Finite Group Databasebeta

← #22599843

Subgroups — #22599843

Order 9088.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14556160C211yesmaximal, characteristic, complemented
2C22×#11765852C211yesmaximal, characteristic, complemented
3Q8×C22⋊D71C211yesmaximal, characteristic, complemented
4#17986640C211yesmaximal, characteristic
5#17986392C211yesmaximal, characteristic, complemented
6#17986248C211yesmaximal, characteristic, complemented
7#17986238C211yesmaximal, characteristic, complemented
8#17986223C211yesmaximal, characteristic, complemented
9#17986203C211yesmaximal, characteristic, complemented
10#17986160C211yesmaximal, characteristic, complemented
11#17986152C211yesmaximal, characteristic, complemented
12#17986139C211yesmaximal, characteristic, complemented
13#17986102C211yesmaximal, characteristic, complemented
14#17986067C211yesmaximal, characteristic, complemented
15#451×C71C211yesmaximal, characteristic, complemented
16C2×#11765866C2211yescharacteristic
17C2×Q8×D71C2211yescharacteristic, complemented
18C22×D284C2211yescharacteristic
19#14556160—24no—
20C2×#11765855C2211yescharacteristic, complemented
21C2×#11765854C2211yescharacteristic
22C2×#11765854C2211yescharacteristic
23C2×#11765852—24no—
24C2×#11765852C2211yescharacteristic
25C2×#11765852C2211yescharacteristic
26C2×#11765852C2211yescharacteristic
27C2×#11765852C2211yescharacteristic
28C2×#11765848C2211yescharacteristic, complemented
29C22×C71⋊C8C2211yescharacteristic
30#14556126—21no—
31#14556126C2211yescharacteristic, complemented
32#14556117C2211yescharacteristic
33#14556113—21no—
34#14556107—21no—
35#14556107C2211yescharacteristic
36#14556104—21no—
37#14556104C2211yescharacteristic, complemented
38#14556103—21no—
39#14556103C2211yescharacteristic
40#14556098—21no—
41Q8×Dic71C2211yescharacteristic, complemented
42#14556090C2211yescharacteristic
43C4×C22⋊D71—21no—
44C4×C22⋊D71C2211yescharacteristic
45C22⋊C4×D71—21no—
46#14556061C2211yescharacteristic, complemented
47Dic71⋊C8C2211yescharacteristic, complemented
48#14556052C2211yescharacteristic, complemented
49#14556051C2211yescharacteristic, complemented
50#14556050C2211yescharacteristic
51#14556050C2211yescharacteristic
52(C2×D71)⋊C8C2211yescharacteristic, complemented
53#14556038C2211yescharacteristic, complemented
54C22×Q8×C71C2211yescharacteristic
55C2×Q16×C71C2211yescharacteristic, complemented
56C2×Q16×C71C2211yescharacteristic, complemented
57C22⋊Q8×C71C2211yescharacteristic
58#103×C71C2211yescharacteristic, complemented
59#103×C71C2211yescharacteristic, complemented
60C22⋊C8×C71C2211yescharacteristic
61C23×D71—21no—
62#11765866—22no—
63#11765866—41no—
64C2×Dic142—21no—
65C2×Dic142C2311yescharacteristic
66Q8×D71—22no—
67Q8×D71—41no—
68C2×D284—22no—
69C2×D284—21no—
70C2×D284—21no—
71C2×D284—21no—
72C2×D284—41no—
73C2×D284C2311yescharacteristic
74C2×D284C2311yescharacteristic
75C2×C22⋊D71D411yescharacteristic
76C2×C4×D71—21no—
77C2×C4×D71—21no—
78C2×C4×D71—21no—
79C2×C4×D71C2311yescharacteristic
80#11765855—22no—
81#11765854—22no—
82#11765854—41no—
83#11765852—22no—
84#11765852—22no—
85#11765852—22no—
86#11765852D414yescomplemented
87#11765848—22no—
88C2×C71⋊C8—22no—
89C2×C71⋊C8C2311yescharacteristic
90C2×C71⋊C8C2311yescharacteristic
91C4⋊Dic71—21no—
92C4⋊Dic71C2311yescharacteristic
93Dic71⋊C4—21no—
94Dic71⋊C4—21no—
95Dic71⋊C4—21no—
96Dic71⋊C4—21no—
97Dic71⋊C4D411yescharacteristic
98C22⋊Dic71—21no—
99C22⋊Dic71D411yescharacteristic
100(C2×D71)⋊C4—21no—
101(C2×D71)⋊C4—21no—
102(C2×D71)⋊C4—21no—
103(C2×D71)⋊C4—21no—
104(C2×D71)⋊C4—21no—
105(C2×D71)⋊C4D411yescharacteristic
106C4×Dic71—21no—
107C4×Dic71C2311yescharacteristic
108C2×Q8×C71—21no—
109C2×Q8×C71—22no—
110C2×Q8×C71—41no—
111C2×Q8×C71C2311yescharacteristic
112C2×Q8×C71C2311yescharacteristic
113C2×Q8×C71C2311yescharacteristic
114C22×C4×C71—21no—
115C22×C4×C71C2311yescharacteristic
116Q16×C71—22no—
117Q16×C71—41no—
118C2×C8×C71C2311yescharacteristic
119C2×C8×C71C2311yescharacteristic
120C4⋊C4×C71—21no—
121C4⋊C4×C71C2311yescharacteristic
122C22⋊C4×C71—21no—
123C23×C71C2×D411yescharacteristic
124C22×D71—21no—
125C22×D71—21no—
126C22×D71—42no—
127C22×D71—42no—
128C22×D71—81no—
129C22×D71C2×D411yescharacteristic
130Q8×C71—21no—
131Q8×C71—21no—
132Q8×C71—21no—
133Q8×C71—21no—
134Q8×C71—21no—
135Q8×C71—42no—
136Q8×C71C2×D412yes—
137C2×C4×C71—21no—
138C2×C4×C71—21no—
139C2×C4×C71—21no—
140C2×C4×C71—21no—
141C2×C4×C71—21no—
142C2×C4×C71—21no—
143C2×C4×C71—42no—
144C2×C4×C71C2×D411yescharacteristic
145C2×C4×C71C2411yescharacteristic
146C22⋊D71—22no—
147C2×Dic71—21no—
148C2×Dic71—21no—
149C2×Dic71C2×D411yescharacteristic
150C2×Dic71C2×D411yescharacteristic
151C2×Dic71C2×D411yescharacteristic
152D284—21no—
153D284—21no—
154D284—21no—
155D284—42no—
156D284—42no—
157D284—81no—
158D284C2×D412yes—
159C4×D71—21no—
160C4×D71—21no—
161C4×D71—42no—
162C4×D71—42no—
163C4×D71—81no—
164Dic142—21no—
165Dic142—21no—
166Dic142—42no—
167C8×C71—21no—
168C8×C71—21no—
169C71⋊C8—21no—
170C71⋊C8C2×D412yes—
171C22×C71—21no—
172C22×C71—21no—
173C22×C71#13712yescomplemented
174C22×C71C22×D411yescharacteristic
175C2×D71—21no—
176C2×D71—22no—
177C2×D71—21no—
178C2×D71—21no—
179C2×D71—41no—
180C2×D71—41no—
181C2×D71—41no—
182C2×D71—41no—
183C2×D71—41no—
184C2×D71—84no—
185C4×C71—22no—
186C4×C71—21no—
187C4×C71—41no—
188C4×C71—41no—
189C4×C71—41no—
190C4×C71C22×D411yescharacteristic
191C4×C71H(5,𝔽2)11yescharacteristic
192Dic71—21no—
193Dic71—21no—
194Dic71—21no—
195Dic71—41no—
196Dic71—41no—
197C2×C71—22no—
198C2×C71D4211yescharacteristic
199C2×C71C2×#13711yescharacteristic
200C2×C71#57711yescharacteristic
201D71—41no—
202D71—42no—
203D71—81no—
204#3266—711nomaximal, Sylow
205C71#326611yescharacteristic, complemented, Sylow
206C2×#137—711no—
207C22×SD16—711no—
208D4×Q8—711no—
209#546—711no—
210#497—711no—
211#468—711no—
212#467—711no—
213#464—711no—
214#461—711no—
215#452—711no—
216#451D7111yescharacteristic, complemented
217#450—711no—
218#448—711no—
219#440—711no—
220#435—711no—
221C2×#41—711no—
222C22×Q8—711no—
223C22×Q8C2×D7111yescharacteristic
224C22×D4—711no—
225#137—1424no—
226C2×Q16—711no—
227C2×Q16C2×D7111yescharacteristic, complemented
228C2×Q16C2×D7111yescharacteristic, complemented
229C2×SD16—711no—
230C2×SD16—711no—
231C2×SD16—711no—
232C2×SD16—711no—
233C2×SD16—711no—
234C2×SD16—711no—
235C2×SD16—1424no—
236C2×M16—711no—
237C22×C8—711no—
238C4⋊Q8—711no—
239C4⋊Q8—1421no—
240#124—711no—
241#123—1421no—
242C22⋊Q8—711no—
243C22⋊Q8—711no—
244C22⋊Q8—1421no—
245C22⋊Q8—1421no—
246C22⋊Q8C2×D7111yescharacteristic
247#121—711no—
248#121—1421no—
249#120—1421no—
250C4×Q8—711no—
251C4×D4—711no—
252C4×D4—711no—
253C4×D4—1421no—
254C2×C22⋊C4—1421no—
255#107—711no—
256C4⋊C8—711no—
257#103—711no—
258#103—711no—
259#103C2×D7111yescharacteristic, complemented
260#103C2×D7111yescharacteristic, complemented
261#102—711no—
262#102—711no—
263C22⋊C8—711no—
264C22⋊C8C2×D7111yescharacteristic
265#97—711no—
266C24—1421no—
267#41—2841no—
268#41—1422no—
269C2×Q8—21no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—2841no—
273C2×Q8—711no—
274C2×Q8—1421no—
275C2×Q8—1422no—
276C2×Q8C22×D7111yescharacteristic
277C2×Q8C22×D7111yescharacteristic
278C2×Q8C22×D7111yescharacteristic
279C2×D4—2841no—
280C2×D4—711no—
281C2×D4—711no—
282C2×D4—711no—
283C2×D4—1421no—
284C2×D4—1422no—
285C2×D4—1421no—
286C2×D4—1421no—
287C22×C4—21no—
288C22×C4—711no—
289C22×C4—1421no—
290C22×C4—1421no—
291C22×C4—1421no—
292C22×C4C22×D7111yescharacteristic
293Q16—22no—
294Q16—41no—
295Q16—1422no—
296SD16—2841no—
297SD16—714no—
298SD16—1422no—
299SD16—1422no—
300SD16—1422no—
301SD16—1422no—
302M16—1422no—
303C2×C8—711no—
304C2×C8—711no—
305C2×C8—1422no—
306C2×C8C22×D7111yescharacteristic
307C2×C8C22×D7111yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—711no—
310C4⋊C4—711no—
311C4⋊C4—1421no—
312C4⋊C4—1421no—
313C4⋊C4—1421no—
314C4⋊C4—1421no—
315C4⋊C4—1421no—
316C4⋊C4C22×D7111yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—711no—
319C22⋊C4—711no—
320C22⋊C4—1421no—
321C22⋊C4—1421no—
322C22⋊C4—1421no—
323C22⋊C4—1421no—
324C22⋊C4—1421no—
325C22⋊C4—1421no—
326C42—711no—
327C42—1421no—
328C23—2842no—
329C23—2842no—
330C23—5681no—
331C23—711no—
332C23—1421no—
333C23—1421no—
334C23D4×D7111yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—21no—
340Q8—42no—
341Q8—2842no—
342Q8—1421no—
343Q8—1421no—
344Q8D4×D7112yes—
345D4—2842no—
346D4—2842no—
347D4—5681no—
348D4—712no—
349D4—1422no—
350D4—1421no—
351D4—1421no—
352D4—1421no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—42no—
360C2×C4—2842no—
361C2×C4—2842no—
362C2×C4—5681no—
363C2×C4—711no—
364C2×C4—711no—
365C2×C4—711no—
366C2×C4—1421no—
367C2×C4—1421no—
368C2×C4—1421no—
369C2×C4—1421no—
370C2×C4D4×D7111yescharacteristic
371C2×C4C23×D7111yescharacteristic
372C8—21no—
373C8—21no—
374C8—712no—
375C8—1421no—
376C22—21no—
377C22—21no—
378C22—2841no—
379C22—2841no—
380C22—2841no—
381C22—2841no—
382C22—2841no—
383C22—5684no—
384C22—1421no—
385C22—1422no—
386C22—1421no—
387C22—1421no—
388C22#1455616012yescomplemented
389C22C2×D4×D7111yescharacteristic
390C4—21no—
391C4—22no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—2841no—
396C4—2841no—
397C4—1421no—
398C4—1421no—
399C4—1421no—
400C4C2×D4×D7111yescharacteristic
401C4#1455617311yescharacteristic
402C2—22no—
403C2—2842no—
404C2—2841no—
405C2—5681no—
406C2#1798663111yescharacteristic
407C2C2×#1455616011yescharacteristic
408C2#1798680811yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#451×C71C2
socleC22×C71C22×D4
hypercenter#451D71
hypocenterC71#3266
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. C2×#14556160C2semidirect
2. C22×#11765852C2semidirect
3. Q8×C22⋊D71C2semidirect
5. #17986392C2semidirect
6. #17986248C2semidirect
7. #17986238C2semidirect
8. #17986223C2semidirect
9. #17986203C2semidirect
10. #17986160C2semidirect
11. #17986152C2semidirect
12. #17986139C2semidirect
13. #17986102C2semidirect
14. #17986067C2semidirect
15. #451×C71C2semidirect
17. C2×Q8×D71C22semidirect
20. C2×#11765855C22semidirect
28. C2×#11765848C22semidirect
31. #14556126C22semidirect
37. #14556104C22semidirect
41. Q8×Dic71C22semidirect
46. #14556061C22semidirect
47. Dic71⋊C8C22semidirect
48. #14556052C22semidirect
49. #14556051C22semidirect
52. (C2×D71)⋊C8C22semidirect
53. #14556038C22semidirect
55. C2×Q16×C71C22semidirect
56. C2×Q16×C71C22semidirect
58. #103×C71C22semidirect
59. #103×C71C22semidirect
86. #11765852D4semidirect
173. C22×C71#137semidirect
205. C71#3266semidirect
216. #451D71semidirect
227. C2×Q16C2×D71semidirect
228. C2×Q16C2×D71semidirect
259. #103C2×D71semidirect
260. #103C2×D71semidirect
388. C22#14556160semidirect