Finite Group Databasebeta

← #22599866

Subgroups — #22599866

Order 9088.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14556153C211yesmaximal, characteristic, complemented
2C2×#14556148C211yesmaximal, characteristic, complemented
3#17986645C211yesmaximal, characteristic, complemented
4#17986391C211yesmaximal, characteristic, complemented
5#17986255C211yesmaximal, characteristic, complemented
6#17986230C211yesmaximal, characteristic, complemented
7#17986214C211yesmaximal, characteristic, complemented
8#17986201C211yesmaximal, characteristic, complemented
9#17986161C211yesmaximal, characteristic, complemented
10#17986159C211yesmaximal, characteristic, complemented
11#17986149C211yesmaximal, characteristic, complemented
12#17986144C211yesmaximal, characteristic, complemented
13#17986109C211yesmaximal, characteristic, complemented
14#17986071C211yesmaximal, characteristic, complemented
15#547×C71C211yesmaximal, characteristic, complemented
16C2×#11765865C2211yescharacteristic
17C22×D284C2211yescharacteristic
18#14556153—24no—
19#14556148—24no—
20C2×#11765855C2211yescharacteristic, complemented
21C2×#11765855C2211yescharacteristic, complemented
22C2×#11765853C2211yescharacteristic, complemented
23C2×#11765852C2211yescharacteristic
24C2×#11765852C2211yescharacteristic
25C2×#11765852C2211yescharacteristic
26C2×#11765852C2211yescharacteristic
27C2×#11765850C2211yescharacteristic
28C2×#11765850C2211yescharacteristic
29C2×#11765849C2211yescharacteristic
30C22×C71⋊C8C2211yescharacteristic, complemented
31#14556116C2211yescharacteristic
32#14556113—21no—
33#14556107C2211yescharacteristic
34#14556102C2211yescharacteristic
35#14556101—21no—
36#14556096—21no—
37C4×D284C2211yescharacteristic
38C22⋊C4×D71—21no—
39#14556059C2211yescharacteristic, complemented
40C4⋊(C71⋊C8)C2211yescharacteristic, complemented
41#14556052C2211yescharacteristic, complemented
42#14556052C2211yescharacteristic, complemented
43#14556051C2211yescharacteristic, complemented
44#14556049C2211yescharacteristic, complemented
45#14556048C2211yescharacteristic
46#14556048C2211yescharacteristic
47C22⋊(C71⋊C8)C2211yescharacteristic
48C22⋊(C71⋊C8)C2211yescharacteristic, complemented
49#14556039C2211yescharacteristic, complemented
50C2×#41×C71C2211yescharacteristic
51C2×#41×C71C2211yescharacteristic, complemented
52C4⋊Q8×C71C2211yescharacteristic, complemented
53#123×C71—21no—
54#123×C71—21no—
55C22⋊Q8×C71C2211yescharacteristic
56C22⋊Q8×C71C2211yescharacteristic, complemented
57#121×C71—21no—
58C4×D4×C71C2211yescharacteristic
59C4×D4×C71C2211yescharacteristic, complemented
60C2×C4⋊C4×C71—21no—
61C23×D71—21no—
62#11765865—22no—
63#11765865—41no—
64C2×Dic142C2311yescharacteristic
65C2×D284—22no—
66C2×D284—21no—
67C2×D284—41no—
68C2×D284C2311yescharacteristic
69C2×D284C2311yescharacteristic
70C2×C22⋊D71—21no—
71C2×C22⋊D71—21no—
72C2×C4×D71—21no—
73C2×C4×D71—21no—
74#11765855—22no—
75#11765855—41no—
76#11765853—22no—
77#11765852—22no—
78#11765852—22no—
79#11765852—22no—
80#11765852—41no—
81#11765850—22no—
82#11765850D414yescomplemented
83#11765849—22no—
84C2×C71⋊C8—22no—
85C2×C71⋊C8C2311yescharacteristic
86C2×C71⋊C8C2311yescharacteristic
87C2×C71⋊C8C2311yescharacteristic
88C2×C71⋊C8C2311yescharacteristic
89C4⋊Dic71C2311yescharacteristic
90Dic71⋊C4—21no—
91C22⋊Dic71—21no—
92(C2×D71)⋊C4—21no—
93(C2×D71)⋊C4—21no—
94(C2×D71)⋊C4—21no—
95(C2×D71)⋊C4—21no—
96#41×C71—22no—
97#41×C71—22no—
98#41×C71—41no—
99#41×C71—41no—
100C2×Q8×C71C2311yescharacteristic
101C2×Q8×C71C2311yescharacteristic
102C2×D4×C71—21no—
103C2×D4×C71—21no—
104C2×D4×C71D411yescharacteristic
105C2×D4×C71C2311yescharacteristic
106C22×C4×C71—21no—
107C22×C4×C71—21no—
108C22×C4×C71—21no—
109C22×C4×C71C2311yescharacteristic
110C22×C4×C71C2311yescharacteristic
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×C71—21no—
120C22⋊C4×C71D411yescharacteristic
121C22⋊C4×C71D411yescharacteristic
122C42×C71C2311yescharacteristic
123C23×C71—21no—
124C23×C71C2×D411yescharacteristic
125C23×C71C2×D411yescharacteristic
126C22×D71—21no—
127C22×D71—21no—
128C22×D71—42no—
129C22×D71—42no—
130C22×D71—81no—
131Q8×C71—21no—
132Q8×C71—21no—
133Q8×C71—21no—
134Q8×C71—21no—
135D4×C71—21no—
136D4×C71—41no—
137D4×C71—42no—
138D4×C71—42no—
139D4×C71C2×D412yes—
140C2×C4×C71—21no—
141C2×C4×C71—21no—
142C2×C4×C71—21no—
143C2×C4×C71—21no—
144C2×C4×C71—21no—
145C2×C4×C71—21no—
146C2×C4×C71—21no—
147C2×C4×C71—42no—
148C2×C4×C71—42no—
149C2×C4×C71—42no—
150C2×C4×C71C2×D411yescharacteristic
151C2×C4×C71C2×D411yescharacteristic
152C2×C4×C71C2×D411yescharacteristic
153C2×C4×C71C2×D411yescharacteristic
154C2×C4×C71C2411yescharacteristic
155C22⋊D71—42no—
156C22⋊D71—81no—
157C2×Dic71—21no—
158C2×Dic71—21no—
159D284—21no—
160D284—21no—
161D284—21no—
162D284—42no—
163D284C2×D412yes—
164C4×D71—42no—
165C4×D71—81no—
166Dic142—21no—
167Dic142—21no—
168C71⋊C8—21no—
169C71⋊C8—21no—
170C71⋊C8—21no—
171C71⋊C8C2×D412yes—
172C22×C71—21no—
173C22×C71—21no—
174C22×C71—22no—
175C22×C71—21no—
176C22×C71—21no—
177C22×C71—21no—
178C22×C71—21no—
179C22×C71—41no—
180C22×C71—41no—
181C22×C71C22×D411yescharacteristic
182C2×D71—22no—
183C2×D71—41no—
184C2×D71—41no—
185C2×D71—41no—
186C2×D71—41no—
187C2×D71—41no—
188C2×D71—84no—
189C4×C71—21no—
190C4×C71—21no—
191C4×C71—21no—
192C4×C71—21no—
193C4×C71—41no—
194C4×C71—41no—
195C4×C71—41no—
196C4×C71C22×D411yescharacteristic
197C4×C71H(5,𝔽2)11yescharacteristic
198Dic71—41no—
199Dic71—41no—
200C2×C71—41no—
201C2×C71—41no—
202C2×C71—42no—
203C2×C71D4211yescharacteristic
204C2×C71#57511yescharacteristic
205C2×C71#57711yescharacteristic
206D71—42no—
207D71—81no—
208#3267—711nomaximal, Sylow
209C71#326711yescharacteristic, complemented, Sylow
210C2×AGL(1,ℤ/8)—711no—
211C2×#135—711no—
212#547D7111yescharacteristic, complemented
213#546—711no—
214#497—711no—
215#470—711no—
216#465—711no—
217#463—711no—
218#461—711no—
219#452—711no—
220#452—711no—
221#450—711no—
222#449—711no—
223#442—711no—
224#435—711no—
225C2×#41—711no—
226C2×#41C2×D7111yescharacteristic
227C2×#41C2×D7111yescharacteristic, complemented
228C22×D4—711no—
229AGL(1,ℤ/8)—1424no—
230#135—1424no—
231C2×Q16—711no—
232C2×Q16—711no—
233C2×SD16—711no—
234C2×SD16—711no—
235C2×SD16—711no—
236C2×SD16—711no—
237C2×SD16—711no—
238C2×D8—711no—
239C2×D8—711no—
240C2×M16—711no—
241C22×C8—711no—
242C4⋊Q8C2×D7111yescharacteristic, complemented
243#124—711no—
244#123—21no—
245#123—21no—
246#123—1421no—
247C22⋊Q8—711no—
248C22⋊Q8C2×D7111yescharacteristic
249C22⋊Q8C2×D7111yescharacteristic, complemented
250#121—21no—
251#121—711no—
252#121—1421no—
253#120—1421no—
254C4×D4—711no—
255C4×D4C2×D7111yescharacteristic
256C4×D4C2×D7111yescharacteristic, complemented
257C2×C4⋊C4—21no—
258C2×C22⋊C4—1421no—
259#106—711no—
260C4⋊C8—711no—
261#103—711no—
262#103—711no—
263#103—711no—
264#102—711no—
265#102—711no—
266#102—711no—
267C22⋊C8—711no—
268C22⋊C8—711no—
269#97—711no—
270C24—1421no—
271#41—22no—
272#41—22no—
273#41—41no—
274#41—41no—
275#41—2841no—
276#41—1422no—
277C2×Q8—711no—
278C2×Q8C22×D7111yescharacteristic
279C2×Q8C22×D7111yescharacteristic
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—2841no—
283C2×D4—711no—
284C2×D4—711no—
285C2×D4—1422no—
286C2×D4—1421no—
287C2×D4—1421no—
288C2×D4—1421no—
289C2×D4C22⋊D7111yescharacteristic
290C2×D4C22×D7111yescharacteristic
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—1421no—
295C22×C4—1421no—
296C22×C4C22×D7111yescharacteristic
297C22×C4C22×D7111yescharacteristic
298Q16—2841no—
299Q16—1422no—
300SD16—2841no—
301SD16—1422no—
302SD16—1422no—
303SD16—1422no—
304SD16—1422no—
305D8—714no—
306D8—1422no—
307M16—1422no—
308C2×C8—711no—
309C2×C8—711no—
310C2×C8—711no—
311C2×C8—711no—
312C2×C8—1422no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—711no—
318C4⋊C4—1421no—
319C4⋊C4C22⋊D7111yescharacteristic
320C4⋊C4C22×D7111yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—21no—
324C22⋊C4—1421no—
325C22⋊C4—1421no—
326C22⋊C4—1421no—
327C22⋊C4—1421no—
328C22⋊C4—1421no—
329C22⋊C4C22⋊D7111yescharacteristic
330C22⋊C4C22⋊D7111yescharacteristic
331C42C22×D7111yescharacteristic
332C23—21no—
333C23—2842no—
334C23—2842no—
335C23—5681no—
336C23—1421no—
337C23—1421no—
338C23C2×C22⋊D7111yescharacteristic
339C23C2×C22⋊D7111yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—1421no—
345Q8—1421no—
346D4—21no—
347D4—41no—
348D4—42no—
349D4—42no—
350D4—2842no—
351D4—2842no—
352D4—5681no—
353D4—712no—
354D4—1421no—
355D4—1421no—
356D4—1421no—
357D4D4×D7112yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—2842no—
369C2×C4—5681no—
370C2×C4—1421no—
371C2×C4—1421no—
372C2×C4C2×C22⋊D7111yescharacteristic
373C2×C4C2×C22⋊D7111yescharacteristic
374C2×C4C2×C22⋊D7111yescharacteristic
375C2×C4C2×C22⋊D7111yescharacteristic
376C2×C4C23×D7111yescharacteristic
377C8—712no—
378C8—1421no—
379C8—1421no—
380C8—1421no—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—41no—
389C22—41no—
390C22—2841no—
391C22—2841no—
392C22—2841no—
393C22—2841no—
394C22—2841no—
395C22—5684no—
396C22—1422no—
397C22C22×C22⋊D7111yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—41no—
403C4—41no—
404C4—41no—
405C4—2841no—
406C4—2841no—
407C4C2×D4×D7111yescharacteristic
408C4#1455617311yescharacteristic
409C2—41no—
410C2—41no—
411C2—42no—
412C2—2842no—
413C2—5681no—
414C2D4×C22⋊D7111yescharacteristic
415C2#1798679711yescharacteristic
416C2#1798680711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#547×C71C2
socleC22×C71C22×D4
hypercenter#547D71
hypocenterC71#3267
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C2×#14556153C2semidirect
2. C2×#14556148C2semidirect
3. #17986645C2semidirect
4. #17986391C2semidirect
5. #17986255C2semidirect
6. #17986230C2semidirect
7. #17986214C2semidirect
8. #17986201C2semidirect
9. #17986161C2semidirect
10. #17986159C2semidirect
11. #17986149C2semidirect
12. #17986144C2semidirect
13. #17986109C2semidirect
14. #17986071C2semidirect
15. #547×C71C2semidirect
20. C2×#11765855C22semidirect
21. C2×#11765855C22semidirect
22. C2×#11765853C22semidirect
30. C22×C71⋊C8C22semidirect
39. #14556059C22semidirect
40. C4⋊(C71⋊C8)C22semidirect
41. #14556052C22semidirect
42. #14556052C22semidirect
43. #14556051C22semidirect
44. #14556049C22semidirect
48. C22⋊(C71⋊C8)C22semidirect
49. #14556039C22semidirect
51. C2×#41×C71C22semidirect
52. C4⋊Q8×C71C22semidirect
56. C22⋊Q8×C71C22semidirect
59. C4×D4×C71C22semidirect
82. #11765850D4semidirect
209. C71#3267semidirect
212. #547D71semidirect
227. C2×#41C2×D71semidirect
242. C4⋊Q8C2×D71semidirect
249. C22⋊Q8C2×D71semidirect
256. C4×D4C2×D71semidirect