Finite Group Databasebeta

← #22599171

Subgroups — #22599171

Order 9088.

414 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17986512C211yesmaximal, characteristic
2C22⋊Q8×D71C211yesmaximal, characteristic, complemented
3#17986343C211yesmaximal, characteristic, complemented
4#17986339C211yesmaximal, characteristic, complemented
5#17986333C211yesmaximal, characteristic, complemented
6#17986319C211yesmaximal, characteristic, complemented
7#17986255C211yesmaximal, characteristic, complemented
8#17986247C211yesmaximal, characteristic, complemented
9#17986245C211yesmaximal, characteristic, complemented
10#17986164C211yesmaximal, characteristic, complemented
11#17986160C211yesmaximal, characteristic, complemented
12#17986153C211yesmaximal, characteristic, complemented
13#17986139C211yesmaximal, characteristic, complemented
14#17985962C211yesmaximal, characteristic, complemented
15#470×C71C211yesmaximal, characteristic, complemented
16C2×#11765866C2211yescharacteristic
17C2×Q8×D71C2211yescharacteristic
18C22×C4×D71C2211yescharacteristic
19C2×#11765855C2211yescharacteristic, complemented
20C2×#11765854C2211yescharacteristic, complemented
21C2×#11765852C2211yescharacteristic, complemented
22C2×#11765849C2211yescharacteristic
23#14556113—21no—
24#14556108C2211yescharacteristic
25#14556107C2211yescharacteristic
26#14556106—21no—
27#14556105—21no—
28#14556104C2211yescharacteristic
29#14556104C2211yescharacteristic
30#14556103C2211yescharacteristic
31#14556102C2211yescharacteristic
32#14556098—21no—
33#14556090—21no—
34(C22⋊D71)⋊C4—21no—
35#14556086C2211yescharacteristic
36C4⋊C4×D71—21no—
37C4⋊C4×D71C2211yescharacteristic
38C22⋊C4×D71—21no—
39#14556062C2211yescharacteristic, complemented
40#14556061C2211yescharacteristic, complemented
41#14556059C2211yescharacteristic, complemented
42#14556053C2211yescharacteristic, complemented
43#14556052C2211yescharacteristic, complemented
44#14556052C2211yescharacteristic, complemented
45#14556051C2211yescharacteristic, complemented
46#14556050C2211yescharacteristic, complemented
47#14556048C2211yescharacteristic, complemented
48(C2×D71)⋊C8C2211yescharacteristic
49(C2×D71)⋊C8C2211yescharacteristic
50(C2×D71)⋊C8C2211yescharacteristic
51(C2×D71)⋊C8C2211yescharacteristic
52C2×Q16×C71C2211yescharacteristic, complemented
53C2×M16×C71C2211yescharacteristic
54C22⋊Q8×C71C2211yescharacteristic
55C22⋊Q8×C71C2211yescharacteristic
56#106×C71C2211yescharacteristic, complemented
57#103×C71C2211yescharacteristic, complemented
58#103×C71C2211yescharacteristic, complemented
59C23×D71D411yescharacteristic
60#11765866—41no—
61#11765866—41no—
62C2×Dic142—21no—
63C2×Dic142C2311yescharacteristic
64Q8×D71—41no—
65Q8×D71—41no—
66C2×D284—21no—
67C2×D284—21no—
68C2×D284C2311yescharacteristic
69C2×C22⋊D71—21no—
70C22×Dic71D411yescharacteristic
71C2×C4×D71—21no—
72C2×C4×D71—21no—
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#11765852—41no—
83#11765849—41no—
84C2×C71⋊C8C2311yescharacteristic
85C2×C71⋊C8C2311yescharacteristic
86C4⋊Dic71—21no—
87C4⋊Dic71C2311yescharacteristic
88C4⋊Dic71C2311yescharacteristic
89Dic71⋊C4—21no—
90Dic71⋊C4—21no—
91Dic71⋊C4—21no—
92Dic71⋊C4—21no—
93C22⋊Dic71—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—
99(C2×D71)⋊C4—21no—
100C4×Dic71—21no—
101C2×Q8×C71C2311yescharacteristic
102C2×Q8×C71C2311yescharacteristic
103C22×C4×C71C2311yescharacteristic
104Q16×C71—41no—
105M16×C71—41no—
106C2×C8×C71C2311yescharacteristic
107C2×C8×C71C2311yescharacteristic
108C4⋊C4×C71—21no—
109C4⋊C4×C71—21no—
110C4⋊C4×C71C2311yescharacteristic
111C4⋊C4×C71C2311yescharacteristic
112C22⋊C4×C71—21no—
113C22⋊C4×C71—21no—
114C23×C71C2×D411yescharacteristic
115C22×D71—22no—
116C22×D71—21no—
117C22×D71—41no—
118C22×D71—41no—
119C22×D71C2×D411yescharacteristic
120C22×D71C2×D411yescharacteristic
121Q8×C71—21no—
122Q8×C71—21no—
123Q8×C71—21no—
124Q8×C71—21no—
125C2×C4×C71—21no—
126C2×C4×C71—21no—
127C2×C4×C71—21no—
128C2×C4×C71—21no—
129C2×C4×C71—21no—
130C2×C4×C71—21no—
131C2×C4×C71C2×D411yescharacteristic
132C2×C4×C71C2411yescharacteristic
133C22⋊D71—42no—
134C2×Dic71—21no—
135C2×Dic71—21no—
136C2×Dic71—21no—
137C2×Dic71—41no—
138C2×Dic71C2×D411yescharacteristic
139C2×Dic71C2×D411yescharacteristic
140D284—21no—
141D284—21no—
142D284—42no—
143D284—42no—
144C4×D71—21no—
145C4×D71—21no—
146C4×D71—21no—
147C4×D71—21no—
148C4×D71—41no—
149C4×D71—41no—
150C4×D71—42no—
151C4×D71—42no—
152C4×D71—42no—
153Dic142—21no—
154Dic142—21no—
155Dic142—42no—
156C8×C71—21no—
157C8×C71—21no—
158C71⋊C8—21no—
159C71⋊C8—21no—
160C22×C71—21no—
161C22×C71—21no—
162C22×C71—21no—
163C22×C71C22×D411yescharacteristic
164C2×D71—21no—
165C2×D71—21no—
166C2×D71—21no—
167C2×D71—21no—
168C2×D71—21no—
169C2×D71—44no—
170C2×D71—41no—
171C2×D71—41no—
172C2×D71—41no—
173C2×D71#13712yes—
174C4×C71—21no—
175C4×C71—41no—
176C4×C71—41no—
177C4×C71—41no—
178C4×C71—41no—
179C4×C71H(5,𝔽2)11yescharacteristic
180C4×C71H(5,𝔽2)11yescharacteristic
181Dic71—21no—
182Dic71—21no—
183Dic71—41no—
184Dic71—41no—
185Dic71—41no—
186C2×C71—41no—
187C2×C71#53411yescharacteristic
188C2×C71C2×#13711yescharacteristic
189C2×C71#57511yescharacteristic
190D71—22no—
191D71—41no—
192D71—81no—
193#3196—711nomaximal, Sylow
194C71#319611yescharacteristic, complemented, Sylow
195#525—711no—
196C2×C22⋊Q8—711no—
197#485—711no—
198#485—711no—
199#484—711no—
200#481—711no—
201#470—711no—
202#470D7111yescharacteristic, complemented
203#468—711no—
204#468—711no—
205#452—711no—
206#452—711no—
207#451—711no—
208#448—711no—
209#407—711no—
210C2×#41—711no—
211C22×Q8—711no—
212C23×C4—711no—
213C2×Q16—711no—
214C2×Q16C2×D7111yescharacteristic, complemented
215C2×SD16—711no—
216C2×SD16—711no—
217C2×M16—711no—
218C2×M16C2×D7111yescharacteristic
219#123—1421no—
220C22⋊Q8—711no—
221C22⋊Q8—711no—
222C22⋊Q8—711no—
223C22⋊Q8—711no—
224C22⋊Q8—1421no—
225C22⋊Q8—1421no—
226C22⋊Q8C2×D7111yescharacteristic
227C22⋊Q8C2×D7111yescharacteristic
228#121—711no—
229#121—711no—
230#120—1421no—
231C4×D4—1421no—
232C4×D4—1421no—
233#117—711no—
234C2×C4⋊C4—711no—
235C2×C4⋊C4—1421no—
236C2×C22⋊C4—1421no—
237#107—711no—
238#107—711no—
239#106—711no—
240#106C2×D7111yescharacteristic, complemented
241#103—711no—
242#103—711no—
243#103—711no—
244#103—711no—
245#103C2×D7111yescharacteristic, complemented
246#103C2×D7111yescharacteristic, complemented
247#102—711no—
248#102—711no—
249C22⋊C8—711no—
250C22⋊C8—711no—
251C22⋊C8—711no—
252C22⋊C8—711no—
253C24—711no—
254#41—2841no—
255#41—2841no—
256C2×Q8—2841no—
257C2×Q8—2841no—
258C2×Q8—711no—
259C2×Q8—1421no—
260C2×Q8C22×D7111yescharacteristic
261C2×Q8C22×D7111yescharacteristic
262C2×D4—711no—
263C2×D4—1421no—
264C2×D4—1421no—
265C2×D4—1421no—
266C22×C4—2841no—
267C22×C4—2841no—
268C22×C4—711no—
269C22×C4—711no—
270C22×C4—711no—
271C22×C4—711no—
272C22×C4—711no—
273C22×C4—1421no—
274C22×C4—1421no—
275C22×C4—1421no—
276C22×C4C22×D7111yescharacteristic
277Q16—41no—
278Q16—2841no—
279SD16—2841no—
280SD16—2841no—
281M16—41no—
282M16—2841no—
283C2×C8—711no—
284C2×C8—711no—
285C2×C8C22×D7111yescharacteristic
286C2×C8C22×D7111yescharacteristic
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—711no—
290C4⋊C4—711no—
291C4⋊C4—1421no—
292C4⋊C4—1421no—
293C4⋊C4—1421no—
294C4⋊C4—1421no—
295C4⋊C4—1421no—
296C4⋊C4C22×D7111yescharacteristic
297C4⋊C4C22×D7111yescharacteristic
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—1421no—
301C22⋊C4—1421no—
302C22⋊C4—1421no—
303C22⋊C4—1421no—
304C22⋊C4—1421no—
305C22⋊C4—1421no—
306C22⋊C4—1421no—
307C42—1421no—
308C23—2841no—
309C23—2841no—
310C23—711no—
311C23—711no—
312C23—1422no—
313C23—1421no—
314C23D4×D7111yescharacteristic
315Q8—21no—
316Q8—21no—
317Q8—21no—
318Q8—21no—
319Q8—2842no—
320Q8—1421no—
321Q8—1421no—
322D4—2842no—
323D4—2842no—
324D4—2842no—
325D4—1421no—
326D4—1421no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—2841no—
334C2×C4—2841no—
335C2×C4—2842no—
336C2×C4—2842no—
337C2×C4—2842no—
338C2×C4—2841no—
339C2×C4—711no—
340C2×C4—711no—
341C2×C4—1421no—
342C2×C4—1421no—
343C2×C4—1421no—
344C2×C4—1421no—
345C2×C4—1421no—
346C2×C4—1421no—
347C2×C4—1421no—
348C2×C4D4×D7111yescharacteristic
349C2×C4C23×D7111yescharacteristic
350C8—21no—
351C8—21no—
352C8—1421no—
353C8—1421no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—2844no—
358C22—2841no—
359C22—2841no—
360C22—2841no—
361C22—712no—
362C22—1421no—
363C22—1421no—
364C22—1421no—
365C22—1421no—
366C22—1421no—
367C22C2×D4×D7111yescharacteristic
368C4—21no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—2841no—
374C4—2841no—
375C4—2841no—
376C4—1421no—
377C4—1421no—
378C4#1455617311yescharacteristic
379C4#1455617411yescharacteristic
380C2—41no—
381C2—2841no—
382C2—5681no—
383C2—1422no—
384C2#1798655711yescharacteristic
385C2#137×D7111yescharacteristic
386C2#1798679811yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#470×C71C2
socleC22×C71C22×D4
hypercenter#470D71
hypocenterC71#3196
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
2. C22⋊Q8×D71C2semidirect
3. #17986343C2semidirect
4. #17986339C2semidirect
5. #17986333C2semidirect
6. #17986319C2semidirect
7. #17986255C2semidirect
8. #17986247C2semidirect
9. #17986245C2semidirect
10. #17986164C2semidirect
11. #17986160C2semidirect
12. #17986153C2semidirect
13. #17986139C2semidirect
14. #17985962C2semidirect
15. #470×C71C2semidirect
19. C2×#11765855C22semidirect
20. C2×#11765854C22semidirect
21. C2×#11765852C22semidirect
39. #14556062C22semidirect
40. #14556061C22semidirect
41. #14556059C22semidirect
42. #14556053C22semidirect
43. #14556052C22semidirect
44. #14556052C22semidirect
45. #14556051C22semidirect
46. #14556050C22semidirect
47. #14556048C22semidirect
52. C2×Q16×C71C22semidirect
56. #106×C71C22semidirect
57. #103×C71C22semidirect
58. #103×C71C22semidirect
194. C71#3196semidirect
202. #470D71semidirect
214. C2×Q16C2×D71semidirect
240. #106C2×D71semidirect
245. #103C2×D71semidirect
246. #103C2×D71semidirect