Finite Group Databasebeta

← #22599911

Subgroups — #22599911

Order 9088.

472 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
1C2×#14556159C211yesmaximal, characteristic, complemented
2C22×#11765855C211yesmaximal, characteristic, complemented
3#17986658C211yesmaximal, characteristic
4#17986391C211yesmaximal, characteristic, complemented
5#17986255C211yesmaximal, characteristic, complemented
6#17986230C211yesmaximal, characteristic, complemented
7#17986221C211yesmaximal, characteristic, complemented
8#17986206C211yesmaximal, characteristic, complemented
9#17986159C211yesmaximal, characteristic, complemented
10#17986155C211yesmaximal, characteristic, complemented
11#17986154C211yesmaximal, characteristic, complemented
12#17986138C211yesmaximal, characteristic, complemented
13#17986104C211yesmaximal, characteristic, complemented
14#17986071C211yesmaximal, characteristic, complemented
15D4×Q8×C71C211yesmaximal, characteristic, complemented
16C2×#11765865C2211yescharacteristic
17C22×Dic142C2211yescharacteristic
18#14556159—24no—
19C2×#11765855—24no—
20C2×#11765855C2211yescharacteristic
21C2×#11765855C2211yescharacteristic
22C2×#11765855C2211yescharacteristic
23C2×#11765855C2211yescharacteristic
24C2×#11765855C2211yescharacteristic
25C2×#11765855C2211yescharacteristic
26C2×#11765852C2211yescharacteristic
27C2×#11765852C2211yescharacteristic
28C2×#11765852C2211yescharacteristic
29C2×#11765849C2211yescharacteristic
30C22×C71⋊C8C2211yescharacteristic
31#14556116C2211yescharacteristic
32#14556114—21no—
33#14556107C2211yescharacteristic
34#14556106—21no—
35#14556102C2211yescharacteristic
36#14556100—21no—
37C4×Dic142C2211yescharacteristic
38(C22⋊D71)⋊C4—21no—
39#14556059C2211yescharacteristic
40C4⋊(C71⋊C8)C2211yescharacteristic
41#14556052C2211yescharacteristic
42#14556052C2211yescharacteristic
43#14556052C2211yescharacteristic
44#14556051C2211yescharacteristic
45#14556051C2211yescharacteristic
46#14556048C2211yescharacteristic
47C22⋊(C71⋊C8)C2211yescharacteristic
48C22⋊(C71⋊C8)C2211yescharacteristic
49#14556039C2211yescharacteristic
50C22×Q8×C71C2211yescharacteristic
51C22×Q8×C71C2211yescharacteristic
52C4⋊Q8×C71—21no—
53C4⋊Q8×C71C2211yescharacteristic
54C22⋊Q8×C71—21no—
55C22⋊Q8×C71—21no—
56C22⋊Q8×C71C2211yescharacteristic
57C22⋊Q8×C71C2211yescharacteristic
58C4×Q8×C71C2211yescharacteristic
59C4×D4×C71—21no—
60C4×D4×C71C2211yescharacteristic
61#11765865—22no—
62#11765865—41no—
63C2×Dic142—22no—
64C2×Dic142—21no—
65C2×Dic142—41no—
66C2×Dic142C2311yescharacteristic
67C2×Dic142C2311yescharacteristic
68C2×D284C2311yescharacteristic
69C2×C22⋊D71—21no—
70C2×C22⋊D71—21no—
71C22×Dic71—21no—
72C2×C4×D71—21no—
73#11765855—22no—
74#11765855—22no—
75#11765855—22no—
76#11765855—22no—
77#11765855—41no—
78#11765855D414yescomplemented
79#11765852—22no—
80#11765852—22no—
81#11765852—41no—
82#11765849—22no—
83C2×C71⋊C8—22no—
84C2×C71⋊C8C2311yescharacteristic
85C2×C71⋊C8C2311yescharacteristic
86C2×C71⋊C8C2311yescharacteristic
87C2×C71⋊C8C2311yescharacteristic
88C4⋊Dic71C2311yescharacteristic
89Dic71⋊C4—21no—
90Dic71⋊C4—21no—
91C22⋊Dic71—21no—
92(C2×D71)⋊C4—21no—
93(C2×D71)⋊C4—21no—
94(C2×D71)⋊C4—21no—
95C4×Dic71—21no—
96C2×Q8×C71—21no—
97C2×Q8×C71—21no—
98C2×Q8×C71—22no—
99C2×Q8×C71—22no—
100C2×Q8×C71—41no—
101C2×Q8×C71—41no—
102C2×Q8×C71C2311yescharacteristic
103C2×Q8×C71C2311yescharacteristic
104C2×Q8×C71C2311yescharacteristic
105C2×D4×C71D411yescharacteristic
106C22×C4×C71—21no—
107C22×C4×C71—21no—
108C22×C4×C71C2311yescharacteristic
109C22×C4×C71C2311yescharacteristic
110C4⋊C4×C71—21no—
111C4⋊C4×C71—21no—
112C4⋊C4×C71—21no—
113C4⋊C4×C71—21no—
114C4⋊C4×C71—21no—
115C4⋊C4×C71D411yescharacteristic
116C4⋊C4×C71C2311yescharacteristic
117C22⋊C4×C71—21no—
118C22⋊C4×C71—21no—
119C22⋊C4×C71D411yescharacteristic
120C22⋊C4×C71D411yescharacteristic
121C42×C71—21no—
122C42×C71C2311yescharacteristic
123C23×C71C2×D411yescharacteristic
124C23×C71C2×D411yescharacteristic
125C22×D71—21no—
126Q8×C71—21no—
127Q8×C71—21no—
128Q8×C71—21no—
129Q8×C71—21no—
130Q8×C71—21no—
131Q8×C71—42no—
132Q8×C71—42no—
133Q8×C71C2×D412yes—
134D4×C71—22no—
135C2×C4×C71—21no—
136C2×C4×C71—21no—
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×C71—42no—
145C2×C4×C71C2×D411yescharacteristic
146C2×C4×C71C2×D411yescharacteristic
147C2×C4×C71C2×D411yescharacteristic
148C2×C4×C71C2×D411yescharacteristic
149C2×C4×C71C2411yescharacteristic
150C22⋊D71—42no—
151C22⋊D71—42no—
152C2×Dic71—21no—
153C2×Dic71—21no—
154C2×Dic71—21no—
155C2×Dic71—42no—
156D284—21no—
157D284—21no—
158C4×D71—42no—
159Dic142—21no—
160Dic142—21no—
161Dic142—21no—
162Dic142—42no—
163Dic142C2×D412yes—
164C71⋊C8—21no—
165C71⋊C8—21no—
166C71⋊C8—21no—
167C71⋊C8C2×D412yes—
168C22×C71—21no—
169C22×C71—21no—
170C22×C71—21no—
171C22×C71—21no—
172C22×C71—21no—
173C22×C71#13712yescomplemented
174C22×C71C22×D411yescharacteristic
175C2×D71—41no—
176C2×D71—41no—
177C2×D71—41no—
178C4×C71—21no—
179C4×C71—21no—
180C4×C71—22no—
181C4×C71—21no—
182C4×C71—21no—
183C4×C71—41no—
184C4×C71—41no—
185C4×C71—41no—
186C4×C71C22×D411yescharacteristic
187C4×C71H(5,𝔽2)11yescharacteristic
188Dic71—22no—
189Dic71—41no—
190Dic71—41no—
191C2×C71—22no—
192C2×C71—41no—
193C2×C71D4211yescharacteristic
194C2×C71C2×#13711yescharacteristic
195C2×C71#57811yescharacteristic
196D71—81no—
197#3270—711nomaximal, Sylow
198C71#327011yescharacteristic, complemented, Sylow
199C2×#137—711no—
200C22×Q16—711no—
201D4×Q8D7111yescharacteristic, complemented
202#548—711no—
203#497—711no—
204#470—711no—
205#465—711no—
206#464—711no—
207#462—711no—
208#452—711no—
209#451—711no—
210#451—711no—
211#448—711no—
212#441—711no—
213#435—711no—
214C2×#41—711no—
215C22×Q8—711no—
216C22×Q8C2×D7111yescharacteristic
217C22×Q8C2×D7111yescharacteristic
218#137—1424no—
219C2×Q16—711no—
220C2×Q16—711no—
221C2×Q16—711no—
222C2×Q16—711no—
223C2×Q16—711no—
224C2×Q16—711no—
225C2×Q16—1424no—
226C2×SD16—711no—
227C2×SD16—711no—
228C2×SD16—711no—
229C2×M16—711no—
230C22×C8—711no—
231C4⋊Q8—21no—
232C4⋊Q8C2×D7111yescharacteristic
233#124—711no—
234#124—1421no—
235C22⋊Q8—21no—
236C22⋊Q8—21no—
237C22⋊Q8—711no—
238C22⋊Q8—1421no—
239C22⋊Q8C2×D7111yescharacteristic
240C22⋊Q8C2×D7111yescharacteristic
241#121—711no—
242#121—1421no—
243C4×Q8—711no—
244C4×Q8C2×D7111yescharacteristic
245C4×D4—21no—
246C4×D4—1421no—
247C4×D4C2×D7111yescharacteristic
248#106—711no—
249C4⋊C8—711no—
250#103—711no—
251#103—711no—
252#103—711no—
253#103—711no—
254#103—711no—
255#102—711no—
256C22⋊C8—711no—
257C22⋊C8—711no—
258#97—711no—
259#41—2841no—
260#41—1422no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—2841no—
268C2×Q8—711no—
269C2×Q8—711no—
270C2×Q8—1422no—
271C2×Q8—1421no—
272C2×Q8C22×D7111yescharacteristic
273C2×Q8C22×D7111yescharacteristic
274C2×Q8C22×D7111yescharacteristic
275C2×D4—711no—
276C2×D4—1421no—
277C2×D4—1421no—
278C2×D4C22⋊D7111yescharacteristic
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—1421no—
282C22×C4—1421no—
283C22×C4C22×D7111yescharacteristic
284C22×C4C22×D7111yescharacteristic
285Q16—2841no—
286Q16—714no—
287Q16—1422no—
288Q16—1422no—
289Q16—1422no—
290Q16—1422no—
291SD16—2841no—
292SD16—1422no—
293SD16—1422no—
294M16—1422no—
295C2×C8—711no—
296C2×C8—711no—
297C2×C8—711no—
298C2×C8—711no—
299C2×C8—1422no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—711no—
306C4⋊C4—1421no—
307C4⋊C4—1421no—
308C4⋊C4C22⋊D7111yescharacteristic
309C4⋊C4C22×D7111yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—1421no—
313C22⋊C4—1421no—
314C22⋊C4—1421no—
315C22⋊C4—1421no—
316C22⋊C4C22⋊D7111yescharacteristic
317C22⋊C4C22⋊D7111yescharacteristic
318C42—21no—
319C42—1421no—
320C42C22×D7111yescharacteristic
321C23—1421no—
322C23C2×C22⋊D7111yescharacteristic
323C23C2×C22⋊D7111yescharacteristic
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—42no—
330Q8—42no—
331Q8—2842no—
332Q8—712no—
333Q8—1421no—
334Q8—1421no—
335Q8—1421no—
336Q8D4×D7112yes—
337D4—22no—
338D4—2842no—
339D4—2842no—
340D4—1421no—
341D4—1421no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—2842no—
353C2×C4—2842no—
354C2×C4—1421no—
355C2×C4—1421no—
356C2×C4—1421no—
357C2×C4C2×C22⋊D7111yescharacteristic
358C2×C4C2×C22⋊D7111yescharacteristic
359C2×C4C2×C22⋊D7111yescharacteristic
360C2×C4C2×C22⋊D7111yescharacteristic
361C2×C4C23×D7111yescharacteristic
362C8—712no—
363C8—1421no—
364C8—1421no—
365C8—1421no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—2841no—
372C22—2841no—
373C22—2841no—
374C22#1455615912yescomplemented
375C22C22×C22⋊D7111yescharacteristic
376C4—21no—
377C4—21no—
378C4—22no—
379C4—21no—
380C4—21no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—2841no—
385C4—2841no—
386C4—1422no—
387C4C2×D4×D7111yescharacteristic
388C4#1455617311yescharacteristic
389C2—22no—
390C2—41no—
391C2—5681no—
392C2D4×C22⋊D7111yescharacteristic
393C2C2×#1455615911yescharacteristic
394C2#1798681411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
FittingD4×Q8×C71C2
socleC22×C71C22×D4
hypercenterD4×Q8D71
hypocenterC71#3270
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#14556159C2semidirect
2. C22×#11765855C2semidirect
4. #17986391C2semidirect
5. #17986255C2semidirect
6. #17986230C2semidirect
7. #17986221C2semidirect
8. #17986206C2semidirect
9. #17986159C2semidirect
10. #17986155C2semidirect
11. #17986154C2semidirect
12. #17986138C2semidirect
13. #17986104C2semidirect
14. #17986071C2semidirect
15. D4×Q8×C71C2semidirect
78. #11765855D4semidirect
173. C22×C71#137semidirect
198. C71#3270semidirect
201. D4×Q8D71semidirect
374. C22#14556159semidirect