Finite Group Databasebeta

← #10611972

Subgroups — #10611972

Order 640.

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
1#66163C211yesmaximal, characteristic
2C2×#4097C211yesmaximal, characteristic, complemented
3C2×D5×SD16C211yesmaximal, characteristic, complemented
4#65966C211yesmaximal, characteristic, complemented
5C5×#452C211yesmaximal, characteristic, complemented
6#65479C211yesmaximal, characteristic, complemented
7#65475C211yesmaximal, characteristic, complemented
8#65451C211yesmaximal, characteristic, complemented
9#65338C211yesmaximal, characteristic, complemented
10#65102C211yesmaximal, characteristic, complemented
11#65087C211yesmaximal, characteristic, complemented
12#65065C211yesmaximal, characteristic, complemented
13#65046C211yesmaximal, characteristic, complemented
14#65027C211yesmaximal, characteristic, complemented
15#65017C211yesmaximal, characteristic, complemented
16C2×C5×#41C2211yescharacteristic
17C2×Q8×D5C2211yescharacteristic
18C2×D4×D5C2211yescharacteristic
19C22×Dic10C2211yescharacteristic, complemented
20C2×C5×Q16C2211yescharacteristic, complemented
21C2×C5×SD16C2211yescharacteristic
22C5×C22⋊Q8C2211yescharacteristic, complemented
23#4097—24no—
24#4093C2211yescharacteristic
25#4092C2211yescharacteristic
26#4090C2211yescharacteristic, complemented
27C2×#689C2211yescharacteristic, complemented
28C2×#689C2211yescharacteristic, complemented
29C2×#688C2211yescharacteristic
30#4085—21no—
31#4083—21no—
32D4×Dic5C2211yescharacteristic
33C2×#687C2211yescharacteristic
34C2×#687C2211yescharacteristic
35#4076C2211yescharacteristic
36#4075—21no—
37C4×C22⋊D5C2211yescharacteristic
38C2×#685—21no—
39C2×#681C2211yescharacteristic, complemented
40D5×SD16—24no—
41C2×#677C2211yescharacteristic
42C2×#677C2211yescharacteristic
43C2×C8×D5C2211yescharacteristic
44#4043C2211yescharacteristic
45#4042—21no—
46#4034—21no—
47#4033C2211yescharacteristic, complemented
48#4027—21no—
49#4024—21no—
50C5×#103C2211yescharacteristic, complemented
51C5×#102C2211yescharacteristic
52C5×C22⋊C8C2211yescharacteristic, complemented
53#3967C2211yescharacteristic, complemented
54#3952C2211yescharacteristic
55#3948C2211yescharacteristic, complemented
56#3948C2211yescharacteristic, complemented
57#3947C2211yescharacteristic, complemented
58#3946C2211yescharacteristic, complemented
59#3941C2211yescharacteristic
60#3939C2211yescharacteristic, complemented
61#3266—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⋊D5—21no—
73C2×C22⋊D5D411yescharacteristic
74C22×Dic5—21no—
75C22×Dic5—21no—
76Q8×D5—22no—
77Q8×D5—41no—
78D4×D5—22no—
79D4×D5—41no—
80C2×D20C2311yescharacteristic
81C2×C4×D5—21no—
82C2×C4×D5C2311yescharacteristic
83C2×Dic10—22no—
84C2×Dic10—21no—
85C2×Dic10—21no—
86C2×Dic10—41no—
87C2×Dic10C2311yescharacteristic
88C2×Dic10C2311yescharacteristic
89C5×Q16—41no—
90C5×SD16—22no—
91C2×C5×C8C2311yescharacteristic
92C2×C5×C8C2311yescharacteristic
93C5×C4⋊C4—21no—
94C5×C4⋊C4C2311yescharacteristic
95C5×C22⋊C4—21no—
96C22⋊Dic5—21no—
97C22⋊Dic5D411yescharacteristic
98#689—22no—
99#689—22no—
100#688—22no—
101#687—22no—
102#687D414yescomplemented
103#685—21no—
104#685—21no—
105#685—21no—
106#685—21no—
107#685—21no—
108#685D411yescharacteristic
109C4⋊Dic5—21no—
110C4⋊Dic5C2311yescharacteristic
111#683—21no—
112#683—21no—
113#683—21no—
114#683—21no—
115#683D411yescharacteristic
116C4×Dic5—21no—
117C4×Dic5C2311yescharacteristic
118#681—22no—
119C2×#168C2311yescharacteristic
120C2×#168C2311yescharacteristic
121#677—22no—
122#677—41no—
123C8×D5—22no—
124C2×#137—51no—
125C22×SD16—51no—
126D4×Q8—51no—
127#546—51no—
128#497—51no—
129#468—51no—
130#467—51no—
131#464—51no—
132#461—51no—
133#452D511yescharacteristic, complemented
134#451—51no—
135#450—51no—
136#448—51no—
137#440—51no—
138#435—51no—
139C23×C5—21no—
140C23×C5C2×D411yescharacteristic
141C22×D5—21no—
142C22×D5—42no—
143C22×D5—42no—
144C22×D5—81no—
145C22×D5C2×D411yescharacteristic
146C5×Q8—21no—
147C5×Q8—21no—
148C5×Q8—21no—
149C5×Q8—21no—
150C5×D4—21no—
151C5×D4—42no—
152C5×D4C2×D412yes—
153C2×C4×C5—21no—
154C2×C4×C5—21no—
155C2×C4×C5—21no—
156C2×C4×C5—21no—
157C2×C4×C5—21no—
158C2×C4×C5—42no—
159C2×C4×C5C2×D411yescharacteristic
160C2×C4×C5C2411yescharacteristic
161C22⋊D5—22no—
162C22⋊D5—42no—
163C22⋊D5—81no—
164C2×Dic5—21no—
165C2×Dic5—21no—
166C2×Dic5—21no—
167C2×Dic5—42no—
168C2×Dic5—81no—
169C2×Dic5C2×D411yescharacteristic
170C2×Dic5C2×D411yescharacteristic
171C2×Dic5C2×D411yescharacteristic
172D20—21no—
173D20—21no—
174C4×D5—21no—
175C4×D5—21no—
176C4×D5—42no—
177Dic10—21no—
178Dic10—21no—
179Dic10—21no—
180Dic10—42no—
181Dic10—42no—
182Dic10C2×D412yes—
183C5×C8—21no—
184C5×C8—21no—
185#168—21no—
186#168C2×D412yes—
187C2×#41C2×D511yescharacteristic
188C22×Q8—51no—
189C22×Q8—51no—
190C22×D4—51no—
191#137—104no—
192C2×Q16—51no—
193C2×Q16—51no—
194C2×Q16C2×D511yescharacteristic, complemented
195C2×SD16—51no—
196C2×SD16—51no—
197C2×SD16—51no—
198C2×SD16—51no—
199C2×SD16—51no—
200C2×SD16—104no—
201C2×SD16C2×D511yescharacteristic
202C2×M16—51no—
203C22×C8—51no—
204C4⋊Q8—51no—
205C4⋊Q8—101no—
206#124—51no—
207#123—101no—
208C22⋊Q8—51no—
209C22⋊Q8—51no—
210C22⋊Q8—101no—
211C22⋊Q8—101no—
212C22⋊Q8C2×D511yescharacteristic, complemented
213#121—51no—
214#121—101no—
215#120—101no—
216C4×Q8—51no—
217C4×D4—51no—
218C4×D4—51no—
219C4×D4—101no—
220C2×C22⋊C4—101no—
221#107—51no—
222C4⋊C8—51no—
223#103—51no—
224#103—51no—
225#103—51no—
226#103C2×D511yescharacteristic, complemented
227#102—51no—
228#102C2×D511yescharacteristic
229C22⋊C8—51no—
230C22⋊C8C2×D511yescharacteristic, complemented
231#97—51no—
232C22×C5—21no—
233C22×C5—22no—
234C22×C5—21no—
235C22×C5—21no—
236C22×C5—41no—
237C22×C5—41no—
238C22×C5C22×D411yescharacteristic
239C2×D5—21no—
240C2×D5—21no—
241C2×D5—41no—
242C2×D5—41no—
243C2×D5—41no—
244C2×D5—84no—
245C2×D5#13712yescomplemented
246C4×C5—21no—
247C4×C5—41no—
248C4×C5—41no—
249C4×C5—41no—
250C4×C5C22×D411yescharacteristic
251C4×C5H(5,𝔽2)11yescharacteristic
252Dic5—22no—
253Dic5—21no—
254Dic5—21no—
255Dic5—21no—
256Dic5—41no—
257Dic5—41no—
258C24—101no—
259#41—22no—
260#41—41no—
261C2×Q8—51no—
262C2×Q8—51no—
263C2×Q8—101no—
264C2×Q8—102no—
265C2×Q8—101no—
266C2×Q8—102no—
267C2×Q8—201no—
268C2×Q8—201no—
269C2×Q8C22×D511yescharacteristic
270C2×Q8C22×D511yescharacteristic
271C2×D4—21no—
272C2×D4—51no—
273C2×D4—51no—
274C2×D4—101no—
275C2×D4—102no—
276C2×D4—101no—
277C2×D4—201no—
278C2×D4C22×D511yescharacteristic
279C22×C4—21no—
280C22×C4—51no—
281C22×C4—101no—
282C22×C4—101no—
283C22×C4—101no—
284C22×C4C22×D511yescharacteristic
285Q16—41no—
286Q16—102no—
287Q16—102no—
288SD16—22no—
289SD16—54no—
290SD16—102no—
291SD16—102no—
292SD16—102no—
293SD16—201no—
294M16—102no—
295C2×C8—51no—
296C2×C8—51no—
297C2×C8—102no—
298C2×C8C22×D511yescharacteristic
299C2×C8C22×D511yescharacteristic
300C4⋊C4—21no—
301C4⋊C4—51no—
302C4⋊C4—51no—
303C4⋊C4—101no—
304C4⋊C4—101no—
305C4⋊C4—101no—
306C4⋊C4—101no—
307C4⋊C4—101no—
308C4⋊C4C22×D511yescharacteristic
309C22⋊C4—21no—
310C22⋊C4—51no—
311C22⋊C4—51no—
312C22⋊C4—101no—
313C22⋊C4—101no—
314C22⋊C4—101no—
315C22⋊C4—101no—
316C22⋊C4—101no—
317C22⋊C4—101no—
318C42—51no—
319C42—101no—
320C2×C5—42no—
321C2×C5—41no—
322C2×C5D4211yescharacteristic
323C2×C5C2×#13711yescharacteristic
324C2×C5#57711yescharacteristic
325D5—22no—
326D5—81no—
327C23—21no—
328C23—51no—
329C23—101no—
330C23—202no—
331C23—202no—
332C23—401no—
333C23D4×D511yescharacteristic
334Q8—21no—
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—52no—
339Q8—101no—
340Q8—101no—
341Q8—101no—
342Q8—202no—
343Q8—202no—
344D4—21no—
345D4—42no—
346D4—102no—
347D4—101no—
348D4—101no—
349D4—202no—
350D4—401no—
351D4D4×D512yes—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—42no—
358C2×C4—51no—
359C2×C4—51no—
360C2×C4—51no—
361C2×C4—101no—
362C2×C4—101no—
363C2×C4—101no—
364C2×C4—101no—
365C2×C4—101no—
366C2×C4—202no—
367C2×C4—202no—
368C2×C4—401no—
369C2×C4D4×D511yescharacteristic
370C2×C4C23×D511yescharacteristic
371C8—21no—
372C8—21no—
373C8—52no—
374C8—101no—
375C5#326611yescharacteristic, complemented, Sylow
376C22—21no—
377C22—21no—
378C22—21no—
379C22—22no—
380C22—41no—
381C22—41no—
382C22—52no—
383C22—101no—
384C22—101no—
385C22—201no—
386C22—201no—
387C22—201no—
388C22—404no—
389C22C2×D4×D511yescharacteristic
390C4—21no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—101no—
395C4—101no—
396C4—101no—
397C4—102no—
398C4—201no—
399C4—201no—
400C4C2×D4×D511yescharacteristic
401C4#414411yescharacteristic
402C2—41no—
403C2—42no—
404C2—102no—
405C2—401no—
406C2#6592411yescharacteristic
407C2#6610411yescharacteristic
408C2D5×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#452C2
socleC22×C5C22×D4
prime radical#66163C2
hypercenter#452D5
hypocenterC5#3266
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
2. C2×#4097C2semidirect
3. C2×D5×SD16C2semidirect
4. #65966C2semidirect
5. C5×#452C2semidirect
6. #65479C2semidirect
7. #65475C2semidirect
8. #65451C2semidirect
9. #65338C2semidirect
10. #65102C2semidirect
11. #65087C2semidirect
12. #65065C2semidirect
13. #65046C2semidirect
14. #65027C2semidirect
15. #65017C2semidirect
19. C22×Dic10C22semidirect
20. C2×C5×Q16C22semidirect
22. C5×C22⋊Q8C22semidirect
26. #4090C22semidirect
27. C2×#689C22semidirect
28. C2×#689C22semidirect
39. C2×#681C22semidirect
47. #4033C22semidirect
50. C5×#103C22semidirect
52. C5×C22⋊C8C22semidirect
53. #3967C22semidirect
55. #3948C22semidirect
56. #3948C22semidirect
57. #3947C22semidirect
58. #3946C22semidirect
60. #3939C22semidirect
102. #687D4semidirect
133. #452D5semidirect
194. C2×Q16C2×D5semidirect
212. C22⋊Q8C2×D5semidirect
226. #103C2×D5semidirect
230. C22⋊C8C2×D5semidirect
245. C2×D5#137semidirect
375. C5#3266semidirect

Maximal subgroups

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

subgroupindex
#661632
C2×#40972
C2×D5×SD162
#659662
C5×#4522
#654792
#654752
#654512
#653382
#651022
#650872
#650652
#650462
#650272
#650172
#32665