Finite Group Databasebeta

← #10612570

Subgroups — #10612570

Order 640.

520 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4060C211yesmaximal, characteristic, complemented
2C2×#4054C211yesmaximal, characteristic, complemented
3#65964C211yesmaximal, characteristic, complemented
4#65946C211yesmaximal, characteristic, complemented
5C5×#468C211yesmaximal, characteristic, complemented
6#65457C211yesmaximal, characteristic, complemented
7#65417C211yesmaximal, characteristic, complemented
8#65336C211yesmaximal, characteristic, complemented
9#65325C211yesmaximal, characteristic, complemented
10#65175C211yesmaximal, characteristic, complemented
11#65161C211yesmaximal, characteristic, complemented
12#65108C211yesmaximal, characteristic, complemented
13#65099C211yesmaximal, characteristic, complemented
14#65076C211yesmaximal, characteristic, complemented
15#65062C211yesmaximal, characteristic, complemented
16C2×#713C2211yescharacteristic
17C2×D4×D5C2211yescharacteristic, complemented
18C2×#709C2211yescharacteristic
19C22×D20C2211yescharacteristic, complemented
20C2×C5×SD16C2211yescharacteristic, complemented
21C2×C5×M16C2211yescharacteristic, complemented
22C5×C22⋊Q8C2211yescharacteristic, complemented
23C5×#121C2211yescharacteristic
24#4093C2211yescharacteristic, complemented
25C2×#689C2211yescharacteristic, complemented
26C2×#688C2211yescharacteristic, complemented
27#4084—21no—
28#4083—21no—
29#4082C2211yescharacteristic, complemented
30C2×#686C2211yescharacteristic
31C2×#686C2211yescharacteristic
32C4×C22⋊D5C2211yescharacteristic
33#4060—24no—
34#4054—24no—
35C2×D40C2211yescharacteristic
36C2×D40C2211yescharacteristic
37C2×#677C2211yescharacteristic, complemented
38C2×#677C2211yescharacteristic, complemented
39C2×#676C2211yescharacteristic, complemented
40#4043C2211yescharacteristic, complemented
41#4041—21no—
42#4041C2211yescharacteristic
43#4040—21no—
44#4039C2211yescharacteristic, complemented
45#4039C2211yescharacteristic, complemented
46#4029—21no—
47#4028—21no—
48D5×C22⋊C4—21no—
49D5×C22⋊C4—21no—
50C5×#107C2211yescharacteristic, complemented
51C5×#103C2211yescharacteristic, complemented
52C5×#102C2211yescharacteristic, complemented
53C22⋊#168C2211yescharacteristic, complemented
54#3952C2211yescharacteristic, complemented
55#3946C2211yescharacteristic, complemented
56C8×Dic5C2211yescharacteristic, complemented
57#3942C2211yescharacteristic, complemented
58#3941C2211yescharacteristic, complemented
59#3941C2211yescharacteristic, complemented
60#3941C2211yescharacteristic, complemented
61#3273—51nomaximal, Sylow
62C23×D5—21no—
63C23×D5—21no—
64C2×C5×Q8C2311yescharacteristic
65C2×C5×D4—21no—
66C2×C5×D4C2311yescharacteristic
67C22×C4×C5C2311yescharacteristic
68C2×C22⋊D5—21no—
69C2×C22⋊D5—21no—
70C2×C22⋊D5D411yescharacteristic
71#713—22no—
72#713—41no—
73D4×D5—22no—
74D4×D5—41no—
75#709—22no—
76#709—41no—
77C2×D20—22no—
78C2×D20—21no—
79C2×D20—21no—
80C2×D20—21no—
81C2×D20—41no—
82C2×D20C2311yescharacteristic
83C2×D20C2311yescharacteristic
84C2×D20C2311yescharacteristic
85C2×C4×D5—21no—
86C2×C4×D5—21no—
87C2×C4×D5—21no—
88C2×C4×D5—21no—
89C2×C4×D5C2311yescharacteristic
90C2×Dic10C2311yescharacteristic
91C5×SD16—22no—
92C5×M16—22no—
93C2×C5×C8C2311yescharacteristic
94C2×C5×C8C2311yescharacteristic
95C5×C4⋊C4—21no—
96C5×C4⋊C4C2311yescharacteristic
97C5×C4⋊C4C2311yescharacteristic
98C5×C22⋊C4—21no—
99C5×C22⋊C4—21no—
100C22⋊Dic5—21no—
101C22⋊Dic5—21no—
102C22⋊Dic5D411yescharacteristic
103#689—41no—
104#688—22no—
105#686—22no—
106#686—41no—
107#685—21no—
108#685—21no—
109#685—21no—
110#685—21no—
111#685—21no—
112#685—21no—
113#685D411yescharacteristic
114C4⋊Dic5—21no—
115#683D411yescharacteristic
116C4×Dic5C2311yescharacteristic
117C2×#168C2311yescharacteristic
118C2×#168C2311yescharacteristic
119D40—22no—
120D40D414yescomplemented
121#677—22no—
122#677—22no—
123#676—22no—
124C2×AGL(1,ℤ/8)—51no—
125C2×AGL(1,ℤ/8)—51no—
126#546—51no—
127#546—51no—
128#495—51no—
129#468—51no—
130#468D511yescharacteristic, complemented
131#463—51no—
132#463—51no—
133#452—51no—
134#452—51no—
135#447—51no—
136#447—51no—
137C4×D8—51no—
138#436—51no—
139C23×C5—21no—
140C23×C5C2×D411yescharacteristic
141C22×D5—21no—
142C22×D5—21no—
143C22×D5—21no—
144C22×D5—42no—
145C22×D5—42no—
146C22×D5—42no—
147C22×D5—42no—
148C22×D5—81no—
149C22×D5—81no—
150C22×D5C2×D411yescharacteristic
151C5×Q8—21no—
152C5×Q8—21no—
153C5×D4—21no—
154C5×D4—21no—
155C5×D4—81no—
156C2×C4×C5—21no—
157C2×C4×C5—21no—
158C2×C4×C5—21no—
159C2×C4×C5—21no—
160C2×C4×C5—21no—
161C2×C4×C5C2×D411yescharacteristic
162C2×C4×C5C2411yescharacteristic
163C22⋊D5—41no—
164C22⋊D5—42no—
165C22⋊D5—42no—
166C2×Dic5—21no—
167C2×Dic5C2×D411yescharacteristic
168C2×Dic5C2×D411yescharacteristic
169C2×Dic5C2×D411yescharacteristic
170D20—21no—
171D20—21no—
172D20—21no—
173D20—21no—
174D20—42no—
175D20—42no—
176D20—81no—
177D20C2×D412yes—
178D20C2×D412yes—
179C4×D5—21no—
180C4×D5—21no—
181C4×D5—42no—
182C4×D5—42no—
183C4×D5—81no—
184C4×D5—81no—
185Dic10—21no—
186Dic10—21no—
187C5×C8—21no—
188C5×C8C2×D412yes—
189#168—21no—
190#168—21no—
191C2×#41—51no—
192C2×#41—51no—
193C22×D4—51no—
194C22×D4—51no—
195AGL(1,ℤ/8)—104no—
196AGL(1,ℤ/8)—104no—
197C2×Q16—51no—
198C2×SD16—51no—
199C2×SD16—51no—
200C2×SD16—51no—
201C2×SD16C2×D511yescharacteristic, complemented
202C2×D8—51no—
203C2×D8—51no—
204C2×D8—51no—
205C2×D8—51no—
206C2×M16—51no—
207C2×M16C2×D511yescharacteristic, complemented
208#124—51no—
209#124—51no—
210#123—101no—
211#123—101no—
212C22⋊Q8—51no—
213C22⋊Q8C2×D511yescharacteristic, complemented
214#121—51no—
215#121—101no—
216#121—101no—
217#121C2×D511yescharacteristic
218#120—101no—
219#120—101no—
220C4×D4—51no—
221C4×D4—51no—
222C4×D4—51no—
223C2×C22⋊C4—101no—
224C2×C22⋊C4—101no—
225#107C2×D511yescharacteristic, complemented
226C4⋊C8—51no—
227#103—51no—
228#103C2×D511yescharacteristic, complemented
229#102—51no—
230#102—51no—
231#102—51no—
232#102C2×D511yescharacteristic, complemented
233C22⋊C8—51no—
234C22⋊C8—51no—
235C4×C8—51no—
236C22×C5—21no—
237C22×C5—21no—
238C22×C5—21no—
239C22×C5—41no—
240C22×C5—41no—
241C22×C5—41no—
242C22×C5C22×D411yescharacteristic
243C2×D5—21no—
244C2×D5—22no—
245C2×D5—22no—
246C2×D5—21no—
247C2×D5—21no—
248C2×D5—41no—
249C2×D5—41no—
250C2×D5—41no—
251C2×D5—41no—
252C2×D5—41no—
253C2×D5—41no—
254C2×D5—41no—
255C2×D5—84no—
256C2×D5—84no—
257C4×C5—21no—
258C4×C5—41no—
259C4×C5—41no—
260C4×C5—41no—
261C4×C5C22×D411yescharacteristic
262C4×C5H(5,𝔽2)11yescharacteristic
263Dic5—21no—
264Dic5—21no—
265Dic5—21no—
266Dic5—41no—
267C24—101no—
268C24—101no—
269#41—102no—
270#41—102no—
271#41—201no—
272#41—201no—
273C2×Q8—51no—
274C2×Q8C22×D511yescharacteristic
275C2×D4—21no—
276C2×D4—51no—
277C2×D4—51no—
278C2×D4—51no—
279C2×D4—51no—
280C2×D4—102no—
281C2×D4—101no—
282C2×D4—101no—
283C2×D4—101no—
284C2×D4—102no—
285C2×D4—101no—
286C2×D4—101no—
287C2×D4—201no—
288C2×D4—201no—
289C2×D4C22×D511yescharacteristic
290C22×C4—51no—
291C22×C4—101no—
292C22×C4—101no—
293C22×C4—101no—
294C22×C4—101no—
295C22×C4C22×D511yescharacteristic
296Q16—201no—
297SD16—22no—
298SD16—102no—
299SD16—102no—
300SD16—102no—
301D8—54no—
302D8—102no—
303D8—102no—
304D8—201no—
305M16—22no—
306M16—102no—
307C2×C8—51no—
308C2×C8—51no—
309C2×C8C22×D511yescharacteristic
310C2×C8C22×D511yescharacteristic
311C4⋊C4—21no—
312C4⋊C4—51no—
313C4⋊C4—101no—
314C4⋊C4C22×D511yescharacteristic
315C4⋊C4C22×D511yescharacteristic
316C22⋊C4—21no—
317C22⋊C4—21no—
318C22⋊C4—51no—
319C22⋊C4—51no—
320C22⋊C4—101no—
321C22⋊C4—101no—
322C22⋊C4—101no—
323C22⋊C4—101no—
324C22⋊C4—101no—
325C22⋊C4—101no—
326C22⋊C4—101no—
327C22⋊C4—101no—
328C42—51no—
329C2×C5—41no—
330C2×C5—81no—
331C2×C5D4211yescharacteristic
332C2×C5#57511yescharacteristic
333C2×C5#57611yescharacteristic
334D5—41no—
335D5—42no—
336D5—42no—
337D5—81no—
338C23—21no—
339C23—51no—
340C23—101no—
341C23—101no—
342C23—101no—
343C23—202no—
344C23—202no—
345C23—202no—
346C23—202no—
347C23—401no—
348C23—401no—
349C23D4×D511yescharacteristic
350Q8—21no—
351Q8—21no—
352Q8—101no—
353Q8—101no—
354D4—21no—
355D4—21no—
356D4—52no—
357D4—52no—
358D4—81no—
359D4—101no—
360D4—101no—
361D4—101no—
362D4—101no—
363D4—202no—
364D4—202no—
365D4—201no—
366D4—202no—
367D4—202no—
368D4—401no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—51no—
375C2×C4—51no—
376C2×C4—51no—
377C2×C4—101no—
378C2×C4—101no—
379C2×C4—101no—
380C2×C4—202no—
381C2×C4—202no—
382C2×C4—401no—
383C2×C4—401no—
384C2×C4D4×D511yescharacteristic
385C2×C4C23×D511yescharacteristic
386C8—21no—
387C8—101no—
388C8—101no—
389C8D4×D512yes—
390C5#327311yescharacteristic, complemented, Sylow
391C22—21no—
392C22—21no—
393C22—21no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—101no—
398C22—101no—
399C22—101no—
400C22—102no—
401C22—102no—
402C22—201no—
403C22—201no—
404C22—201no—
405C22—201no—
406C22—201no—
407C22—201no—
408C22—201no—
409C22—404no—
410C22—404no—
411C22C2×D4×D511yescharacteristic
412C4—21no—
413C4—41no—
414C4—41no—
415C4—41no—
416C4—101no—
417C4—101no—
418C4—101no—
419C4—201no—
420C4C2×D4×D511yescharacteristic
421C4#414911yescharacteristic
422C2—41no—
423C2—81no—
424C2—201no—
425C2—202no—
426C2—202no—
427C2—401no—
428C2#6594311yescharacteristic
429C2#6610811yescharacteristic
430C2#6611111yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#468C2
socleC22×C5C22×D4
hypercenter#468D5
hypocenterC5#3273
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#4060C2semidirect
2. C2×#4054C2semidirect
3. #65964C2semidirect
4. #65946C2semidirect
5. C5×#468C2semidirect
6. #65457C2semidirect
7. #65417C2semidirect
8. #65336C2semidirect
9. #65325C2semidirect
10. #65175C2semidirect
11. #65161C2semidirect
12. #65108C2semidirect
13. #65099C2semidirect
14. #65076C2semidirect
15. #65062C2semidirect
17. C2×D4×D5C22semidirect
19. C22×D20C22semidirect
20. C2×C5×SD16C22semidirect
21. C2×C5×M16C22semidirect
22. C5×C22⋊Q8C22semidirect
24. #4093C22semidirect
25. C2×#689C22semidirect
26. C2×#688C22semidirect
29. #4082C22semidirect
37. C2×#677C22semidirect
38. C2×#677C22semidirect
39. C2×#676C22semidirect
40. #4043C22semidirect
44. #4039C22semidirect
45. #4039C22semidirect
50. C5×#107C22semidirect
51. C5×#103C22semidirect
52. C5×#102C22semidirect
53. C22⋊#168C22semidirect
54. #3952C22semidirect
55. #3946C22semidirect
56. C8×Dic5C22semidirect
57. #3942C22semidirect
58. #3941C22semidirect
59. #3941C22semidirect
60. #3941C22semidirect
120. D40D4semidirect
130. #468D5semidirect
201. C2×SD16C2×D5semidirect
207. C2×M16C2×D5semidirect
213. C22⋊Q8C2×D5semidirect
225. #107C2×D5semidirect
228. #103C2×D5semidirect
232. #102C2×D5semidirect
390. C5#3273semidirect

Maximal subgroups

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

subgroupindex
C2×#40602
C2×#40542
#659642
#659462
C5×#4682
#654572
#654172
#653362
#653252
#651752
#651612
#651082
#650992
#650762
#650622
#32735