Finite Group Databasebeta

← #10618616

Subgroups — #10618616

Order 640.

1540 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4149C211yesmaximal, characteristic, complemented
2C2×#4147C211yesmaximal, characteristic, complemented
3C5×#548C211yesmaximal, characteristic, complemented
4#66165C211yesmaximal, characteristic, complemented
5#66151C211yesmaximal, characteristic, complemented
6#66010C213yesmaximal, complemented
7D5×#124C213yesmaximal, complemented
8#65964C213yesmaximal, complemented
9#65962C213yesmaximal, complemented
10#65946C213yesmaximal, complemented
11#65941C213yesmaximal, complemented
12#65910C211yesmaximal, characteristic, complemented
13C4×#713C211yesmaximal, characteristic, complemented
14#65884C213yesmaximal, complemented
15#65882C213yesmaximal, complemented
16C2×C5×#41C2211yescharacteristic
17C22×C5×Q8C2211yescharacteristic, complemented
18#4149—28no—
19#4147—28no—
20C2×#713C2211yescharacteristic
21C2×#713C2211yescharacteristic
22C2×#713C2213yescomplemented
23C2×Q8×D5C2211yescharacteristic, complemented
24C2×Q8×D5C2213yescomplemented
25C2×D4×D5C2213yes—
26C2×D4×D5C2213yes—
27C2×#709C2213yes—
28C2×#709C2213yescomplemented
29C2×#709C2213yescomplemented
30C22×D20C2213yes—
31C5×#124C2213yescomplemented
32C5×C22⋊Q8C2213yescomplemented
33C5×#121C2213yescomplemented
34C4×C5×Q8C2211yescharacteristic, complemented
35C4×C5×D4C2213yescomplemented
36#4093C2213yescomplemented
37#4092C2213yescomplemented
38Q8×Dic5C2211yescharacteristic, complemented
39#4086C2213yescomplemented
40#4084C2213yescomplemented
41#4084C2213yescomplemented
42#4083C2216yescomplemented
43#4082C2213yescomplemented
44#4076C2213yescomplemented
45C4×C22⋊D5C2213yescomplemented
46C4×C22⋊D5C2213yescomplemented
47#4070C2213yescomplemented
48#4043C2213yescomplemented
49#4041C2213yescomplemented
50#4041C2213yescomplemented
51#4040C2216yescomplemented
52#4039C2213yescomplemented
53#4038C2213yescomplemented
54#4034C2213yescomplemented
55#4031C2211yescharacteristic, complemented
56#4031C2213yescomplemented
57#4031C2216yescomplemented
58#4030C2213yescomplemented
59#4030C2213yescomplemented
60#4029C2216yescomplemented
61#4028C2216yescomplemented
62D5×C22⋊C4C2216yescomplemented
63D5×C22⋊C4C2216yescomplemented
64#4021C2213yescomplemented
65C4×D20C2213yescomplemented
66C4×D20C2213yescomplemented
67C42×D5C2213yescomplemented
68#3500—51nomaximal, Sylow
69C23×D5C2316yes—
70C5×#41—24no—
71C2×C5×Q8—24no—
72C2×C5×Q8C2313yes—
73C2×C5×Q8C2311yescharacteristic
74C2×C5×D4C2313yes—
75C2×C5×D4C2313yes—
76C22×C4×C5C2313yes—
77C22×C4×C5C2313yes—
78C2×C22⋊D5C2313yes—
79C2×C22⋊D5C2313yes—
80C2×C22⋊D5C2313yes—
81C2×C22⋊D5C2311yescharacteristic
82C2×C22⋊D5C2313yes—
83C2×C22⋊D5C2313yes—
84#713—24no—
85#713—212no—
86#713D418yescomplemented
87Q8×D5—24no—
88Q8×D5—212no—
89D4×D5—212no—
90D4×D5—212no—
91#709—212no—
92#709—212no—
93#709—212no—
94C2×D20—212no—
95C2×D20C2313yes—
96C2×D20C2313yes—
97C2×D20C2313yes—
98C2×D20C2313yes—
99C2×D20C2313yes—
100C2×C4×D5C2313yes—
101C2×C4×D5C2313yes—
102C2×C4×D5C2316yes—
103C2×C4×D5C2313yes—
104C2×C4×D5C2313yes—
105C2×C4×D5C2313yes—
106C2×Dic10C2311yescharacteristic
107C2×Dic10C2313yes—
108C2×Dic10C2313yes—
109C5×C4⋊C4C2313yes—
110C5×C4⋊C4C2313yes—
111C5×C22⋊C4C2311yescharacteristic
112C5×C22⋊C4C2313yes—
113C5×C22⋊C4C2316yes—
114C42×C5C2313yes—
115C22⋊Dic5C2313yes—
116C22⋊Dic5C2316yes—
117C22⋊Dic5C2311yescharacteristic
118#685C2311yescharacteristic
119#685C2313yes—
120#685C2316yes—
121#685C2316yes—
122#685C2313yes—
123#685C2316yes—
124#685C2311yescharacteristic
125C4⋊Dic5C2313yes—
126C4⋊Dic5C2313yes—
127#683C2313yes—
128#683C2313yes—
129C4×Dic5C2311yescharacteristic
130C4×Dic5C2313yes—
131C4×Dic5C2313yes—
132C2×2−1+4—51no—
133C2×H(5,𝔽2)—51no—
134#548—51no—
135#548—51no—
136#548—51no—
137#548D511yescharacteristic, complemented
138#546—53no—
139#546—53no—
140#546—53no—
141#546—53no—
142#536—53no—
143#535—53no—
144#532—53no—
145C2×#124—53no—
146C4×#41—51no—
147C23×C5C2413yes—
148C23×C5C2411yescharacteristic
149C22×D5—212no—
150C22×D5—46no—
151C22×D5—46no—
152C22×D5C2413yes—
153C22×D5C2411yescharacteristic
154C22×D5C2413yes—
155C5×Q8—26no—
156C5×Q8C2×D414yes—
157C5×D4—26no—
158C5×D4—43no—
159C2×C4×C5—26no—
160C2×C4×C5—26no—
161C2×C4×C5C2413yes—
162C2×C4×C5C2411yescharacteristic
163C2×C4×C5C2411yescharacteristic
164C2×C4×C5C2413yes—
165C2×C4×C5C2413yes—
166C22⋊D5—26no—
167C22⋊D5—26no—
168C22⋊D5—22no—
169C22⋊D5—26no—
170C22⋊D5—43no—
171C22⋊D5—43no—
172C2×Dic5C2411yescharacteristic
173C2×Dic5C2413yes—
174C2×Dic5C2411yescharacteristic
175C2×Dic5C2413yes—
176C2×Dic5C2411yescharacteristic
177D20—26no—
178D20—26no—
179D20—26no—
180D20—43no—
181D20C2×D4112yes—
182C4×D5—212no—
183C4×D5—26no—
184C4×D5—26no—
185C4×D5—43no—
186C4×D5—43no—
187C4×D5C2×D4112yes—
188Dic10—26no—
189Dic10—22no—
190Dic10—26no—
1912−1+4—108no—
192H(5,𝔽2)—108no—
193C2×#41—51no—
194C2×#41—53no—
195C2×#41—51no—
196C2×#41—53no—
197C2×#41—53no—
198C2×#41—53no—
199C2×#41C2×D511yescharacteristic
200C22×Q8—53no—
201C22×Q8—51no—
202C22×Q8C2×D511yescharacteristic, complemented
203C22×D4—53no—
204C22×D4—53no—
205C22×D4—53no—
206C4⋊Q8—53no—
207#127—53no—
208#124—53no—
209#124—53no—
210#124—53no—
211#124—56no—
212#124—51no—
213#124—53no—
214#124C2×D513yescomplemented
215#123—56no—
216#123—56no—
217C22⋊Q8—53no—
218C22⋊Q8—53no—
219C22⋊Q8—53no—
220C22⋊Q8C2×D513yescomplemented
221#121—53no—
222#121—53no—
223#121—53no—
224#121—53no—
225#121—53no—
226#121—53no—
227#121—53no—
228#121C2×D513yescomplemented
229#120—56no—
230#120—56no—
231C4×Q8—51no—
232C4×Q8C2×D511yescharacteristic, complemented
233C4×D4—53no—
234C4×D4—53no—
235C4×D4—53no—
236C4×D4—53no—
237C4×D4—53no—
238C4×D4C2×D513yescomplemented
239#117—53no—
240C2×C22⋊C4—56no—
241C2×C22⋊C4—56no—
242C2×C42—53no—
243C22×C5—21no—
244C22×C5—23no—
245C22×C5—21no—
246C22×C5—23no—
247C22×C5—21no—
248C22×C5—23no—
249C22×C5C2511yescharacteristic
250C2×D5—23no—
251C2×D5—23no—
252C2×D5—21no—
253C2×D5—23no—
254C2×D5—21no—
255C2×D5—23no—
256C2×D5—21no—
257C2×D5—23no—
258C2×D5—812no—
259C2×D5C22×D416yes—
260C4×C5—23no—
261C4×C5—21no—
262C4×C5—21no—
263C4×C5—23no—
264C4×C5C22×D416yes—
265Dic5—23no—
266Dic5—21no—
267Dic5—21no—
268Dic5—23no—
269Dic5C22×D412yes—
270C24—56no—
271#41—24no—
272#41—58no—
273#41—1012no—
274#41—1012no—
275#41—1012no—
276#41—1012no—
277#41—104no—
278C2×Q8—24no—
279C2×Q8—53no—
280C2×Q8—53no—
281C2×Q8—51no—
282C2×Q8—1012no—
283C2×Q8—104no—
284C2×Q8C22×D513yes—
285C2×Q8C22×D511yescharacteristic
286C2×D4—53no—
287C2×D4—53no—
288C2×D4—53no—
289C2×D4—51no—
290C2×D4—53no—
291C2×D4—53no—
292C2×D4—53no—
293C2×D4—53no—
294C2×D4—53no—
295C2×D4—53no—
296C2×D4—53no—
297C2×D4—1012no—
298C2×D4—1012no—
299C2×D4—1012no—
300C2×D4C22×D513yes—
301C2×D4C22×D513yes—
302C22×C4—53no—
303C22×C4—56no—
304C22×C4—53no—
305C22×C4—53no—
306C22×C4—53no—
307C22×C4—53no—
308C22×C4C22×D513yes—
309C22×C4C22×D513yes—
310C4⋊C4—53no—
311C4⋊C4—53no—
312C4⋊C4—53no—
313C4⋊C4—53no—
314C4⋊C4C22×D513yes—
315C4⋊C4C22×D513yes—
316C22⋊C4—53no—
317C22⋊C4—56no—
318C22⋊C4—51no—
319C22⋊C4—53no—
320C22⋊C4—53no—
321C22⋊C4—56no—
322C22⋊C4—56no—
323C22⋊C4—56no—
324C22⋊C4—51no—
325C22⋊C4—51no—
326C22⋊C4C22×D513yes—
327C22⋊C4C22×D516yes—
328C22⋊C4C22×D511yescharacteristic
329C42—51no—
330C42—53no—
331C42—53no—
332C42C22×D513yes—
333C2×C5—43no—
334C2×C5—41no—
335C2×C5C23×D411yescharacteristic
336C2×C5#58511yescharacteristic
337C2×C5#58511yescharacteristic
338D5—26no—
339D5—41no—
340D5—43no—
341C23—53no—
342C23—53no—
343C23—51no—
344C23—1012no—
345C23—206no—
346C23—206no—
347C23C23×D513yes—
348C23C23×D511yescharacteristic
349Q8—26no—
350Q8—106no—
351Q8—102no—
352Q8—106no—
353Q8D4×D514yes—
354D4—26no—
355D4—43no—
356D4—512no—
357D4—106no—
358D4—106no—
359D4—106no—
360D4—102no—
361D4—106no—
362D4—106no—
363D4—106no—
364D4—203no—
365D4—203no—
366D4—203no—
367C2×C4—26no—
368C2×C4—26no—
369C2×C4—51no—
370C2×C4—512no—
371C2×C4—51no—
372C2×C4—53no—
373C2×C4—53no—
374C2×C4—51no—
375C2×C4—1012no—
376C2×C4—106no—
377C2×C4—106no—
378C2×C4—203no—
379C2×C4—203no—
380C2×C4C23×D511yescharacteristic
381C2×C4C23×D513yes—
382C2×C4C23×D513yes—
383C2×C4C23×D511yescharacteristic
384C2×C4C23×D513yes—
385C5#350011yescharacteristic, complemented, Sylow
386C22—21no—
387C22—23no—
388C22—21no—
389C22—23no—
390C22—21no—
391C22—23no—
392C22—56no—
393C22—103no—
394C22—103no—
395C22—101no—
396C22—103no—
397C22—101no—
398C22—103no—
399C22—101no—
400C22—103no—
401C22—4012no—
402C22C24×D511yescharacteristic
403C4—21no—
404C4—23no—
405C4—23no—
406C4—21no—
407C4—52no—
408C4—101no—
409C4—103no—
410C4—101no—
411C4—103no—
412C4C2×D4×D516yes—
413C2—43no—
414C2—41no—
415C2—106no—
416C2—203no—
417C2—201no—
418C2C22×D4×D511yescharacteristic
419C2#6628311yescharacteristic
420C2#6628711yescharacteristic
subgroupG/H
center, FrattiniC22C24×D5
commutator, socle, cosocleC22×C5C25
FittingC5×#548C2
hypercenter#548D5
hypocenterC5#3500
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#4149C2semidirect
2. C2×#4147C2semidirect
3. C5×#548C2semidirect
4. #66165C2semidirect
5. #66151C2semidirect
6. #66010C2semidirect
7. D5×#124C2semidirect
8. #65964C2semidirect
9. #65962C2semidirect
10. #65946C2semidirect
11. #65941C2semidirect
12. #65910C2semidirect
13. C4×#713C2semidirect
14. #65884C2semidirect
15. #65882C2semidirect
17. C22×C5×Q8C22semidirect
22. C2×#713C22semidirect
23. C2×Q8×D5C22semidirect
24. C2×Q8×D5C22semidirect
28. C2×#709C22semidirect
29. C2×#709C22semidirect
31. C5×#124C22semidirect
32. C5×C22⋊Q8C22semidirect
33. C5×#121C22semidirect
34. C4×C5×Q8C22semidirect
35. C4×C5×D4C22semidirect
36. #4093C22semidirect
37. #4092C22semidirect
38. Q8×Dic5C22semidirect
39. #4086C22semidirect
40. #4084C22semidirect
41. #4084C22semidirect
42. #4083C22semidirect
43. #4082C22semidirect
44. #4076C22semidirect
45. C4×C22⋊D5C22semidirect
46. C4×C22⋊D5C22semidirect
47. #4070C22semidirect
48. #4043C22semidirect
49. #4041C22semidirect
50. #4041C22semidirect
51. #4040C22semidirect
52. #4039C22semidirect
53. #4038C22semidirect
54. #4034C22semidirect
55. #4031C22semidirect
56. #4031C22semidirect
57. #4031C22semidirect
58. #4030C22semidirect
59. #4030C22semidirect
60. #4029C22semidirect
61. #4028C22semidirect
62. D5×C22⋊C4C22semidirect
63. D5×C22⋊C4C22semidirect
64. #4021C22semidirect
65. C4×D20C22semidirect
66. C4×D20C22semidirect
67. C42×D5C22semidirect
86. #713D4semidirect
137. #548D5semidirect
202. C22×Q8C2×D5semidirect
214. #124C2×D5semidirect
220. C22⋊Q8C2×D5semidirect
228. #121C2×D5semidirect
232. C4×Q8C2×D5semidirect
238. C4×D4C2×D5semidirect
385. C5#3500semidirect

Maximal subgroups

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

subgroupindex
C2×#41492
C2×#41472
C5×#5482
#661652
#661512
#660102
D5×#1242
#659642
#659622
#659462
#659412
#659102
C4×#7132
#658842
#658822
#35005