Finite Group Databasebeta

← #22599181

Subgroups — #22599181

Order 9088.

418 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17986512C211yesmaximal, characteristic, complemented
2#17986506C211yesmaximal, characteristic, complemented
3#17986339C211yesmaximal, characteristic, complemented
4#17986337C211yesmaximal, characteristic, complemented
5#17986324C211yesmaximal, characteristic, complemented
6#17986322C211yesmaximal, characteristic, complemented
7#17986254C211yesmaximal, characteristic, complemented
8#17986251C211yesmaximal, characteristic, complemented
9#17986248C211yesmaximal, characteristic, complemented
10#17986242C211yesmaximal, characteristic, complemented
11#17986160C211yesmaximal, characteristic, complemented
12#17986144C211yesmaximal, characteristic, complemented
13#17986143C211yesmaximal, characteristic, complemented
14#17985961C211yesmaximal, characteristic, complemented
15#452×C71C211yesmaximal, characteristic, complemented
16C2×#11765866C2211yescharacteristic
17C22×C4×D71C2211yescharacteristic
18C2×#11765852C2211yescharacteristic, complemented
19C2×#11765850C2211yescharacteristic, complemented
20C2×#11765848C2211yescharacteristic
21C2×#11765848C2211yescharacteristic
22#14556113—21no—
23#14556109—21no—
24#14556104C2211yescharacteristic
25#14556104C2211yescharacteristic
26#14556104C2211yescharacteristic
27#14556103C2211yescharacteristic
28#14556102C2211yescharacteristic
29#14556099C2211yescharacteristic
30#14556099C2211yescharacteristic
31#14556098—21no—
32#14556096—21no—
33#14556090—21no—
34C4×C22⋊D71—21no—
35C4×C22⋊D71—21no—
36(C22⋊D71)⋊C4—21no—
37#14556087C2211yescharacteristic
38#14556086C2211yescharacteristic
39#14556062C2211yescharacteristic, complemented
40#14556061C2211yescharacteristic, complemented
41#14556060C2211yescharacteristic, complemented
42#14556059C2211yescharacteristic, complemented
43#14556052C2211yescharacteristic, complemented
44#14556052C2211yescharacteristic, complemented
45#14556052C2211yescharacteristic, complemented
46#14556050C2211yescharacteristic, complemented
47#14556050C2211yescharacteristic, complemented
48#14556049C2211yescharacteristic, complemented
49C22⋊(C71⋊C8)C2211yescharacteristic
50(C2×D71)⋊C8C2211yescharacteristic
51(C2×D71)⋊C8C2211yescharacteristic
52C2×#41×C71C2211yescharacteristic
53C2×Q16×C71C2211yescharacteristic, complemented
54C2×SD16×C71C2211yescharacteristic, complemented
55C22⋊Q8×C71C2211yescharacteristic
56#103×C71C2211yescharacteristic, complemented
57#102×C71C2211yescharacteristic, complemented
58C22⋊C8×C71C2211yescharacteristic
59C23×D71D411yescharacteristic
60#11765866—41no—
61#11765866—41no—
62C2×D284—21no—
63C2×D284—21no—
64C2×D284C2311yescharacteristic
65C2×C22⋊D71—21no—
66C2×C22⋊D71—21no—
67C2×C22⋊D71—21no—
68C22×Dic71D411yescharacteristic
69C2×C4×D71—21no—
70C2×C4×D71—41no—
71C2×C4×D71—41no—
72C2×C4×D71D411yescharacteristic
73C2×C4×D71D411yescharacteristic
74C2×C4×D71C2311yescharacteristic
75C2×C4×D71C2311yescharacteristic
76#11765852—41no—
77#11765850—41no—
78#11765848—41no—
79#11765848—41no—
80C2×C71⋊C8C2311yescharacteristic
81C2×C71⋊C8C2311yescharacteristic
82C4⋊Dic71C2311yescharacteristic
83C4⋊Dic71C2311yescharacteristic
84C4⋊Dic71C2311yescharacteristic
85Dic71⋊C4—21no—
86Dic71⋊C4—21no—
87Dic71⋊C4—21no—
88C22⋊Dic71—21no—
89C22⋊Dic71—21no—
90C22⋊Dic71—21no—
91(C2×D71)⋊C4—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—
97C4×Dic71—21no—
98C4×Dic71—21no—
99#41×C71—41no—
100#41×C71—41no—
101C2×Q8×C71C2311yescharacteristic
102C2×Q8×C71C2311yescharacteristic
103C2×D4×C71—21no—
104C2×D4×C71C2311yescharacteristic
105C22×C4×C71—21no—
106C22×C4×C71C2311yescharacteristic
107Q16×C71—41no—
108SD16×C71—41no—
109C2×C8×C71C2311yescharacteristic
110C2×C8×C71C2311yescharacteristic
111C4⋊C4×C71—21no—
112C4⋊C4×C71C2311yescharacteristic
113C22⋊C4×C71—21no—
114C23×C71—21no—
115C23×C71C2×D411yescharacteristic
116C22×D71—21no—
117C22×D71—41no—
118C22×D71—41no—
119C22×D71—41no—
120C22×D71C2×D411yescharacteristic
121C22×D71C2×D411yescharacteristic
122Q8×C71—21no—
123Q8×C71—21no—
124Q8×C71—21no—
125Q8×C71—21no—
126D4×C71—21no—
127D4×C71—21no—
128D4×C71—42no—
129C2×C4×C71—21no—
130C2×C4×C71—21no—
131C2×C4×C71—21no—
132C2×C4×C71—21no—
133C2×C4×C71—21no—
134C2×C4×C71—42no—
135C2×C4×C71C2×D411yescharacteristic
136C2×C4×C71C2411yescharacteristic
137C22⋊D71—42no—
138C22⋊D71—42no—
139C22⋊D71—42no—
140C2×Dic71—21no—
141C2×Dic71—21no—
142C2×Dic71—21no—
143C2×Dic71—41no—
144C2×Dic71C2×D411yescharacteristic
145C2×Dic71C2×D411yescharacteristic
146D284—21no—
147D284—21no—
148D284—42no—
149D284—42no—
150C4×D71—21no—
151C4×D71—21no—
152C4×D71—21no—
153C4×D71—21no—
154C4×D71—41no—
155C4×D71—41no—
156C4×D71—42no—
157C8×C71—21no—
158C8×C71—21no—
159C71⋊C8—21no—
160C71⋊C8—21no—
161C22×C71—21no—
162C22×C71—21no—
163C22×C71—21no—
164C22×C71—41no—
165C22×C71—41no—
166C22×C71—41no—
167C22×C71C22×D411yescharacteristic
168C2×D71—21no—
169C2×D71—21no—
170C2×D71—21no—
171C2×D71—21no—
172C2×D71—21no—
173C2×D71—21no—
174C2×D71—44no—
175C2×D71—41no—
176C2×D71—41no—
177C2×D71—41no—
178C4×C71—21no—
179C4×C71—41no—
180C4×C71—41no—
181C4×C71—41no—
182C4×C71H(5,𝔽2)11yescharacteristic
183C4×C71H(5,𝔽2)11yescharacteristic
184Dic71—21no—
185Dic71—21no—
186Dic71—41no—
187Dic71—41no—
188Dic71—41no—
189C2×C71—41no—
190C2×C71—81no—
191C2×C71#53411yescharacteristic
192C2×C71#57511yescharacteristic
193C2×C71#57511yescharacteristic
194D71—41no—
195D71—41no—
196D71—81no—
197#3197—711nomaximal, Sylow
198C71#319711yescharacteristic, complemented, Sylow
199#525—711no—
200#525—711no—
201#485—711no—
202#485—711no—
203#482—711no—
204#482—711no—
205#470—711no—
206#469—711no—
207#468—711no—
208#468—711no—
209#452—711no—
210#452D7111yescharacteristic, complemented
211#449—711no—
212#449—711no—
213#407—711no—
214C2×#41—711no—
215C2×#41C2×D7111yescharacteristic
216C23×C4—711no—
217C2×Q16C2×D7111yescharacteristic, complemented
218C2×SD16—711no—
219C2×SD16C2×D7111yescharacteristic, complemented
220C2×D8—711no—
221C2×M16—711no—
222C2×M16—711no—
223#123—1421no—
224#123—1421no—
225C22⋊Q8—711no—
226C22⋊Q8—711no—
227C22⋊Q8—711no—
228C22⋊Q8C2×D7111yescharacteristic
229#121—711no—
230#121—711no—
231#121—711no—
232#121—711no—
233#120—1421no—
234#120—1421no—
235C4×D4—1421no—
236C4×D4—1421no—
237C4×D4—1421no—
238C4×D4—1421no—
239#117—711no—
240#117—711no—
241#107—711no—
242#107—711no—
243#106—711no—
244#106—711no—
245#103—711no—
246#103—711no—
247#103—711no—
248#103C2×D7111yescharacteristic, complemented
249#102—711no—
250#102—711no—
251#102—711no—
252#102C2×D7111yescharacteristic, complemented
253C22⋊C8—711no—
254C22⋊C8—711no—
255C22⋊C8—711no—
256C22⋊C8C2×D7111yescharacteristic
257C24—711no—
258#41—41no—
259#41—41no—
260#41—2841no—
261#41—2841no—
262C2×Q8C22×D7111yescharacteristic
263C2×Q8C22×D7111yescharacteristic
264C2×D4—21no—
265C2×D4—711no—
266C2×D4—1421no—
267C2×D4—1421no—
268C2×D4—1421no—
269C2×D4—1421no—
270C2×D4—1421no—
271C2×D4C22×D7111yescharacteristic
272C22×C4—21no—
273C22×C4—2841no—
274C22×C4—2841no—
275C22×C4—711no—
276C22×C4—711no—
277C22×C4—711no—
278C22×C4—711no—
279C22×C4—711no—
280C22×C4—1421no—
281C22×C4C22×D7111yescharacteristic
282Q16—41no—
283SD16—41no—
284SD16—2841no—
285D8—2841no—
286M16—2841no—
287M16—2841no—
288C2×C8—711no—
289C2×C8—711no—
290C2×C8C22×D7111yescharacteristic
291C2×C8C22×D7111yescharacteristic
292C4⋊C4—21no—
293C4⋊C4—711no—
294C4⋊C4—711no—
295C4⋊C4—711no—
296C4⋊C4—1421no—
297C4⋊C4—1421no—
298C4⋊C4—1421no—
299C4⋊C4C22×D7111yescharacteristic
300C22⋊C4—21no—
301C22⋊C4—1421no—
302C22⋊C4—1421no—
303C22⋊C4—1421no—
304C22⋊C4—1421no—
305C22⋊C4—1421no—
306C22⋊C4—1421no—
307C22⋊C4—1421no—
308C22⋊C4—1421no—
309C22⋊C4—1421no—
310C42—1421no—
311C42—1421no—
312C23—21no—
313C23—2841no—
314C23—2841no—
315C23—2841no—
316C23—711no—
317C23—711no—
318C23—1421no—
319C23D4×D7111yescharacteristic
320Q8—21no—
321Q8—21no—
322Q8—21no—
323Q8—21no—
324D4—21no—
325D4—21no—
326D4—42no—
327D4—2842no—
328D4—2842no—
329D4—2842no—
330D4—2842no—
331D4—2842no—
332D4—1421no—
333D4—1421no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—42no—
340C2×C4—2841no—
341C2×C4—2841no—
342C2×C4—2842no—
343C2×C4—2841no—
344C2×C4—711no—
345C2×C4—711no—
346C2×C4—1421no—
347C2×C4—1421no—
348C2×C4—1421no—
349C2×C4—1421no—
350C2×C4—1421no—
351C2×C4—1421no—
352C2×C4—1421no—
353C2×C4D4×D7111yescharacteristic
354C2×C4C23×D7111yescharacteristic
355C8—21no—
356C8—21no—
357C8—1421no—
358C8—1421no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—2844no—
366C22—2841no—
367C22—2841no—
368C22—2841no—
369C22—1421no—
370C22—1421no—
371C22—1421no—
372C22—1421no—
373C22—1421no—
374C22—1421no—
375C22C2×D4×D7111yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—2841no—
381C4—2841no—
382C4—2841no—
383C4—1421no—
384C4—1421no—
385C4#1455617311yescharacteristic
386C4#1455617311yescharacteristic
387C2—41no—
388C2—81no—
389C2—2841no—
390C2—2841no—
391C2—5681no—
392C2#1798655911yescharacteristic
393C2#1798679611yescharacteristic
394C2#1798679811yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#452×C71C2
socleC22×C71C22×D4
hypercenter#452D71
hypocenterC71#3197
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. #17986512C2semidirect
2. #17986506C2semidirect
3. #17986339C2semidirect
4. #17986337C2semidirect
5. #17986324C2semidirect
6. #17986322C2semidirect
7. #17986254C2semidirect
8. #17986251C2semidirect
9. #17986248C2semidirect
10. #17986242C2semidirect
11. #17986160C2semidirect
12. #17986144C2semidirect
13. #17986143C2semidirect
14. #17985961C2semidirect
15. #452×C71C2semidirect
18. C2×#11765852C22semidirect
19. C2×#11765850C22semidirect
39. #14556062C22semidirect
40. #14556061C22semidirect
41. #14556060C22semidirect
42. #14556059C22semidirect
43. #14556052C22semidirect
44. #14556052C22semidirect
45. #14556052C22semidirect
46. #14556050C22semidirect
47. #14556050C22semidirect
48. #14556049C22semidirect
53. C2×Q16×C71C22semidirect
54. C2×SD16×C71C22semidirect
56. #103×C71C22semidirect
57. #102×C71C22semidirect
198. C71#3197semidirect
210. #452D71semidirect
217. C2×Q16C2×D71semidirect
219. C2×SD16C2×D71semidirect
248. #103C2×D71semidirect
252. #102C2×D71semidirect