Finite Group Databasebeta

← #10611856

Subgroups — #10611856

Order 640.

556 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#66168C211yesmaximal, characteristic, complemented
2C2×#4095C211yesmaximal, characteristic, complemented
3C2×#4060C211yesmaximal, characteristic, complemented
4#65943C211yesmaximal, characteristic, complemented
5C5×#449C211yesmaximal, characteristic, complemented
6#65452C211yesmaximal, characteristic, complemented
7#65445C211yesmaximal, characteristic, complemented
8#65440C211yesmaximal, characteristic, complemented
9#65328C211yesmaximal, characteristic, complemented
10#65100C211yesmaximal, characteristic, complemented
11#65077C211yesmaximal, characteristic, complemented
12#65075C211yesmaximal, characteristic, complemented
13#65071C211yesmaximal, characteristic, complemented
14#65021C211yesmaximal, characteristic, complemented
15#65018C211yesmaximal, characteristic, complemented
16C2×C5×#41C2211yescharacteristic, complemented
17C2×#713C2211yescharacteristic, complemented
18C2×D4×D5—21no—
19C2×D4×D5C2211yescharacteristic
20C22×D20C2211yescharacteristic
21C2×C5×SD16C2211yescharacteristic, complemented
22C2×C5×D8C2211yescharacteristic, complemented
23C5×#121C2211yescharacteristic
24#4095—24no—
25Q8×Dic5C2211yescharacteristic, complemented
26C2×#688C2211yescharacteristic
27C2×#688C2211yescharacteristic, complemented
28#4086—21no—
29#4086C2211yescharacteristic
30#4086C2211yescharacteristic, complemented
31#4084—21no—
32#4084C2211yescharacteristic, complemented
33#4083—21no—
34C2×#686C2211yescharacteristic
35C2×#686C2211yescharacteristic, complemented
36C2×#686C2211yescharacteristic, complemented
37#4076—21no—
38#4076C2211yescharacteristic, complemented
39C4×C22⋊D5—21no—
40C4×C22⋊D5C2211yescharacteristic
41C2×#681C2211yescharacteristic, complemented
42#4060—24no—
43C2×D40C2211yescharacteristic
44C2×D40C2211yescharacteristic, complemented
45C2×#676C2211yescharacteristic, complemented
46#4041C2211yescharacteristic
47#4039C2211yescharacteristic
48#4030—21no—
49#4028—21no—
50C5×#103C2211yescharacteristic, complemented
51C5×#102C2211yescharacteristic, complemented
52C5×C22⋊C8C2211yescharacteristic, complemented
53#3964C2211yescharacteristic, complemented
54#3953C2211yescharacteristic
55#3953C2211yescharacteristic, complemented
56#3952C2211yescharacteristic, complemented
57#3946C2211yescharacteristic, complemented
58C8×Dic5C2211yescharacteristic, complemented
59#3941C2211yescharacteristic, complemented
60#3940C2211yescharacteristic, complemented
61#3274—51nomaximal, Sylow
62C23×D5—21no—
63C23×D5—21no—
64C5×#41—22no—
65C5×#41—41no—
66C2×C5×Q8C2311yescharacteristic
67C2×C5×D4—21no—
68C2×C5×D4—21no—
69C2×C5×D4C2311yescharacteristic
70C2×C5×D4C2311yescharacteristic
71C22×C4×C5—21no—
72C22×C4×C5C2311yescharacteristic
73C2×C22⋊D5—21no—
74C2×C22⋊D5—21no—
75C2×C22⋊D5—21no—
76C2×C22⋊D5—21no—
77C2×C22⋊D5—21no—
78C2×C22⋊D5—21no—
79C2×C22⋊D5—21no—
80C2×C22⋊D5D411yescharacteristic
81#713—22no—
82#713—41no—
83D4×D5—22no—
84D4×D5—41no—
85D4×D5—44no—
86C2×D20—22no—
87C2×D20—21no—
88C2×D20—21no—
89C2×D20—21no—
90C2×D20—41no—
91C2×D20C2311yescharacteristic
92C2×D20C2311yescharacteristic
93C2×D20C2311yescharacteristic
94C2×C4×D5—21no—
95C2×C4×D5—21no—
96C2×C4×D5C2311yescharacteristic
97C5×SD16—22no—
98C5×D8—41no—
99C2×C5×C8C2311yescharacteristic
100C2×C5×C8C2311yescharacteristic
101C5×C4⋊C4C2311yescharacteristic
102C5×C22⋊C4—21no—
103C22⋊Dic5—21no—
104C22⋊Dic5D411yescharacteristic
105#688—22no—
106#688D414yescomplemented
107#686—22no—
108#686—22no—
109#686—22no—
110#685—21no—
111#685—21no—
112#685—21no—
113#685D411yescharacteristic
114C4⋊Dic5—21no—
115C4⋊Dic5C2311yescharacteristic
116#683D411yescharacteristic
117C4×Dic5—21no—
118C4×Dic5C2311yescharacteristic
119#681—22no—
120C2×#168C2311yescharacteristic
121C2×#168C2311yescharacteristic
122D40—22no—
123D40—41no—
124#676—22no—
125C2×AGL(1,ℤ/8)—51no—
126C2×AGL(1,ℤ/8)—51no—
127#550—51no—
128D42—51no—
129#493—51no—
130#469—51no—
131#469—51no—
132#460—51no—
133#459—51no—
134#449—51no—
135#449D511yescharacteristic, complemented
136#447—51no—
137#447—51no—
138C4×SD16—51no—
139#436—51no—
140C23×C5—21no—
141C23×C5—21no—
142C23×C5C2×D411yescharacteristic
143C22×D5—21no—
144C22×D5—21no—
145C22×D5—21no—
146C22×D5—42no—
147C22×D5—42no—
148C22×D5—42no—
149C22×D5—42no—
150C22×D5—81no—
151C22×D5—81no—
152C22×D5C2×D411yescharacteristic
153C5×Q8—21no—
154C5×Q8C2×D412yescomplemented
155C5×D4—21no—
156C5×D4—21no—
157C5×D4—21no—
158C5×D4—21no—
159C5×D4—42no—
160C5×D4—42no—
161C2×C4×C5—21no—
162C2×C4×C5—21no—
163C2×C4×C5—21no—
164C2×C4×C5—21no—
165C2×C4×C5—42no—
166C2×C4×C5C2×D411yescharacteristic
167C2×C4×C5C2411yescharacteristic
168C22⋊D5—22no—
169C22⋊D5—42no—
170C22⋊D5—42no—
171C22⋊D5—42no—
172C22⋊D5—81no—
173C22⋊D5—81no—
174C22⋊D5—81no—
175C22⋊D5—81no—
176C2×Dic5—21no—
177C2×Dic5C2×D411yescharacteristic
178C2×Dic5C2×D411yescharacteristic
179C2×Dic5C2×D411yescharacteristic
180D20—21no—
181D20—21no—
182D20—21no—
183D20—21no—
184D20—21no—
185D20—42no—
186D20—42no—
187D20—42no—
188D20C2×D412yes—
189C4×D5—21no—
190C4×D5—21no—
191C4×D5—42no—
192C4×D5—42no—
193C5×C8—21no—
194C5×C8—21no—
195#168—21no—
196#168C2×D412yescomplemented
197C2×#41—51no—
198C2×#41C2×D511yescharacteristic, complemented
199C22×D4—51no—
200C22×D4—51no—
201C22×D4—101no—
202AGL(1,ℤ/8)—104no—
203AGL(1,ℤ/8)—104no—
204C2×SD16—51no—
205C2×SD16—51no—
206C2×SD16C2×D511yescharacteristic, complemented
207C2×D8—51no—
208C2×D8—51no—
209C2×D8—51no—
210C2×D8—51no—
211C2×D8—51no—
212C2×D8C2×D511yescharacteristic, complemented
213C2×M16—51no—
214C2×M16—51no—
215#127—51no—
216#127—51no—
217#127—101no—
218#121—51no—
219#121—51no—
220#121—51no—
221#121—101no—
222#121—101no—
223#121—101no—
224#121C2×D511yescharacteristic
225#120—101no—
226#120—101no—
227C4×Q8—51no—
228C4×D4—51no—
229C4×D4—51no—
230C4×D4—101no—
231#106—51no—
232C4⋊C8—51no—
233#103C2×D511yescharacteristic, complemented
234#102—51no—
235#102—51no—
236#102—51no—
237#102—51no—
238#102C2×D511yescharacteristic, complemented
239C22⋊C8—51no—
240C22⋊C8C2×D511yescharacteristic, complemented
241C4×C8—51no—
242C22×C5—21no—
243C22×C5—21no—
244C22×C5—21no—
245C22×C5—41no—
246C22×C5—41no—
247C22×C5—41no—
248C22×C5—41no—
249C22×C5—41no—
250C22×C5—41no—
251C22×C5C22×D411yescharacteristic
252C2×D5—21no—
253C2×D5—22no—
254C2×D5—21no—
255C2×D5—21no—
256C2×D5—41no—
257C2×D5—41no—
258C2×D5—41no—
259C2×D5—41no—
260C2×D5—41no—
261C2×D5—41no—
262C2×D5—41no—
263C2×D5—41no—
264C2×D5—84no—
265C2×D5—84no—
266C4×C5—22no—
267C4×C5—21no—
268C4×C5—41no—
269C4×C5C22×D411yescharacteristic
270C4×C5H(5,𝔽2)11yescharacteristic
271Dic5—21no—
272Dic5—21no—
273Dic5—41no—
274Dic5AGL(1,ℤ/8)12yescomplemented
275C24—101no—
276C24—101no—
277#41—22no—
278#41—41no—
279#41—102no—
280#41—201no—
281C2×Q8C22×D511yescharacteristic
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—51no—
285C2×D4—51no—
286C2×D4—51no—
287C2×D4—51no—
288C2×D4—102no—
289C2×D4—101no—
290C2×D4—101no—
291C2×D4—101no—
292C2×D4—101no—
293C2×D4—101no—
294C2×D4—101no—
295C2×D4—101no—
296C2×D4—102no—
297C2×D4—101no—
298C2×D4—101no—
299C2×D4—101no—
300C2×D4—201no—
301C2×D4—201no—
302C2×D4—204no—
303C2×D4C22×D511yescharacteristic
304C2×D4C22×D511yescharacteristic
305C22×C4—21no—
306C22×C4—51no—
307C22×C4—101no—
308C22×C4—101no—
309C22×C4C22×D511yescharacteristic
310SD16—22no—
311SD16—54no—
312SD16—102no—
313D8—41no—
314D8—102no—
315D8—102no—
316D8—102no—
317D8—102no—
318D8—201no—
319M16—102no—
320M16—102no—
321C2×C8—51no—
322C2×C8—51no—
323C2×C8C22×D511yescharacteristic
324C2×C8C22×D511yescharacteristic
325C4⋊C4—51no—
326C4⋊C4—51no—
327C4⋊C4—101no—
328C4⋊C4C22×D511yescharacteristic
329C22⋊C4—21no—
330C22⋊C4—51no—
331C22⋊C4—51no—
332C22⋊C4—101no—
333C22⋊C4—101no—
334C22⋊C4—101no—
335C22⋊C4—101no—
336C42—51no—
337C42—101no—
338C2×C5—41no—
339C2×C5—81no—
340C2×C5—81no—
341C2×C5D4211yescharacteristic
342C2×C5C2×AGL(1,ℤ/8)11yescharacteristic
343C2×C5#57611yescharacteristic
344D5—41no—
345D5—42no—
346D5—81no—
347D5—81no—
348C23—21no—
349C23—21no—
350C23—51no—
351C23—101no—
352C23—101no—
353C23—101no—
354C23—202no—
355C23—202no—
356C23—202no—
357C23—202no—
358C23—401no—
359C23—401no—
360C23D4×D511yescharacteristic
361Q8—21no—
362Q8D4×D512yescomplemented
363D4—21no—
364D4—21no—
365D4—21no—
366D4—21no—
367D4—42no—
368D4—42no—
369D4—52no—
370D4—102no—
371D4—101no—
372D4—101no—
373D4—101no—
374D4—101no—
375D4—101no—
376D4—202no—
377D4—202no—
378D4—202no—
379D4—202no—
380D4—202no—
381D4—202no—
382D4—401no—
383D4—401no—
384D4—401no—
385D4—401no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—42no—
391C2×C4—51no—
392C2×C4—51no—
393C2×C4—51no—
394C2×C4—101no—
395C2×C4—101no—
396C2×C4—101no—
397C2×C4—202no—
398C2×C4—202no—
399C2×C4D4×D511yescharacteristic
400C2×C4C23×D511yescharacteristic
401C8—21no—
402C8—21no—
403C8—52no—
404C8—101no—
405C5#327411yescharacteristic, complemented, Sylow
406C22—21no—
407C22—21no—
408C22—21no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—101no—
416C22—101no—
417C22—101no—
418C22—102no—
419C22—201no—
420C22—201no—
421C22—201no—
422C22—201no—
423C22—201no—
424C22—201no—
425C22—201no—
426C22—201no—
427C22—404no—
428C22—404no—
429C22C2×D4×D511yescharacteristic
430C4—21no—
431C4—22no—
432C4—41no—
433C4—52no—
434C4—101no—
435C4—101no—
436C4—201no—
437C4C2×D4×D511yescharacteristic
438C4#414411yescharacteristic
439C2—41no—
440C2—81no—
441C2—81no—
442C2—201no—
443C2—202no—
444C2—401no—
445C2—401no—
446C2#6592411yescharacteristic
447C2#6610311yescharacteristic
448C2D5×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#449C2
socleC22×C5C22×D4
hypercenter#449D5
hypocenterC5#3274
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #66168C2semidirect
2. C2×#4095C2semidirect
3. C2×#4060C2semidirect
4. #65943C2semidirect
5. C5×#449C2semidirect
6. #65452C2semidirect
7. #65445C2semidirect
8. #65440C2semidirect
9. #65328C2semidirect
10. #65100C2semidirect
11. #65077C2semidirect
12. #65075C2semidirect
13. #65071C2semidirect
14. #65021C2semidirect
15. #65018C2semidirect
16. C2×C5×#41C22semidirect
17. C2×#713C22semidirect
21. C2×C5×SD16C22semidirect
22. C2×C5×D8C22semidirect
25. Q8×Dic5C22semidirect
27. C2×#688C22semidirect
30. #4086C22semidirect
32. #4084C22semidirect
35. C2×#686C22semidirect
36. C2×#686C22semidirect
38. #4076C22semidirect
41. C2×#681C22semidirect
44. C2×D40C22semidirect
45. C2×#676C22semidirect
50. C5×#103C22semidirect
51. C5×#102C22semidirect
52. C5×C22⋊C8C22semidirect
53. #3964C22semidirect
55. #3953C22semidirect
56. #3952C22semidirect
57. #3946C22semidirect
58. C8×Dic5C22semidirect
59. #3941C22semidirect
60. #3940C22semidirect
106. #688D4semidirect
135. #449D5semidirect
154. C5×Q8C2×D4semidirect
196. #168C2×D4semidirect
198. C2×#41C2×D5semidirect
206. C2×SD16C2×D5semidirect
212. C2×D8C2×D5semidirect
233. #103C2×D5semidirect
238. #102C2×D5semidirect
240. C22⋊C8C2×D5semidirect
274. Dic5AGL(1,ℤ/8)semidirect
362. Q8D4×D5semidirect
405. C5#3274semidirect

Maximal subgroups

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

subgroupindex
#661682
C2×#40952
C2×#40602
#659432
C5×#4492
#654522
#654452
#654402
#653282
#651002
#650772
#650752
#650712
#650212
#650182
#32745