Finite Group Databasebeta

← #22599428

Subgroups — #22599428

Order 9088.

408 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17986568C211yesmaximal, characteristic, complemented
2#17986389C211yesmaximal, characteristic, complemented
3#17986381C211yesmaximal, characteristic, complemented
4#17986361C211yesmaximal, characteristic, complemented
5#17986347C211yesmaximal, characteristic, complemented
6#17986221C211yesmaximal, characteristic, complemented
7#17986214C211yesmaximal, characteristic, complemented
8#17986199C211yesmaximal, characteristic, complemented
9#17986194C211yesmaximal, characteristic, complemented
10#17986159C211yesmaximal, characteristic, complemented
11#17986149C211yesmaximal, characteristic, complemented
12#17986142C211yesmaximal, characteristic, complemented
13#17986132C211yesmaximal, characteristic, complemented
14#17986060C211yesmaximal, characteristic, complemented
15#542×C71C211yesmaximal, characteristic, complemented
16C2×#11765865C2211yescharacteristic
17C22×D284C2211yescharacteristic
18C2×#11765855C2211yescharacteristic, complemented
19C2×#11765853C2211yescharacteristic, complemented
20C2×#11765852C2211yescharacteristic
21C2×#11765852C2211yescharacteristic
22C2×#11765852C2211yescharacteristic, complemented
23C2×#11765850C2211yescharacteristic
24C2×#11765850C2211yescharacteristic
25C2×#11765850C2211yescharacteristic, complemented
26#14556125C2211yescharacteristic
27#14556125C2211yescharacteristic
28#14556116C2211yescharacteristic
29#14556116C2211yescharacteristic
30#14556103—21no—
31#14556103—21no—
32#14556098—21no—
33#14556098—21no—
34C4⋊(C71⋊C8)C2211yescharacteristic, complemented
35C4⋊(C71⋊C8)C2211yescharacteristic, complemented
36#14556051C2211yescharacteristic, complemented
37#14556051C2211yescharacteristic, complemented
38#14556048C2211yescharacteristic
39#14556048C2211yescharacteristic
40#14556048C2211yescharacteristic
41#14556048C2211yescharacteristic
42#14556048C2211yescharacteristic, complemented
43#14556048C2211yescharacteristic, complemented
44C22⋊(C71⋊C8)C2211yescharacteristic
45C22⋊(C71⋊C8)C2211yescharacteristic
46#14556039C2211yescharacteristic, complemented
47C4×C71⋊C8C2211yescharacteristic, complemented
48C4⋊Q8×C71C2211yescharacteristic, complemented
49#126×C71—21no—
50#124×C71—21no—
51#124×C71C2211yescharacteristic, complemented
52#123×C71—21no—
53C22⋊Q8×C71—21no—
54C22⋊Q8×C71C2211yescharacteristic
55#121×C71C2211yescharacteristic
56C4×Q8×C71C2211yescharacteristic, complemented
57C4×D4×C71C2211yescharacteristic, complemented
58#117×C71C2211yescharacteristic
59C23×D71—21no—
60#11765865—41no—
61#11765865—41no—
62C2×Dic142C2311yescharacteristic
63C2×D284—21no—
64C2×D284—21no—
65C2×D284—21no—
66C2×D284—21no—
67C2×D284—21no—
68C2×D284—41no—
69C2×D284—41no—
70C2×D284C2311yescharacteristic
71C2×D284C2311yescharacteristic
72C2×D284C2311yescharacteristic
73C2×C22⋊D71—21no—
74C2×C4×D71—21no—
75#11765855—41no—
76#11765853—41no—
77#11765852—41no—
78#11765852—41no—
79#11765852—41no—
80#11765850—41no—
81#11765850—41no—
82#11765850—41no—
83C2×C71⋊C8C2311yescharacteristic
84C2×C71⋊C8C2311yescharacteristic
85C2×C71⋊C8C2311yescharacteristic
86C2×C71⋊C8C2311yescharacteristic
87(C2×D71)⋊C4—21no—
88(C2×D71)⋊C4—21no—
89(C2×D71)⋊C4—21no—
90(C2×D71)⋊C4—21no—
91C2×Q8×C71—21no—
92C2×Q8×C71C2311yescharacteristic
93C2×D4×C71—21no—
94C2×D4×C71C2311yescharacteristic
95C22×C4×C71—21no—
96C22×C4×C71C2311yescharacteristic
97C4⋊C4×C71—21no—
98C4⋊C4×C71—21no—
99C4⋊C4×C71—21no—
100C4⋊C4×C71D411yescharacteristic
101C4⋊C4×C71D411yescharacteristic
102C4⋊C4×C71C2311yescharacteristic
103C4⋊C4×C71C2311yescharacteristic
104C22⋊C4×C71—21no—
105C22⋊C4×C71—21no—
106C22⋊C4×C71—21no—
107C22⋊C4×C71—21no—
108C22⋊C4×C71—21no—
109C22⋊C4×C71D411yescharacteristic
110C22⋊C4×C71D411yescharacteristic
111C42×C71—21no—
112C42×C71C2311yescharacteristic
113C42×C71C2311yescharacteristic
114C23×C71—21no—
115C23×C71C2×D411yescharacteristic
116C22×D71—21no—
117C22×D71—21no—
118C22×D71—21no—
119C22×D71—42no—
120C22×D71—81no—
121C22×D71—81no—
122Q8×C71—21no—
123Q8×C71—21no—
124Q8×C71—81no—
125D4×C71—21no—
126D4×C71—21no—
127D4×C71—81no—
128C2×C4×C71—21no—
129C2×C4×C71—21no—
130C2×C4×C71—21no—
131C2×C4×C71—21no—
132C2×C4×C71—21no—
133C2×C4×C71—81no—
134C2×C4×C71C2×D411yescharacteristic
135C2×C4×C71C2×D411yescharacteristic
136C2×C4×C71C2×D411yescharacteristic
137C2×C4×C71C2×D411yescharacteristic
138C2×C4×C71C2×D411yescharacteristic
139C2×C4×C71C2411yescharacteristic
140C22⋊D71—42no—
141C2×Dic71—21no—
142D284—21no—
143D284—21no—
144D284—21no—
145D284—21no—
146D284—21no—
147D284—21no—
148D284—42no—
149D284—81no—
150D284—81no—
151D284—81no—
152D284—81no—
153C4×D71—42no—
154Dic142—21no—
155Dic142—21no—
156C71⋊C8—21no—
157C71⋊C8—21no—
158C71⋊C8—21no—
159C71⋊C8—21no—
160C22×C71—21no—
161C22×C71—21no—
162C22×C71—21no—
163C22×C71—41no—
164C22×C71—41no—
165C22×C71—41no—
166C22×C71C22×D411yescharacteristic
167C2×D71—41no—
168C2×D71—41no—
169C2×D71—41no—
170C2×D71—41no—
171C2×D71—41no—
172C2×D71—41no—
173C2×D71—41no—
174C2×D71—41no—
175C2×D71—41no—
176C2×D71—84no—
177C4×C71—21no—
178C4×C71—21no—
179C4×C71—21no—
180C4×C71—21no—
181C4×C71—21no—
182C4×C71—41no—
183C4×C71—41no—
184C4×C71—41no—
185C4×C71H(5,𝔽2)11yescharacteristic
186C4×C71H(5,𝔽2)11yescharacteristic
187Dic71—41no—
188C2×C71—41no—
189C2×C71—81no—
190C2×C71#53511yescharacteristic
191C2×C71#57611yescharacteristic
192C2×C71#57711yescharacteristic
193D71—81no—
194D71—81no—
195D71—81no—
196#3224—711nomaximal, Sylow
197C71#322411yescharacteristic, complemented, Sylow
198#542D7111yescharacteristic, complemented
199#535—711no—
200#496—711no—
201#495—711no—
202#489—711no—
203#486—711no—
204#464—711no—
205#463—711no—
206#460—711no—
207#459—711no—
208#452—711no—
209#450—711no—
210#449—711no—
211#447—711no—
212#433—711no—
213C2×#41—711no—
214C22×D4—711no—
215C2×Q16—711no—
216C2×SD16—711no—
217C2×SD16—711no—
218C2×SD16—711no—
219C2×SD16—711no—
220C2×D8—711no—
221C2×D8—711no—
222C2×D8—711no—
223C4⋊Q8C2×D7111yescharacteristic, complemented
224#127—711no—
225#127—711no—
226#126—21no—
227#124—21no—
228#124—711no—
229#124—711no—
230#124C2×D7111yescharacteristic, complemented
231#123—21no—
232C22⋊Q8—21no—
233C22⋊Q8C2×D7111yescharacteristic
234#121—1421no—
235#121—1421no—
236#121C2×D7111yescharacteristic
237#120—1421no—
238#120—1421no—
239C4×Q8C2×D7111yescharacteristic, complemented
240C4×D4C2×D7111yescharacteristic, complemented
241#117C2×D7111yescharacteristic
242C4⋊C8—711no—
243C4⋊C8—711no—
244#103—711no—
245#103—711no—
246#102—711no—
247#102—711no—
248#102—711no—
249#102—711no—
250#102—711no—
251#102—711no—
252C22⋊C8—711no—
253C22⋊C8—711no—
254#97—711no—
255C4×C8—711no—
256C24—1421no—
257#41—2841no—
258#41—2841no—
259C2×Q8—21no—
260C2×Q8—711no—
261C2×Q8C22×D7111yescharacteristic
262C2×D4—21no—
263C2×D4—2841no—
264C2×D4—2841no—
265C2×D4—711no—
266C2×D4—711no—
267C2×D4—711no—
268C2×D4—1421no—
269C2×D4—1421no—
270C2×D4—1421no—
271C2×D4—1421no—
272C2×D4—1421no—
273C2×D4—1421no—
274C2×D4C22×D7111yescharacteristic
275C22×C4—21no—
276C22×C4—1421no—
277C22×C4C22×D7111yescharacteristic
278Q16—2841no—
279SD16—2841no—
280SD16—2841no—
281SD16—2841no—
282SD16—2841no—
283D8—2841no—
284D8—2841no—
285D8—2841no—
286C2×C8—711no—
287C2×C8—711no—
288C2×C8—711no—
289C2×C8—711no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4C22⋊D7111yescharacteristic
294C4⋊C4C22⋊D7111yescharacteristic
295C4⋊C4C22×D7111yescharacteristic
296C4⋊C4C22×D7111yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—1421no—
303C22⋊C4—1421no—
304C22⋊C4—1421no—
305C22⋊C4—1421no—
306C22⋊C4C22⋊D7111yescharacteristic
307C22⋊C4C22⋊D7111yescharacteristic
308C42—21no—
309C42C22×D7111yescharacteristic
310C42C22×D7111yescharacteristic
311C23—21no—
312C23—2842no—
313C23—5681no—
314C23—5681no—
315C23—1421no—
316C23—1421no—
317C23—1421no—
318C23C2×C22⋊D7111yescharacteristic
319Q8—21no—
320Q8—21no—
321Q8—81no—
322Q8—1421no—
323Q8—1421no—
324D4—21no—
325D4—21no—
326D4—81no—
327D4—2842no—
328D4—2842no—
329D4—5681no—
330D4—5681no—
331D4—5681no—
332D4—5681no—
333D4—1421no—
334D4—1421no—
335D4—1421no—
336D4—1421no—
337D4—1421no—
338D4—1421no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—81no—
345C2×C4—2842no—
346C2×C4—1421no—
347C2×C4C2×C22⋊D7111yescharacteristic
348C2×C4C2×C22⋊D7111yescharacteristic
349C2×C4C2×C22⋊D7111yescharacteristic
350C2×C4C2×C22⋊D7111yescharacteristic
351C2×C4C2×C22⋊D7111yescharacteristic
352C2×C4C23×D7111yescharacteristic
353C8—1421no—
354C8—1421no—
355C8—1421no—
356C8—1421no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—2841no—
364C22—2841no—
365C22—2841no—
366C22—2841no—
367C22—2841no—
368C22—2841no—
369C22—2841no—
370C22—2841no—
371C22—2841no—
372C22—5684no—
373C22C22×C22⋊D7111yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—2841no—
383C4#1455617411yescharacteristic
384C4#1455617411yescharacteristic
385C2—41no—
386C2—81no—
387C2—5681no—
388C2—5681no—
389C2—5681no—
390C2#1798656311yescharacteristic
391C2#1798680211yescharacteristic
392C2#1798680711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#542×C71C2
socleC22×C71C22×D4
hypercenter#542D71
hypocenterC71#3224
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ #117 ◁ #542

Complements

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

NH ≅ G/Ntype
1. #17986568C2semidirect
2. #17986389C2semidirect
3. #17986381C2semidirect
4. #17986361C2semidirect
5. #17986347C2semidirect
6. #17986221C2semidirect
7. #17986214C2semidirect
8. #17986199C2semidirect
9. #17986194C2semidirect
10. #17986159C2semidirect
11. #17986149C2semidirect
12. #17986142C2semidirect
13. #17986132C2semidirect
14. #17986060C2semidirect
15. #542×C71C2semidirect
18. C2×#11765855C22semidirect
19. C2×#11765853C22semidirect
22. C2×#11765852C22semidirect
25. C2×#11765850C22semidirect
34. C4⋊(C71⋊C8)C22semidirect
35. C4⋊(C71⋊C8)C22semidirect
36. #14556051C22semidirect
37. #14556051C22semidirect
42. #14556048C22semidirect
43. #14556048C22semidirect
46. #14556039C22semidirect
47. C4×C71⋊C8C22semidirect
48. C4⋊Q8×C71C22semidirect
51. #124×C71C22semidirect
56. C4×Q8×C71C22semidirect
57. C4×D4×C71C22semidirect
197. C71#3224semidirect
198. #542D71semidirect
223. C4⋊Q8C2×D71semidirect
230. #124C2×D71semidirect
239. C4×Q8C2×D71semidirect
240. C4×D4C2×D71semidirect