1421no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—2842no—
320C2×C4—2842no—
321C2×C4—5681no—
322C2×C4—711no—
323C2×C4—711no—
324C2×C4—1421no—
325C2×C4—1421no—
326C2×C4—1421no—
327C2×C4—1421no—
328C2×C4—1421no—
329C2×C4—1421no—
330C2×C4—1421no—
331C2×C4D4×D7111yescharacteristic
332C2×C4D4×D7111yescharacteristic
333C2×C4C23×D7111yescharacteristic
334C8—21no—
335C8—21no—
336C8—1421no—
337C8—1421no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—2841no—
342C22—2841no—
343C22—5684no—
344C22—1422no—
345C22—1421no—
346C22—1421no—
347C22—1421no—
348C22—1421no—
349C22—1421no—
350C22—1421no—
351C22C2×D4×D7111yescharacteristic
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—2841no—
358C4—2841no—
359C4—2841no—
360C4—1421no—
361C4—1421no—
362C4#41×D7111yescharacteristic
363C4#1455617311yescharacteristic
364C2—81no—
365C2—2842no—
366C2—2841no—
367C2—2841no—
368C2#1798664611yescharacteristic
369C2#1798679611yescharacteristic
370C2#1798680811yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#464×C71C2
socleC22×C71C22×D4
hypercenter#464D71
hypocenterC71#3323
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. #17986690C2semidirect
3. #17986353C2semidirect
4. #17986339C2semidirect
5. #17986316C2semidirect
6. #17986300C2semidirect
7. #17986254C2semidirect
8. #17986242C2semidirect
9. #17986160C2semidirect
10. #17986148C2semidirect
11. C4×#11765852C2semidirect
12. #17986076C2semidirect
13. #17986002C2semidirect
14. #17985999C2semidirect
15. #464×C71C2semidirect
20. C2×#11765848C22semidirect
22. #14556120C22semidirect
28. #14556104C22semidirect
36. C4⋊C4×D71C22semidirect
38. #14556062C22semidirect
39. #14556060C22semidirect
40. #14556059C22semidirect
41. #14556052C22semidirect
42. #14556052C22semidirect
43. #14556052C22semidirect
50. (C2×D71)⋊C8C22semidirect
51. C4×C71⋊C8C22semidirect
52. C2×Q16×C71C22semidirect
55. C4×Q8×C71C22semidirect
56. C4⋊C8×C71C22semidirect
57. #103×C71C22semidirect
186. C71#3323semidirect
195. #464D71semidirect
204. C2×Q16C2×D71semidirect
221. C4×Q8C2×D71semidirect
231. C4⋊C8C2×D71semidirect
235. #103C2×D71semidirect