Finite Group Databasebeta

← #22599848

Subgroups — #22599848

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×#14556158C211yesmaximal, characteristic, complemented
2C2×SD16×D71C211yesmaximal, characteristic, complemented
3#17986662C211yesmaximal, characteristic, complemented
4#17986637C211yesmaximal, characteristic
5#17986392C211yesmaximal, characteristic, complemented
6#17986247C211yesmaximal, characteristic, complemented
7#17986239C211yesmaximal, characteristic, complemented
8#17986224C211yesmaximal, characteristic, complemented
9#17986202C211yesmaximal, characteristic, complemented
10#17986153C211yesmaximal, characteristic, complemented
11#17986151C211yesmaximal, characteristic, complemented
12#17986140C211yesmaximal, characteristic, complemented
13#17986096C211yesmaximal, characteristic, complemented
14#17986073C211yesmaximal, characteristic, complemented
15#452×C71C211yesmaximal, characteristic, complemented
16C22×Dic142C2211yescharacteristic, complemented
17C2×Q8×D71C2211yescharacteristic
18C2×D4×D71C2211yescharacteristic
19#14556158—24no—
20C2×#11765855C2211yescharacteristic, complemented
21C2×#11765855C2211yescharacteristic, complemented
22C2×#11765854C2211yescharacteristic
23C2×#11765854C2211yescharacteristic
24C2×#11765853C2211yescharacteristic
25C2×#11765853C2211yescharacteristic
26C2×#11765852C2211yescharacteristic
27SD16×D71—24no—
28C2×#11765849C2211yescharacteristic, complemented
29C2×C8×D71C2211yescharacteristic
30#14556127—21no—
31#14556126C2211yescharacteristic, complemented
32#14556117C2211yescharacteristic
33#14556111—21no—
34#14556108C2211yescharacteristic
35#14556106—21no—
36#14556105—21no—
37#14556104C2211yescharacteristic
38#14556102C2211yescharacteristic
39#14556100—21no—
40#14556096—21no—
41#14556093C2211yescharacteristic, complemented
42D4×Dic71C2211yescharacteristic
43C4×C22⋊D71C2211yescharacteristic
44(C22⋊D71)⋊C4—21no—
45C2×(C2×D71)⋊C4—21no—
46#14556061C2211yescharacteristic, complemented
47Dic71⋊C8C2211yescharacteristic, complemented
48#14556053C2211yescharacteristic, complemented
49#14556053C2211yescharacteristic, complemented
50#14556052C2211yescharacteristic, complemented
51#14556048C2211yescharacteristic
52(C2×D71)⋊C8C2211yescharacteristic
53#14556038C2211yescharacteristic, complemented
54C2×#41×C71C2211yescharacteristic
55C2×Q16×C71C2211yescharacteristic, complemented
56C2×SD16×C71C2211yescharacteristic
57C22⋊Q8×C71C2211yescharacteristic, complemented
58#103×C71C2211yescharacteristic, complemented
59#102×C71C2211yescharacteristic
60C22⋊C8×C71C2211yescharacteristic, complemented
61C23×D71—21no—
62C2×Dic142—22no—
63C2×Dic142—21no—
64C2×Dic142—21no—
65C2×Dic142—41no—
66C2×Dic142C2311yescharacteristic
67C2×Dic142C2311yescharacteristic
68Q8×D71—22no—
69Q8×D71—41no—
70C2×D284C2311yescharacteristic
71C2×C22⋊D71—21no—
72C2×C22⋊D71—21no—
73C2×C22⋊D71D411yescharacteristic
74D4×D71—22no—
75D4×D71—41no—
76C22×Dic71—21no—
77C22×Dic71—21no—
78C2×C4×D71—21no—
79C2×C4×D71C2311yescharacteristic
80#11765855—22no—
81#11765855—22no—
82#11765854—22no—
83#11765854—41no—
84#11765853—22no—
85#11765853D414yescomplemented
86#11765852—22no—
87#11765849—22no—
88C2×C71⋊C8C2311yescharacteristic
89C2×C71⋊C8C2311yescharacteristic
90C8×D71—22no—
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
108#41×C71—22no—
109#41×C71—41no—
110C2×Q8×C71C2311yescharacteristic
111C2×Q8×C71C2311yescharacteristic
112C2×D4×C71—21no—
113C2×D4×C71C2311yescharacteristic
114C22×C4×C71—21no—
115C22×C4×C71C2311yescharacteristic
116Q16×C71—41no—
117SD16×C71—22no—
118C2×C8×C71C2311yescharacteristic
119C2×C8×C71C2311yescharacteristic
120C4⋊C4×C71—21no—
121C4⋊C4×C71C2311yescharacteristic
122C22⋊C4×C71—21no—
123C23×C71—21no—
124C23×C71C2×D411yescharacteristic
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—
134D4×C71—21no—
135D4×C71—42no—
136D4×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—42no—
143C2×C4×C71C2×D411yescharacteristic
144C2×C4×C71C2411yescharacteristic
145C22⋊D71—22no—
146C22⋊D71—42no—
147C22⋊D71—81no—
148C2×Dic71—21no—
149C2×Dic71—21no—
150C2×Dic71—21no—
151C2×Dic71—42no—
152C2×Dic71—81no—
153C2×Dic71C2×D411yescharacteristic
154C2×Dic71C2×D411yescharacteristic
155C2×Dic71C2×D411yescharacteristic
156D284—21no—
157D284—21no—
158C4×D71—21no—
159C4×D71—21no—
160C4×D71—42no—
161Dic142—21no—
162Dic142—21no—
163Dic142—21no—
164Dic142—42no—
165Dic142—42no—
166Dic142C2×D412yes—
167C8×C71—21no—
168C8×C71—21no—
169C71⋊C8—21no—
170C71⋊C8C2×D412yes—
171C22×C71—21no—
172C22×C71—22no—
173C22×C71—21no—
174C22×C71—21no—
175C22×C71—41no—
176C22×C71—41no—
177C22×C71C22×D411yescharacteristic
178C2×D71—21no—
179C2×D71—21no—
180C2×D71—41no—
181C2×D71—41no—
182C2×D71—41no—
183C2×D71—84no—
184C2×D71#13712yescomplemented
185C4×C71—21no—
186C4×C71—41no—
187C4×C71—41no—
188C4×C71—41no—
189C4×C71C22×D411yescharacteristic
190C4×C71H(5,𝔽2)11yescharacteristic
191Dic71—22no—
192Dic71—21no—
193Dic71—21no—
194Dic71—21no—
195Dic71—41no—
196Dic71—41no—
197C2×C71—42no—
198C2×C71—41no—
199C2×C71D4211yescharacteristic
200C2×C71C2×#13711yescharacteristic
201C2×C71#57711yescharacteristic
202D71—22no—
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#452D7111yescharacteristic, complemented
216#451—711no—
217#450—711no—
218#448—711no—
219#440—711no—
220#435—711no—
221C2×#41C2×D7111yescharacteristic
222C22×Q8—711no—
223C22×Q8—711no—
224C22×D4—711no—
225#137—1424no—
226C2×Q16—711no—
227C2×Q16—711no—
228C2×Q16C2×D7111yescharacteristic, complemented
229C2×SD16—711no—
230C2×SD16—711no—
231C2×SD16—711no—
232C2×SD16—711no—
233C2×SD16—711no—
234C2×SD16—1424no—
235C2×SD16C2×D7111yescharacteristic
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, complemented
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#103—711no—
260#103C2×D7111yescharacteristic, complemented
261#102—711no—
262#102C2×D7111yescharacteristic
263C22⋊C8—711no—
264C22⋊C8C2×D7111yescharacteristic, complemented
265#97—711no—
266C24—1421no—
267#41—22no—
268#41—41no—
269C2×Q8—2841no—
270C2×Q8—2841no—
271C2×Q8—711no—
272C2×Q8—711no—
273C2×Q8—1422no—
274C2×Q8—1421no—
275C2×Q8—1422no—
276C2×Q8—1421no—
277C2×Q8C22×D7111yescharacteristic
278C2×Q8C22×D7111yescharacteristic
279C2×D4—21no—
280C2×D4—2841no—
281C2×D4—711no—
282C2×D4—711no—
283C2×D4—1421no—
284C2×D4—1422no—
285C2×D4—1421no—
286C2×D4C22×D7111yescharacteristic
287C22×C4—21no—
288C22×C4—711no—
289C22×C4—1421no—
290C22×C4—1421no—
291C22×C4—1421no—
292C22×C4C22×D7111yescharacteristic
293Q16—41no—
294Q16—1422no—
295Q16—1422no—
296SD16—22no—
297SD16—2841no—
298SD16—714no—
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—21no—
329C23—2842no—
330C23—2842no—
331C23—5681no—
332C23—711no—
333C23—1421no—
334C23D4×D7111yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—2842no—
340Q8—2842no—
341Q8—712no—
342Q8—1421no—
343Q8—1421no—
344Q8—1421no—
345D4—21no—
346D4—42no—
347D4—2842no—
348D4—5681no—
349D4—1422no—
350D4—1421no—
351D4—1421no—
352D4D4×D7112yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—42no—
359C2×C4—2842no—
360C2×C4—2842no—
361C2×C4—5681no—
362C2×C4—711no—
363C2×C4—711no—
364C2×C4—711no—
365C2×C4—1421no—
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—22no—
378C22—21no—
379C22—21no—
380C22—41no—
381C22—41no—
382C22—2841no—
383C22—2841no—
384C22—2841no—
385C22—5684no—
386C22—712no—
387C22—1421no—
388C22—1421no—
389C22C2×D4×D7111yescharacteristic
390C4—21no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—2841no—
395C4—2841no—
396C4—1422no—
397C4—1421no—
398C4—1421no—
399C4—1421no—
400C4C2×D4×D7111yescharacteristic
401C4#1455617311yescharacteristic
402C2—41no—
403C2—42no—
404C2—5681no—
405C2—1422no—
406C2#1798663111yescharacteristic
407C2#137×D7111yescharacteristic
408C2#1798680611yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#452×C71C2
socleC22×C71C22×D4
hypercenter#452D71
hypocenterC71#3266
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
1. C2×#14556158C2semidirect
2. C2×SD16×D71C2semidirect
3. #17986662C2semidirect
5. #17986392C2semidirect
6. #17986247C2semidirect
7. #17986239C2semidirect
8. #17986224C2semidirect
9. #17986202C2semidirect
10. #17986153C2semidirect
11. #17986151C2semidirect
12. #17986140C2semidirect
13. #17986096C2semidirect
14. #17986073C2semidirect
15. #452×C71C2semidirect
16. C22×Dic142C22semidirect
20. C2×#11765855C22semidirect
21. C2×#11765855C22semidirect
28. C2×#11765849C22semidirect
31. #14556126C22semidirect
41. #14556093C22semidirect
46. #14556061C22semidirect
47. Dic71⋊C8C22semidirect
48. #14556053C22semidirect
49. #14556053C22semidirect
50. #14556052C22semidirect
53. #14556038C22semidirect
55. C2×Q16×C71C22semidirect
57. C22⋊Q8×C71C22semidirect
58. #103×C71C22semidirect
60. C22⋊C8×C71C22semidirect
85. #11765853D4semidirect
184. C2×D71#137semidirect
205. C71#3266semidirect
215. #452D71semidirect
228. C2×Q16C2×D71semidirect
246. C22⋊Q8C2×D71semidirect
260. #103C2×D71semidirect
264. C22⋊C8C2×D71semidirect