Finite Group Databasebeta

← #22599178

Subgroups — #22599178

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#17986509C211yesmaximal, characteristic, complemented
3#17986342C211yesmaximal, characteristic, complemented
4#17986339C211yesmaximal, characteristic, complemented
5#17986326C211yesmaximal, characteristic, complemented
6#17986321C211yesmaximal, characteristic, complemented
7#17986255C211yesmaximal, characteristic, complemented
8#17986250C211yesmaximal, characteristic, complemented
9#17986245C211yesmaximal, characteristic, complemented
10#17986164C211yesmaximal, characteristic, complemented
11#17986158C211yesmaximal, characteristic, complemented
12#17986145C211yesmaximal, characteristic, complemented
13#17986143C211yesmaximal, characteristic, complemented
14#17985962C211yesmaximal, characteristic, complemented
15#468×C71C211yesmaximal, characteristic, complemented
16C2×#11765866C2211yescharacteristic
17C2×#11765864C2211yescharacteristic
18C22×C4×D71C2211yescharacteristic
19C2×#11765855C2211yescharacteristic, complemented
20C2×#11765854C2211yescharacteristic, complemented
21C2×#11765850C2211yescharacteristic, complemented
22C2×#11765849C2211yescharacteristic
23#14556113—21no—
24#14556110—21no—
25#14556108C2211yescharacteristic
26#14556107C2211yescharacteristic
27#14556104C2211yescharacteristic
28#14556103C2211yescharacteristic
29#14556102C2211yescharacteristic
30#14556099C2211yescharacteristic
31#14556098—21no—
32#14556096—21no—
33D4×Dic71—21no—
34#14556090—21no—
35(C22⋊D71)⋊C4—21no—
36(C22⋊D71)⋊C4—21no—
37#14556086C2211yescharacteristic
38#14556086C2211yescharacteristic
39#14556062C2211yescharacteristic, complemented
40#14556059C2211yescharacteristic, complemented
41#14556059C2211yescharacteristic, complemented
42#14556053C2211yescharacteristic, complemented
43#14556052C2211yescharacteristic, complemented
44#14556051C2211yescharacteristic, complemented
45#14556050C2211yescharacteristic, complemented
46#14556049C2211yescharacteristic, complemented
47#14556048C2211yescharacteristic, complemented
48(C2×D71)⋊C8C2211yescharacteristic
49(C2×D71)⋊C8C2211yescharacteristic
50(C2×D71)⋊C8C2211yescharacteristic
51(C2×D71)⋊C8C2211yescharacteristic
52C2×SD16×C71C2211yescharacteristic, complemented
53C2×M16×C71C2211yescharacteristic
54C22⋊Q8×C71C2211yescharacteristic
55#121×C71C2211yescharacteristic
56#107×C71C2211yescharacteristic, complemented
57#103×C71C2211yescharacteristic, complemented
58#102×C71C2211yescharacteristic, complemented
59C23×D71D411yescharacteristic
60#11765866—41no—
61#11765866—41no—
62#11765864—41no—
63#11765864—41no—
64C2×Dic142C2311yescharacteristic
65C2×D284—21no—
66C2×D284—21no—
67C2×D284C2311yescharacteristic
68C2×C22⋊D71—21no—
69C2×C22⋊D71—21no—
70C2×C22⋊D71—21no—
71C22×Dic71—21no—
72C22×Dic71D411yescharacteristic
73C2×C4×D71—21no—
74C2×C4×D71—41no—
75C2×C4×D71—41no—
76C2×C4×D71D411yescharacteristic
77C2×C4×D71D411yescharacteristic
78C2×C4×D71C2311yescharacteristic
79C2×C4×D71C2311yescharacteristic
80#11765855—41no—
81#11765854—41no—
82#11765850—41no—
83#11765849—41no—
84C2×C71⋊C8C2311yescharacteristic
85C2×C71⋊C8C2311yescharacteristic
86C4⋊Dic71—21no—
87C4⋊Dic71C2311yescharacteristic
88C4⋊Dic71C2311yescharacteristic
89Dic71⋊C4—21no—
90Dic71⋊C4—21no—
91C22⋊Dic71—21no—
92C22⋊Dic71—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—
98(C2×D71)⋊C4—21no—
99C4×Dic71—21no—
100C4×Dic71—21no—
101C2×Q8×C71C2311yescharacteristic
102C2×D4×C71—21no—
103C2×D4×C71C2311yescharacteristic
104C22×C4×C71C2311yescharacteristic
105SD16×C71—41no—
106M16×C71—41no—
107C2×C8×C71C2311yescharacteristic
108C2×C8×C71C2311yescharacteristic
109C4⋊C4×C71—21no—
110C4⋊C4×C71C2311yescharacteristic
111C4⋊C4×C71C2311yescharacteristic
112C22⋊C4×C71—21no—
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—
124D4×C71—21no—
125D4×C71—21no—
126D4×C71—42no—
127C2×C4×C71—21no—
128C2×C4×C71—21no—
129C2×C4×C71—21no—
130C2×C4×C71—21no—
131C2×C4×C71—21no—
132C2×C4×C71C2×D411yescharacteristic
133C2×C4×C71C2411yescharacteristic
134C22⋊D71—42no—
135C22⋊D71—42no—
136C22⋊D71—42no—
137C2×Dic71—21no—
138C2×Dic71—21no—
139C2×Dic71—21no—
140C2×Dic71—41no—
141C2×Dic71—42no—
142C2×Dic71C2×D411yescharacteristic
143C2×Dic71C2×D411yescharacteristic
144D284—21no—
145D284—21no—
146D284—42no—
147D284—42no—
148C4×D71—21no—
149C4×D71—21no—
150C4×D71—21no—
151C4×D71—21no—
152C4×D71—41no—
153C4×D71—41no—
154C4×D71—42no—
155Dic142—21no—
156Dic142—21no—
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#468D7111yescharacteristic, complemented
209#452—711no—
210#452—711no—
211#449—711no—
212#449—711no—
213#407—711no—
214C2×#41—711no—
215C2×#41—711no—
216C23×C4—711no—
217C2×Q16—711no—
218C2×SD16—711no—
219C2×SD16C2×D7111yescharacteristic, complemented
220C2×D8—711no—
221C2×M16—711no—
222C2×M16C2×D7111yescharacteristic
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#121C2×D7111yescharacteristic
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#107C2×D7111yescharacteristic, complemented
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⋊C8—711no—
257C24—711no—
258#41—2841no—
259#41—2841no—
260#41—2841no—
261#41—2841no—
262C2×Q8—711no—
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—2841no—
273C22×C4—2841no—
274C22×C4—711no—
275C22×C4—711no—
276C22×C4—711no—
277C22×C4—711no—
278C22×C4—711no—
279C22×C4—1421no—
280C22×C4—1421no—
281C22×C4C22×D7111yescharacteristic
282Q16—2841no—
283SD16—41no—
284SD16—2841no—
285D8—2841no—
286M16—41no—
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—1421no—
296C4⋊C4—1421no—
297C4⋊C4—1421no—
298C4⋊C4C22×D7111yescharacteristic
299C4⋊C4C22×D7111yescharacteristic
300C22⋊C4—21no—
301C22⋊C4—21no—
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—1421no—
323Q8—1421no—
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—2841no—
340C2×C4—2841no—
341C2×C4—2842no—
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#1455617411yescharacteristic
387C2—41no—
388C2—81no—
389C2—2841no—
390C2—2841no—
391C2—5681no—
392C2#1798655711yescharacteristic
393C2#1798679511yescharacteristic
394C2#1798679811yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#468×C71C2
socleC22×C71C22×D4
hypercenter#468D71
hypocenterC71#3197
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. #17986512C2semidirect
2. #17986509C2semidirect
3. #17986342C2semidirect
4. #17986339C2semidirect
5. #17986326C2semidirect
6. #17986321C2semidirect
7. #17986255C2semidirect
8. #17986250C2semidirect
9. #17986245C2semidirect
10. #17986164C2semidirect
11. #17986158C2semidirect
12. #17986145C2semidirect
13. #17986143C2semidirect
14. #17985962C2semidirect
15. #468×C71C2semidirect
19. C2×#11765855C22semidirect
20. C2×#11765854C22semidirect
21. C2×#11765850C22semidirect
39. #14556062C22semidirect
40. #14556059C22semidirect
41. #14556059C22semidirect
42. #14556053C22semidirect
43. #14556052C22semidirect
44. #14556051C22semidirect
45. #14556050C22semidirect
46. #14556049C22semidirect
47. #14556048C22semidirect
52. C2×SD16×C71C22semidirect
56. #107×C71C22semidirect
57. #103×C71C22semidirect
58. #102×C71C22semidirect
198. C71#3197semidirect
208. #468D71semidirect
219. C2×SD16C2×D71semidirect
242. #107C2×D71semidirect
248. #103C2×D71semidirect
252. #102C2×D71semidirect