Finite Group Databasebeta

← #22599194

Subgroups — #22599194

Order 9088.

422 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17986577C211yesmaximal, characteristic, complemented
2#17986563C211yesmaximal, characteristic, complemented
3#17986341C211yesmaximal, characteristic, complemented
4#17986324C211yesmaximal, characteristic, complemented
5#17986318C211yesmaximal, characteristic, complemented
6#17986313C211yesmaximal, characteristic, complemented
7#17986252C211yesmaximal, characteristic, complemented
8#17986245C211yesmaximal, characteristic, complemented
9#17986235C211yesmaximal, characteristic, complemented
10#17986228C211yesmaximal, characteristic, complemented
11#17986152C211yesmaximal, characteristic, complemented
12#17986146C211yesmaximal, characteristic, complemented
13#17986146C211yesmaximal, characteristic, complemented
14#17986135C211yesmaximal, characteristic, complemented
15#408×C71C211yesmaximal, characteristic, complemented
16C2×#11765865C2211yescharacteristic, complemented
17C22×D284C2211yescharacteristic
18C2×#11765854C2211yescharacteristic
19C2×#11765854C2211yescharacteristic
20C2×D568C2211yescharacteristic
21C2×D568C2211yescharacteristic
22#14556124—21no—
23#14556117—21no—
24#14556107C2211yescharacteristic
25#14556107C2211yescharacteristic, complemented
26#14556104—21no—
27#14556102C2211yescharacteristic
28#14556102C2211yescharacteristic
29#14556102C2211yescharacteristic
30#14556102C2211yescharacteristic
31#14556102C2211yescharacteristic
32#14556102C2211yescharacteristic, complemented
33#14556100—21no—
34#14556100—21no—
35#14556099—21no—
36#14556096—21no—
37#14556096—21no—
38#14556087C2211yescharacteristic
39C2×C4⋊Dic71C2211yescharacteristic, complemented
40#14556062C2211yescharacteristic, complemented
41#14556062C2211yescharacteristic, complemented
42#14556060C2211yescharacteristic, complemented
43#14556060C2211yescharacteristic, complemented
44#14556053C2211yescharacteristic, complemented
45#14556053C2211yescharacteristic, complemented
46#14556050C2211yescharacteristic
47#14556050C2211yescharacteristic
48#14556050C2211yescharacteristic
49#14556050C2211yescharacteristic
50#14556050C2211yescharacteristic, complemented
51#14556050C2211yescharacteristic, complemented
52C2×#41×C71C2211yescharacteristic
53C2×M16×C71C2211yescharacteristic, complemented
54C22×C8×C71C2211yescharacteristic, complemented
55C22⋊C8×C71C2211yescharacteristic
56C22⋊C8×C71C2211yescharacteristic
57C22⋊C8×C71C2211yescharacteristic, complemented
58C22⋊C8×C71C2211yescharacteristic, complemented
59C23×D71—21no—
60#11765865—41no—
61#11765865—41no—
62C2×Dic142C2311yescharacteristic
63C2×D284—21no—
64C2×D284—41no—
65C2×D284—41no—
66C2×D284C2311yescharacteristic
67C2×D284C2311yescharacteristic
68C2×D284C2311yescharacteristic
69C2×C22⋊D71—21no—
70C2×C22⋊D71—21no—
71C2×C22⋊D71—21no—
72C2×C22⋊D71—21no—
73C2×C22⋊D71—21no—
74C2×C22⋊D71—21no—
75C2×C22⋊D71—21no—
76C22×Dic71—21no—
77C2×C4×D71—21no—
78#11765854—41no—
79#11765854—41no—
80D568—41no—
81D568—41no—
82C4⋊Dic71—21no—
83C4⋊Dic71C2311yescharacteristic
84C4⋊Dic71C2311yescharacteristic
85C4⋊Dic71C2311yescharacteristic
86C4⋊Dic71C2311yescharacteristic
87Dic71⋊C4—21no—
88Dic71⋊C4—21no—
89C22⋊Dic71—21no—
90C22⋊Dic71—21no—
91C22⋊Dic71—21no—
92(C2×D71)⋊C4—21no—
93(C2×D71)⋊C4—21no—
94(C2×D71)⋊C4—21no—
95(C2×D71)⋊C4—21no—
96(C2×D71)⋊C4—21no—
97(C2×D71)⋊C4—21no—
98C4×Dic71—21no—
99#41×C71—41no—
100#41×C71—41no—
101C2×Q8×C71D411yescharacteristic
102C2×D4×C71D411yescharacteristic
103C2×D4×C71D411yescharacteristic
104C2×D4×C71D411yescharacteristic
105C22×C4×C71C2311yescharacteristic
106C22×C4×C71C2311yescharacteristic
107C22×C4×C71C2311yescharacteristic
108M16×C71—41no—
109C2×C8×C71—41no—
110C2×C8×C71C2311yescharacteristic
111C2×C8×C71C2311yescharacteristic
112C2×C8×C71C2311yescharacteristic
113C2×C8×C71C2311yescharacteristic
114C23×C71C2×D411yescharacteristic
115C23×C71C2×D411yescharacteristic
116C23×C71C2×D411yescharacteristic
117C22×D71—21no—
118C22×D71—21no—
119C22×D71—21no—
120C22×D71—42no—
121C22×D71—81no—
122C22×D71—81no—
123Q8×C71—41no—
124D4×C71—41no—
125D4×C71—41no—
126D4×C71—41no—
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×C71C2×D411yescharacteristic
134C2×C4×C71C2×D411yescharacteristic
135C2×C4×C71C2×D411yescharacteristic
136C2×C4×C71C2411yescharacteristic
137C22⋊D71—42no—
138C22⋊D71—81no—
139C22⋊D71—81no—
140C22⋊D71—81no—
141C22⋊D71—81no—
142C22⋊D71—81no—
143C22⋊D71—81no—
144C2×Dic71—21no—
145C2×Dic71—21no—
146C2×Dic71—21no—
147C2×Dic71—21no—
148C2×Dic71—21no—
149C2×Dic71—81no—
150D284—21no—
151D284—21no—
152D284—21no—
153D284—21no—
154D284—21no—
155D284—21no—
156D284—42no—
157C4×D71—42no—
158Dic142—21no—
159Dic142—21no—
160C8×C71—21no—
161C8×C71—21no—
162C8×C71—21no—
163C8×C71—21no—
164C22×C71—21no—
165C22×C71—21no—
166C22×C71—21no—
167C22×C71—21no—
168C22×C71—21no—
169C22×C71—21no—
170C22×C71—21no—
171C22×C71—21no—
172C22×C71—21no—
173C22×C71C22×D411yescharacteristic
174C2×D71—41no—
175C2×D71—41no—
176C2×D71—41no—
177C2×D71—41no—
178C2×D71—41no—
179C2×D71—41no—
180C2×D71—41no—
181C2×D71—41no—
182C2×D71—41no—
183C2×D71—84no—
184C4×C71—21no—
185C4×C71—21no—
186C4×C71—21no—
187C4×C71H(5,𝔽2)11yescharacteristic
188C4×C71H(5,𝔽2)11yescharacteristic
189Dic71—41no—
190Dic71—41no—
191Dic71—41no—
192Dic71—41no—
193Dic71—41no—
194C2×C71—41no—
195C2×C71—41no—
196C2×C71—41no—
197C2×C71#53411yescharacteristic
198C2×C71#57611yescharacteristic
199C2×C71#57711yescharacteristic
200D71—81no—
201D71—81no—
202D71—81no—
203#3198—711nomaximal, Sylow
204C71#319811yescharacteristic, complemented, Sylow
205#537—711no—
206#535—711no—
207#485—711no—
208#482—711no—
209#481—711no—
210#480—711no—
211#469—711no—
212#468—711no—
213#466—711no—
214#465—711no—
215#450—711no—
216#449—711no—
217#449—711no—
218#447—711no—
219#408D7111yescharacteristic, complemented
220C2×#41—711no—
221C2×#41C2×D7111yescharacteristic
222C22×D4—711no—
223C2×SD16—711no—
224C2×SD16—711no—
225C2×D8—711no—
226C2×D8—711no—
227C2×M16C2×D7111yescharacteristic, complemented
228C22×C8C2×D7111yescharacteristic, complemented
229#127—1421no—
230#124—1421no—
231C22⋊Q8—711no—
232C22⋊Q8—711no—
233C22⋊Q8—1421no—
234#121—711no—
235#121—711no—
236#121—711no—
237#121—711no—
238#121—711no—
239#121—711no—
240#121—1421no—
241#121—1421no—
242#121—1421no—
243#120—1421no—
244#120—1421no—
245#117—711no—
246C2×C4⋊C4—711no—
247#107—711no—
248#107—711no—
249#106—711no—
250#106—711no—
251#103—711no—
252#103—711no—
253#102—711no—
254#102—711no—
255#102—711no—
256#102—711no—
257#102—711no—
258#102—711no—
259C22⋊C8C2×D7111yescharacteristic
260C22⋊C8C2×D7111yescharacteristic
261C22⋊C8C2×D7111yescharacteristic, complemented
262C22⋊C8C2×D7111yescharacteristic, complemented
263C24—1421no—
264#41—41no—
265#41—41no—
266#41—2841no—
267#41—2841no—
268C2×Q8—711no—
269C2×Q8D28411yescharacteristic
270C2×D4—2841no—
271C2×D4—2841no—
272C2×D4—711no—
273C2×D4—711no—
274C2×D4—711no—
275C2×D4—1421no—
276C2×D4—1421no—
277C2×D4—1421no—
278C2×D4—1421no—
279C2×D4—1421no—
280C2×D4—1421no—
281C2×D4—1421no—
282C2×D4—1421no—
283C2×D4D28411yescharacteristic
284C2×D4D28411yescharacteristic
285C2×D4D28411yescharacteristic
286C22×C4—1421no—
287C22×C4—1421no—
288C22×C4C22×D7111yescharacteristic
289C22×C4C22×D7111yescharacteristic
290C22×C4C22×D7111yescharacteristic
291SD16—2841no—
292SD16—2841no—
293D8—2841no—
294D8—2841no—
295M16—41no—
296C2×C8—41no—
297C2×C8C22×D7111yescharacteristic
298C2×C8C22×D7111yescharacteristic
299C2×C8C22×D7111yescharacteristic
300C2×C8C22×D7111yescharacteristic
301C4⋊C4—711no—
302C4⋊C4—711no—
303C4⋊C4—711no—
304C4⋊C4—711no—
305C4⋊C4—1421no—
306C4⋊C4—1421no—
307C4⋊C4—1421no—
308C22⋊C4—1421no—
309C22⋊C4—1421no—
310C22⋊C4—1421no—
311C22⋊C4—1421no—
312C22⋊C4—1421no—
313C22⋊C4—1421no—
314C22⋊C4—1421no—
315C22⋊C4—1421no—
316C22⋊C4—1421no—
317C42—1421no—
318C23—2842no—
319C23—5681no—
320C23—5681no—
321C23—1421no—
322C23—1421no—
323C23—1421no—
324C23C2×D28411yescharacteristic
325C23C2×D28411yescharacteristic
326C23C2×D28411yescharacteristic
327Q8—41no—
328Q8—1421no—
329Q8—1421no—
330D4—41no—
331D4—41no—
332D4—41no—
333D4—2842no—
334D4—2842no—
335D4—5681no—
336D4—5681no—
337D4—5681no—
338D4—5681no—
339D4—5681no—
340D4—5681no—
341D4—1421no—
342D4—1421no—
343D4—1421no—
344D4—1421no—
345D4—1421no—
346D4—1421no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—2842no—
354C2×C4—5681no—
355C2×C4—1421no—
356C2×C4—1421no—
357C2×C4—1421no—
358C2×C4—1421no—
359C2×C4—1421no—
360C2×C4C2×D28411yescharacteristic
361C2×C4C2×D28411yescharacteristic
362C2×C4C2×D28411yescharacteristic
363C2×C4C23×D7111yescharacteristic
364C8—21no—
365C8—21no—
366C8—21no—
367C8—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—2841no—
378C22—2841no—
379C22—2841no—
380C22—2841no—
381C22—2841no—
382C22—2841no—
383C22—2841no—
384C22—2841no—
385C22—2841no—
386C22—5684no—
387C22C22×D28411yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—2841no—
392C4—2841no—
393C4—2841no—
394C4—2841no—
395C4—2841no—
396C4#1455617311yescharacteristic
397C4#1455617311yescharacteristic
398C2—41no—
399C2—41no—
400C2—41no—
401C2—5681no—
402C2—5681no—
403C2—5681no—
404C2#1798656111yescharacteristic
405C2#1798680411yescharacteristic
406C2#1798681111yescharacteristic
subgroupG/H
centerC22C22×D284
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#408×C71C2
socleC22×C71C22×D4
hypercenter#408D71
hypocenterC71#3198
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
1. #17986577C2semidirect
2. #17986563C2semidirect
3. #17986341C2semidirect
4. #17986324C2semidirect
5. #17986318C2semidirect
6. #17986313C2semidirect
7. #17986252C2semidirect
8. #17986245C2semidirect
9. #17986235C2semidirect
10. #17986228C2semidirect
11. #17986152C2semidirect
12. #17986146C2semidirect
13. #17986146C2semidirect
14. #17986135C2semidirect
15. #408×C71C2semidirect
16. C2×#11765865C22semidirect
25. #14556107C22semidirect
32. #14556102C22semidirect
39. C2×C4⋊Dic71C22semidirect
40. #14556062C22semidirect
41. #14556062C22semidirect
42. #14556060C22semidirect
43. #14556060C22semidirect
44. #14556053C22semidirect
45. #14556053C22semidirect
50. #14556050C22semidirect
51. #14556050C22semidirect
53. C2×M16×C71C22semidirect
54. C22×C8×C71C22semidirect
57. C22⋊C8×C71C22semidirect
58. C22⋊C8×C71C22semidirect
204. C71#3198semidirect
219. #408D71semidirect
227. C2×M16C2×D71semidirect
228. C22×C8C2×D71semidirect
261. C22⋊C8C2×D71semidirect
262. C22⋊C8C2×D71semidirect