Finite Group Databasebeta

← #22599119

Subgroups — #22599119

Order 9088.

420 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#17986506C211yesmaximal, characteristic
2C22⋊Q8×D71C211yesmaximal, characteristic, complemented
3#17986341C211yesmaximal, characteristic, complemented
4#17986337C211yesmaximal, characteristic, complemented
5#17986328C211yesmaximal, characteristic, complemented
6#17986311C211yesmaximal, characteristic, complemented
7#17986239C211yesmaximal, characteristic, complemented
8#17986237C211yesmaximal, characteristic, complemented
9#17986227C211yesmaximal, characteristic, complemented
10#17986225C211yesmaximal, characteristic, complemented
11#17986161C211yesmaximal, characteristic, complemented
12#17986153C211yesmaximal, characteristic, complemented
13#17986136C211yesmaximal, characteristic, complemented
14C22⋊C8×D71C211yesmaximal, characteristic, complemented
15#452×C71C211yesmaximal, characteristic, complemented
16C2×Q8×D71C2211yescharacteristic
17C22×C4×D71C2211yescharacteristic
18C2×#11765855C2211yescharacteristic, complemented
19C2×#11765853C2211yescharacteristic, complemented
20C2×C8×D71C2211yescharacteristic
21C2×C8×D71C2211yescharacteristic
22#14556109—21no—
23#14556108C2211yescharacteristic
24#14556107C2211yescharacteristic
25#14556106—21no—
26#14556105—21no—
27#14556104C2211yescharacteristic
28#14556104C2211yescharacteristic
29#14556104C2211yescharacteristic
30#14556099C2211yescharacteristic
31#14556099C2211yescharacteristic
32#14556096—21no—
33C4×C22⋊D71—21no—
34C4×C22⋊D71—21no—
35#14556087C2211yescharacteristic
36C4⋊C4×D71—21no—
37C4⋊C4×D71C2211yescharacteristic
38C22⋊C4×D71—21no—
39#14556062C2211yescharacteristic, complemented
40#14556061C2211yescharacteristic, complemented
41#14556060C2211yescharacteristic, complemented
42#14556059C2211yescharacteristic, complemented
43#14556053C2211yescharacteristic, complemented
44#14556053C2211yescharacteristic, complemented
45#14556052C2211yescharacteristic, complemented
46#14556052C2211yescharacteristic, complemented
47#14556052C2211yescharacteristic, 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
60C2×Dic142—21no—
61C2×Dic142C2311yescharacteristic
62Q8×D71—41no—
63Q8×D71—41no—
64C2×C22⋊D71—21no—
65C2×C22⋊D71—21no—
66C22×Dic71D411yescharacteristic
67C2×C4×D71—21no—
68C2×C4×D71—21no—
69C2×C4×D71—41no—
70C2×C4×D71—41no—
71C2×C4×D71D411yescharacteristic
72C2×C4×D71D411yescharacteristic
73C2×C4×D71C2311yescharacteristic
74C2×C4×D71C2311yescharacteristic
75#11765855—41no—
76#11765853—41no—
77C2×C71⋊C8C2311yescharacteristic
78C2×C71⋊C8C2311yescharacteristic
79C8×D71—41no—
80C8×D71—41no—
81C4⋊Dic71—21no—
82C4⋊Dic71C2311yescharacteristic
83C4⋊Dic71C2311yescharacteristic
84C4⋊Dic71C2311yescharacteristic
85Dic71⋊C4—21no—
86Dic71⋊C4—21no—
87Dic71⋊C4—21no—
88Dic71⋊C4—21no—
89Dic71⋊C4—21no—
90C22⋊Dic71—21no—
91C22⋊Dic71—21no—
92C22⋊Dic71—21no—
93C22⋊Dic71—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×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—22no—
117C22×D71—22no—
118C22×D71—41no—
119C22×D71C2×D411yescharacteristic
120C22×D71C2×D411yescharacteristic
121Q8×C71—21no—
122Q8×C71—21no—
123Q8×C71—21no—
124Q8×C71—21no—
125D4×C71—21no—
126D4×C71—21no—
127D4×C71—42no—
128C2×C4×C71—21no—
129C2×C4×C71—21no—
130C2×C4×C71—21no—
131C2×C4×C71—21no—
132C2×C4×C71—21no—
133C2×C4×C71—42no—
134C2×C4×C71C2×D411yescharacteristic
135C2×C4×C71C2411yescharacteristic
136C22⋊D71—42no—
137C22⋊D71—42no—
138C2×Dic71—21no—
139C2×Dic71—21no—
140C2×Dic71—21no—
141C2×Dic71—21no—
142C2×Dic71—41no—
143C2×Dic71C2×D411yescharacteristic
144C2×Dic71C2×D411yescharacteristic
145C4×D71—21no—
146C4×D71—21no—
147C4×D71—21no—
148C4×D71—21no—
149C4×D71—41no—
150C4×D71—41no—
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×C71—41no—
164C22×C71—41no—
165C22×C71—41no—
166C22×C71C22×D411yescharacteristic
167C2×D71—21no—
168C2×D71—21no—
169C2×D71—21no—
170C2×D71—21no—
171C2×D71—44no—
172C2×D71#13512yes—
173C2×D71#13712yes—
174C4×C71—21no—
175C4×C71—41no—
176C4×C71—41no—
177C4×C71—41no—
178C4×C71H(5,𝔽2)11yescharacteristic
179C4×C71H(5,𝔽2)11yescharacteristic
180Dic71—21no—
181Dic71—21no—
182Dic71—41no—
183Dic71—41no—
184Dic71—41no—
185Dic71—41no—
186C2×C71—41no—
187C2×C71—81no—
188C2×C71#53411yescharacteristic
189C2×C71C2×#13511yescharacteristic
190C2×C71C2×#13711yescharacteristic
191D71—22no—
192D71—22no—
193#3190—711nomaximal, Sylow
194C71#319011yescharacteristic, complemented, Sylow
195#525—711no—
196C2×C22⋊Q8—711no—
197#485—711no—
198#485—711no—
199#483—711no—
200#480—711no—
201#467—711no—
202#467—711no—
203#465—711no—
204#465—711no—
205#452—711no—
206#452D7111yescharacteristic, complemented
207#451—711no—
208#448—711no—
209C2×C22⋊C8—711no—
210C2×#41C2×D7111yescharacteristic
211C22×Q8—711no—
212C23×C4—711no—
213C2×Q16—711no—
214C2×Q16C2×D7111yescharacteristic, complemented
215C2×SD16—711no—
216C2×SD16C2×D7111yescharacteristic, complemented
217C22×C8—711no—
218C22×C8—711no—
219#123—1421no—
220C22⋊Q8—711no—
221C22⋊Q8—711no—
222C22⋊Q8—711no—
223C22⋊Q8—711no—
224C22⋊Q8—711no—
225C22⋊Q8—1421no—
226C22⋊Q8—1421no—
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#106—711no—
241#103—711no—
242#103—711no—
243#103—711no—
244#103—711no—
245#103—711no—
246#103C2×D7111yescharacteristic, complemented
247#102—711no—
248#102C2×D7111yescharacteristic, complemented
249C22⋊C8—711no—
250C22⋊C8—711no—
251C22⋊C8—711no—
252C22⋊C8C2×D7111yescharacteristic
253C24—711no—
254#41—41no—
255#41—41no—
256C2×Q8—2841no—
257C2×Q8—2841no—
258C2×Q8—711no—
259C2×Q8—1421no—
260C2×Q8C22×D7111yescharacteristic
261C2×Q8C22×D7111yescharacteristic
262C2×D4—21no—
263C2×D4—1421no—
264C2×D4—1421no—
265C2×D4C22×D7111yescharacteristic
266C22×C4—21no—
267C22×C4—2841no—
268C22×C4—2841no—
269C22×C4—711no—
270C22×C4—711no—
271C22×C4—711no—
272C22×C4—711no—
273C22×C4—711no—
274C22×C4—1421no—
275C22×C4—1421no—
276C22×C4C22×D7111yescharacteristic
277Q16—41no—
278Q16—2841no—
279SD16—41no—
280SD16—2841no—
281C2×C8—2841no—
282C2×C8—2841no—
283C2×C8—711no—
284C2×C8—711no—
285C2×C8C22×D7111yescharacteristic
286C2×C8C22×D7111yescharacteristic
287C4⋊C4—21no—
288C4⋊C4—711no—
289C4⋊C4—711no—
290C4⋊C4—711no—
291C4⋊C4—1421no—
292C4⋊C4—1421no—
293C4⋊C4—1421no—
294C4⋊C4—1421no—
295C4⋊C4—1421no—
296C4⋊C4—1421no—
297C4⋊C4C22×D7111yescharacteristic
298C22⋊C4—21no—
299C22⋊C4—1421no—
300C22⋊C4—1421no—
301C22⋊C4—1421no—
302C22⋊C4—1421no—
303C22⋊C4—1421no—
304C22⋊C4—1421no—
305C22⋊C4—1421no—
306C22⋊C4—1421no—
307C42—1421no—
308C23—21no—
309C23—2841no—
310C23—711no—
311C23—711no—
312C23—1422no—
313C23—1422no—
314C23D4×D7111yescharacteristic
315Q8—21no—
316Q8—21no—
317Q8—21no—
318Q8—21no—
319Q8—2842no—
320Q8—1421no—
321Q8—1421no—
322D4—21no—
323D4—21no—
324D4—42no—
325D4—2842no—
326D4—2842no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—42no—
333C2×C4—2841no—
334C2×C4—2841no—
335C2×C4—2842no—
336C2×C4—2842no—
337C2×C4—2841no—
338C2×C4—711no—
339C2×C4—711no—
340C2×C4—1421no—
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—41no—
358C22—41no—
359C22—41no—
360C22—2844no—
361C22—712no—
362C22—712no—
363C22—1421no—
364C22—1421no—
365C22—1421no—
366C22—1421no—
367C22C2×D4×D7111yescharacteristic
368C4—21no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—2841no—
373C4—2841no—
374C4—2841no—
375C4—2841no—
376C4—1421no—
377C4—1421no—
378C4#1455617311yescharacteristic
379C4#1455617311yescharacteristic
380C2—41no—
381C2—81no—
382C2—1422no—
383C2—1422no—
384C2#1798655911yescharacteristic
385C2#135×D7111yescharacteristic
386C2#137×D7111yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#452×C71C2
socleC22×C71C22×D4
hypercenter#452D71
hypocenterC71#3190
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
2. C22⋊Q8×D71C2semidirect
3. #17986341C2semidirect
4. #17986337C2semidirect
5. #17986328C2semidirect
6. #17986311C2semidirect
7. #17986239C2semidirect
8. #17986237C2semidirect
9. #17986227C2semidirect
10. #17986225C2semidirect
11. #17986161C2semidirect
12. #17986153C2semidirect
13. #17986136C2semidirect
14. C22⋊C8×D71C2semidirect
15. #452×C71C2semidirect
18. C2×#11765855C22semidirect
19. C2×#11765853C22semidirect
39. #14556062C22semidirect
40. #14556061C22semidirect
41. #14556060C22semidirect
42. #14556059C22semidirect
43. #14556053C22semidirect
44. #14556053C22semidirect
45. #14556052C22semidirect
46. #14556052C22semidirect
47. #14556052C22semidirect
48. #14556049C22semidirect
53. C2×Q16×C71C22semidirect
54. C2×SD16×C71C22semidirect
56. #103×C71C22semidirect
57. #102×C71C22semidirect
194. C71#3190semidirect
206. #452D71semidirect
214. C2×Q16C2×D71semidirect
216. C2×SD16C2×D71semidirect
246. #103C2×D71semidirect
248. #102C2×D71semidirect