Finite Group Databasebeta

← #10611996

Subgroups — #10611996

Order 640.

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
1#66158C211yesmaximal, characteristic, complemented
2C2×#4095C211yesmaximal, characteristic, complemented
3C2×#4062C211yesmaximal, characteristic, complemented
4#65964C211yesmaximal, characteristic, complemented
5C5×#452C211yesmaximal, characteristic, complemented
6#65479C211yesmaximal, characteristic, complemented
7#65476C211yesmaximal, characteristic, complemented
8#65454C211yesmaximal, characteristic, complemented
9#65341C211yesmaximal, characteristic, complemented
10#65107C211yesmaximal, characteristic, complemented
11#65096C211yesmaximal, characteristic, complemented
12#65074C211yesmaximal, characteristic, complemented
13#65070C211yesmaximal, characteristic, complemented
14#65020C211yesmaximal, characteristic, complemented
15#65017C211yesmaximal, characteristic, complemented
16C2×C5×#41C2211yescharacteristic
17C2×#713C2211yescharacteristic
18C2×#711C2211yescharacteristic, complemented
19C22×D20C2211yescharacteristic
20C2×C5×Q16C2211yescharacteristic, complemented
21C2×C5×SD16C2211yescharacteristic, complemented
22C5×C22⋊Q8C2211yescharacteristic
23#4095—24no—
24#4093C2211yescharacteristic
25#4092C2211yescharacteristic, complemented
26#4090C2211yescharacteristic, complemented
27C2×#689C2211yescharacteristic, complemented
28C2×#688C2211yescharacteristic
29C2×#688C2211yescharacteristic
30#4085—21no—
31#4081—21no—
32D4×Dic5C2211yescharacteristic, complemented
33C2×#686C2211yescharacteristic
34C2×#686C2211yescharacteristic
35#4075—21no—
36C4×C22⋊D5C2211yescharacteristic
37#4070C2211yescharacteristic
38C2×#683—21no—
39C2×#681C2211yescharacteristic
40#4062—24no—
41C2×#677C2211yescharacteristic
42C2×#677C2211yescharacteristic
43C2×C8×D5C2211yescharacteristic, complemented
44#4041—21no—
45#4041C2211yescharacteristic
46#4040—21no—
47#4039C2211yescharacteristic
48#4028—21no—
49D5×C22⋊C4—21no—
50C5×#103C2211yescharacteristic, complemented
51C5×#102C2211yescharacteristic, complemented
52C5×C22⋊C8C2211yescharacteristic
53#3967C2211yescharacteristic, complemented
54#3953C2211yescharacteristic
55#3953C2211yescharacteristic
56#3952C2211yescharacteristic, complemented
57#3947C2211yescharacteristic, complemented
58#3946C2211yescharacteristic, complemented
59#3942C2211yescharacteristic, complemented
60#3940C2211yescharacteristic, complemented
61#3267—51nomaximal, Sylow
62C23×D5—21no—
63C5×#41—22no—
64C5×#41—41no—
65C2×C5×Q8C2311yescharacteristic
66C2×C5×Q8C2311yescharacteristic
67C2×C5×D4—21no—
68C2×C5×D4C2311yescharacteristic
69C22×C4×C5—21no—
70C22×C4×C5C2311yescharacteristic
71C2×C22⋊D5—21no—
72C2×C22⋊D5D411yescharacteristic
73C22×Dic5—21no—
74C22×Dic5—21no—
75#713—22no—
76#713—41no—
77#711—22no—
78#711—41no—
79C2×D20—22no—
80C2×D20—21no—
81C2×D20—21no—
82C2×D20—21no—
83C2×D20—41no—
84C2×D20C2311yescharacteristic
85C2×D20C2311yescharacteristic
86C2×C4×D5—21no—
87C2×C4×D5—21no—
88C2×C4×D5C2311yescharacteristic
89C2×Dic10C2311yescharacteristic
90C5×Q16—41no—
91C5×SD16—22no—
92C2×C5×C8C2311yescharacteristic
93C2×C5×C8C2311yescharacteristic
94C5×C4⋊C4—21no—
95C5×C4⋊C4C2311yescharacteristic
96C5×C22⋊C4—21no—
97C22⋊Dic5—21no—
98C22⋊Dic5—21no—
99C22⋊Dic5D411yescharacteristic
100#689—22no—
101#688—22no—
102#688—22no—
103#686—22no—
104#686D414yescomplemented
105#685—21no—
106#685—21no—
107#685—21no—
108#685—21no—
109#685—21no—
110#685D411yescharacteristic
111C4⋊Dic5C2311yescharacteristic
112#683—21no—
113#683—21no—
114#683—21no—
115#683—21no—
116#683D411yescharacteristic
117C4×Dic5C2311yescharacteristic
118#681—22no—
119C2×#168C2311yescharacteristic
120C2×#168C2311yescharacteristic
121#677—22no—
122#677—41no—
123C8×D5—22no—
124C2×AGL(1,ℤ/8)—51no—
125C2×#135—51no—
126#547—51no—
127#546—51no—
128#497—51no—
129#470—51no—
130#465—51no—
131#463—51no—
132#461—51no—
133#452—51no—
134#452D511yescharacteristic, complemented
135#450—51no—
136#449—51no—
137#442—51no—
138#435—51no—
139C23×C5—21no—
140C23×C5C2×D411yescharacteristic
141C22×D5—21no—
142C22×D5—21no—
143C22×D5—42no—
144C22×D5—42no—
145C22×D5—81no—
146C22×D5C2×D411yescharacteristic
147C5×Q8—21no—
148C5×Q8—21no—
149C5×Q8—21no—
150C5×Q8—21no—
151C5×D4—21no—
152C5×D4—42no—
153C5×D4C2×D412yes—
154C2×C4×C5—21no—
155C2×C4×C5—21no—
156C2×C4×C5—21no—
157C2×C4×C5—21no—
158C2×C4×C5—21no—
159C2×C4×C5—42no—
160C2×C4×C5C2×D411yescharacteristic
161C2×C4×C5C2411yescharacteristic
162C22⋊D5—41no—
163C22⋊D5—42no—
164C2×Dic5—21no—
165C2×Dic5—21no—
166C2×Dic5—42no—
167C2×Dic5—42no—
168C2×Dic5C2×D411yescharacteristic
169C2×Dic5C2×D411yescharacteristic
170C2×Dic5C2×D411yescharacteristic
171D20—21no—
172D20—21no—
173D20—21no—
174D20—42no—
175D20—42no—
176D20—81no—
177D20C2×D412yes—
178C4×D5—21no—
179C4×D5—21no—
180C4×D5—42no—
181C4×D5—81no—
182Dic10—21no—
183Dic10—21no—
184C5×C8—21no—
185C5×C8—21no—
186#168—21no—
187#168C2×D412yes—
188C2×#41—51no—
189C2×#41—51no—
190C2×#41C2×D511yescharacteristic
191C22×D4—51no—
192AGL(1,ℤ/8)—104no—
193#135—104no—
194C2×Q16—51no—
195C2×Q16C2×D511yescharacteristic, complemented
196C2×SD16—51no—
197C2×SD16—51no—
198C2×SD16—51no—
199C2×SD16—51no—
200C2×SD16C2×D511yescharacteristic, complemented
201C2×D8—51no—
202C2×D8—51no—
203C2×M16—51no—
204C22×C8—51no—
205C4⋊Q8—51no—
206#124—51no—
207#123—101no—
208#123—101no—
209#123—101no—
210C22⋊Q8—51no—
211C22⋊Q8—51no—
212C22⋊Q8C2×D511yescharacteristic
213#121—51no—
214#121—101no—
215#121—101no—
216#120—101no—
217C4×D4—51no—
218C4×D4—51no—
219C4×D4—51no—
220C2×C4⋊C4—101no—
221C2×C22⋊C4—101no—
222#106—51no—
223C4⋊C8—51no—
224#103—51no—
225#103—51no—
226#103C2×D511yescharacteristic, complemented
227#102—51no—
228#102—51no—
229#102C2×D511yescharacteristic, complemented
230C22⋊C8—51no—
231C22⋊C8C2×D511yescharacteristic
232#97—51no—
233C22×C5—22no—
234C22×C5—21no—
235C22×C5—21no—
236C22×C5—21no—
237C22×C5—41no—
238C22×C5—41no—
239C22×C5C22×D411yescharacteristic
240C2×D5—21no—
241C2×D5—22no—
242C2×D5—21no—
243C2×D5—21no—
244C2×D5—41no—
245C2×D5—41no—
246C2×D5—41no—
247C2×D5—41no—
248C2×D5—41no—
249C2×D5—84no—
250C4×C5—21no—
251C4×C5—41no—
252C4×C5—41no—
253C4×C5—41no—
254C4×C5C22×D411yescharacteristic
255C4×C5H(5,𝔽2)11yescharacteristic
256Dic5—21no—
257Dic5—21no—
258Dic5—21no—
259Dic5—41no—
260Dic5—41no—
261C24—101no—
262#41—22no—
263#41—41no—
264#41—102no—
265#41—102no—
266#41—201no—
267#41—201no—
268C2×Q8—51no—
269C2×Q8C22×D511yescharacteristic
270C2×Q8C22×D511yescharacteristic
271C2×D4—21no—
272C2×D4—51no—
273C2×D4—51no—
274C2×D4—51no—
275C2×D4—102no—
276C2×D4—101no—
277C2×D4—101no—
278C2×D4—101no—
279C2×D4—101no—
280C2×D4—201no—
281C2×D4C22×D511yescharacteristic
282C22×C4—21no—
283C22×C4—51no—
284C22×C4—101no—
285C22×C4—101no—
286C22×C4—101no—
287C22×C4—101no—
288C22×C4C22×D511yescharacteristic
289Q16—41no—
290Q16—102no—
291SD16—22no—
292SD16—102no—
293SD16—102no—
294SD16—102no—
295SD16—201no—
296D8—54no—
297D8—102no—
298M16—102no—
299C2×C8—51no—
300C2×C8—51no—
301C2×C8—102no—
302C2×C8C22×D511yescharacteristic
303C2×C8C22×D511yescharacteristic
304C4⋊C4—21no—
305C4⋊C4—51no—
306C4⋊C4—51no—
307C4⋊C4—101no—
308C4⋊C4—101no—
309C4⋊C4—101no—
310C4⋊C4—101no—
311C4⋊C4C22×D511yescharacteristic
312C22⋊C4—21no—
313C22⋊C4—51no—
314C22⋊C4—51no—
315C22⋊C4—101no—
316C22⋊C4—101no—
317C22⋊C4—101no—
318C22⋊C4—101no—
319C22⋊C4—101no—
320C22⋊C4—101no—
321C22⋊C4—101no—
322C42—51no—
323C2×C5—41no—
324C2×C5—42no—
325C2×C5D4211yescharacteristic
326C2×C5#57511yescharacteristic
327C2×C5#57711yescharacteristic
328D5—41no—
329D5—42no—
330D5—81no—
331C23—21no—
332C23—51no—
333C23—101no—
334C23—101no—
335C23—202no—
336C23—202no—
337C23—401no—
338C23D4×D511yescharacteristic
339Q8—21no—
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—101no—
344Q8—101no—
345D4—21no—
346D4—42no—
347D4—52no—
348D4—101no—
349D4—101no—
350D4—101no—
351D4—202no—
352D4—202no—
353D4—201no—
354D4—202no—
355D4—401no—
356D4D4×D512yes—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—42no—
363C2×C4—51no—
364C2×C4—51no—
365C2×C4—51no—
366C2×C4—101no—
367C2×C4—101no—
368C2×C4—101no—
369C2×C4—101no—
370C2×C4—202no—
371C2×C4—202no—
372C2×C4—202no—
373C2×C4—401no—
374C2×C4D4×D511yescharacteristic
375C2×C4C23×D511yescharacteristic
376C8—21no—
377C8—21no—
378C8—52no—
379C8—101no—
380C5#326711yescharacteristic, complemented, Sylow
381C22—21no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—41no—
386C22—41no—
387C22—101no—
388C22—101no—
389C22—101no—
390C22—102no—
391C22—201no—
392C22—201no—
393C22—201no—
394C22—201no—
395C22—201no—
396C22—404no—
397C22C2×D4×D511yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—41no—
402C4—101no—
403C4—101no—
404C4—101no—
405C4—201no—
406C4—201no—
407C4C2×D4×D511yescharacteristic
408C4#414411yescharacteristic
409C2—41no—
410C2—42no—
411C2—201no—
412C2—202no—
413C2—401no—
414C2#6592411yescharacteristic
415C2#6610411yescharacteristic
416C2#6611111yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#452C2
socleC22×C5C22×D4
hypercenter#452D5
hypocenterC5#3267
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. #66158C2semidirect
2. C2×#4095C2semidirect
3. C2×#4062C2semidirect
4. #65964C2semidirect
5. C5×#452C2semidirect
6. #65479C2semidirect
7. #65476C2semidirect
8. #65454C2semidirect
9. #65341C2semidirect
10. #65107C2semidirect
11. #65096C2semidirect
12. #65074C2semidirect
13. #65070C2semidirect
14. #65020C2semidirect
15. #65017C2semidirect
18. C2×#711C22semidirect
20. C2×C5×Q16C22semidirect
21. C2×C5×SD16C22semidirect
25. #4092C22semidirect
26. #4090C22semidirect
27. C2×#689C22semidirect
32. D4×Dic5C22semidirect
43. C2×C8×D5C22semidirect
50. C5×#103C22semidirect
51. C5×#102C22semidirect
53. #3967C22semidirect
56. #3952C22semidirect
57. #3947C22semidirect
58. #3946C22semidirect
59. #3942C22semidirect
60. #3940C22semidirect
104. #686D4semidirect
134. #452D5semidirect
195. C2×Q16C2×D5semidirect
200. C2×SD16C2×D5semidirect
226. #103C2×D5semidirect
229. #102C2×D5semidirect
380. C5#3267semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#661582
C2×#40952
C2×#40622
#659642
C5×#4522
#654792
#654762
#654542
#653412
#651072
#650962
#650742
#650702
#650202
#650172
#32675