Finite Group Databasebeta

← #10615692

Subgroups — #10615692

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×C5×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
2#66163C211yesmaximal, characteristic, complemented
3#66140C211yesmaximal, characteristic, complemented
4C2×#4057C211yesmaximal, characteristic, complemented
5#65528C211yesmaximal, characteristic, complemented
6#65506C211yesmaximal, characteristic, complemented
7#65463C211yesmaximal, characteristic, complemented
8#65462C211yesmaximal, characteristic, complemented
9#65457C211yesmaximal, characteristic, complemented
10#65454C211yesmaximal, characteristic, complemented
11#65445C211yesmaximal, characteristic, complemented
12#65442C211yesmaximal, characteristic, complemented
13D8×Dic5C211yesmaximal, characteristic, complemented
14#65423C211yesmaximal, characteristic, complemented
15#65417C211yesmaximal, characteristic, complemented
16C2×C5×#41C2211yescharacteristic
17C22×C5×D4C2211yescharacteristic, complemented
18C2×#711C2211yescharacteristic
19C2×D4×D5C2211yescharacteristic, complemented
20C5×AGL(1,ℤ/8)—24no—
21C2×C5×SD16C2211yescharacteristic, complemented
22C2×C5×SD16C2211yescharacteristic, complemented
23C2×C5×D8C2211yescharacteristic
24C2×C5×D8C2211yescharacteristic
25C2×C5×M16C2211yescharacteristic, complemented
26#4099—21no—
27C2×C22⋊Dic5—21no—
28#4093C2211yescharacteristic, complemented
29#4092C2211yescharacteristic, complemented
30C2×#689C2211yescharacteristic, complemented
31#4085—21no—
32#4085—21no—
33#4084C2211yescharacteristic
34#4083—21no—
35#4082C2211yescharacteristic, complemented
36#4081—21no—
37D4×Dic5C2211yescharacteristic, complemented
38D4×Dic5C2211yescharacteristic, complemented
39C2×#687C2211yescharacteristic, complemented
40C2×#686C2211yescharacteristic
41C2×#686C2211yescharacteristic
42#4076C2211yescharacteristic
43#4075—21no—
44C4×C22⋊D5C2211yescharacteristic
45C2×#685—21no—
46#4070C2211yescharacteristic, complemented
47#4057—24no—
48C2×#677C2211yescharacteristic, complemented
49C2×#676C2211yescharacteristic, complemented
50#3967C2211yescharacteristic, complemented
51#3964C2211yescharacteristic, complemented
52#3964C2211yescharacteristic, complemented
53#3964C2211yescharacteristic, complemented
54C22⋊#168C2211yescharacteristic, complemented
55#3953C2211yescharacteristic, complemented
56#3952C2211yescharacteristic, complemented
57#3950C2211yescharacteristic, complemented
58#3948C2211yescharacteristic, complemented
59#3946C2211yescharacteristic, complemented
60C8×Dic5C2211yescharacteristic, complemented
61#3273—51nomaximal, Sylow
62C24×C5—21no—
63C23×D5—21no—
64C5×#41—22no—
65C5×#41—41no—
66C2×C5×Q8C2311yescharacteristic
67C2×C5×D4—22no—
68C2×C5×D4—21no—
69C2×C5×D4—21no—
70C2×C5×D4—41no—
71C2×C5×D4C2311yescharacteristic
72C2×C5×D4C2311yescharacteristic
73C2×C5×D4C2311yescharacteristic
74C22×C4×C5—21no—
75C22×C4×C5C2311yescharacteristic
76C2×C22⋊D5—21no—
77C2×C22⋊D5—21no—
78C2×C22⋊D5—21no—
79C2×C22⋊D5—21no—
80C2×C22⋊D5D411yescharacteristic
81C22×Dic5—21no—
82C22×Dic5—21no—
83C22×Dic5—21no—
84#711—22no—
85#711—41no—
86D4×D5—22no—
87D4×D5—41no—
88C2×D20C2311yescharacteristic
89C2×C4×D5C2311yescharacteristic
90C2×Dic10C2311yescharacteristic
91C5×SD16—22no—
92C5×SD16—22no—
93C5×D8—22no—
94C5×D8D414yescomplemented
95C5×M16—22no—
96C2×C5×C8C2311yescharacteristic
97C2×C5×C8C2311yescharacteristic
98C22⋊Dic5—21no—
99C22⋊Dic5—21no—
100C22⋊Dic5—21no—
101C22⋊Dic5—21no—
102C22⋊Dic5—21no—
103C22⋊Dic5—21no—
104C22⋊Dic5D411yescharacteristic
105#689—41no—
106#687—22no—
107#686—22no—
108#686—41no—
109#685—21no—
110#685—21no—
111#685—21no—
112#685—21no—
113#685D411yescharacteristic
114C4⋊Dic5C2311yescharacteristic
115C4⋊Dic5C2311yescharacteristic
116#683—21no—
117#683—21no—
118#683D411yescharacteristic
119C4×Dic5C2311yescharacteristic
120C2×#168C2311yescharacteristic
121C2×#168C2311yescharacteristic
122#677—22no—
123#676—22no—
124C2×AGL(1,ℤ/8)—51no—
125C2×AGL(1,ℤ/8)D511yescharacteristic, complemented
126#546—51no—
127#546—51no—
128#495—51no—
129#468—51no—
130#468—51no—
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×C5—21no—
141C23×C5—21no—
142C23×C5—42no—
143C23×C5—42no—
144C23×C5—81no—
145C23×C5C2×D411yescharacteristic
146C22×D5—21no—
147C22×D5—42no—
148C22×D5—42no—
149C22×D5—81no—
150C22×D5C2×D411yescharacteristic
151C5×Q8—21no—
152C5×Q8—21no—
153C5×D4—21no—
154C5×D4—21no—
155C5×D4—21no—
156C5×D4—21no—
157C5×D4—42no—
158C5×D4—42no—
159C5×D4C2×D412yes—
160C5×D4C2×D412yes—
161C2×C4×C5—21no—
162C2×C4×C5—21no—
163C2×C4×C5—21no—
164C2×C4×C5—42no—
165C2×C4×C5C2×D411yescharacteristic
166C2×C4×C5C2411yescharacteristic
167C22⋊D5—41no—
168C22⋊D5—42no—
169C22⋊D5—42no—
170C22⋊D5—81no—
171C22⋊D5—81no—
172C2×Dic5—21no—
173C2×Dic5—21no—
174C2×Dic5—21no—
175C2×Dic5—42no—
176C2×Dic5—81no—
177C2×Dic5—81no—
178C2×Dic5C2×D411yescharacteristic
179C2×Dic5C2×D411yescharacteristic
180C2×Dic5C2×D411yescharacteristic
181D20—21no—
182D20—21no—
183C4×D5—21no—
184C4×D5—21no—
185Dic10—21no—
186Dic10—21no—
187C5×C8—21no—
188C5×C8C2×D412yes—
189#168—21no—
190#168—21no—
191C2×#41—51no—
192C2×#41C2×D511yescharacteristic
193C22×D4—51no—
194C22×D4C2×D511yescharacteristic, complemented
195AGL(1,ℤ/8)—24no—
196AGL(1,ℤ/8)—104no—
197C2×Q16—51no—
198C2×SD16—51no—
199C2×SD16—51no—
200C2×SD16C2×D511yescharacteristic, complemented
201C2×SD16C2×D511yescharacteristic, complemented
202C2×D8—51no—
203C2×D8—51no—
204C2×D8C2×D511yescharacteristic
205C2×D8C2×D511yescharacteristic
206C2×M16—51no—
207C2×M16C2×D511yescharacteristic, complemented
208#124—51no—
209#124—51no—
210#123—101no—
211#123—101no—
212C22⋊Q8—51no—
213C22⋊Q8—51no—
214#121—51no—
215#121—51no—
216#121—101no—
217#121—101no—
218#120—101no—
219#120—101no—
220C4×D4—51no—
221C4×D4—51no—
222C4×D4—51no—
223C2×C22⋊C4—101no—
224C2×C22⋊C4—101no—
225#107—51no—
226C4⋊C8—51no—
227#103—51no—
228#103—51no—
229#102—51no—
230#102—51no—
231#102—51no—
232#102—51no—
233C22⋊C8—51no—
234C22⋊C8—51no—
235C4×C8—51no—
236C22×C5—22no—
237C22×C5—22no—
238C22×C5—21no—
239C22×C5—21no—
240C22×C5—21no—
241C22×C5—41no—
242C22×C5—41no—
243C22×C5—41no—
244C22×C5—41no—
245C22×C5—41no—
246C22×C5—41no—
247C22×C5—41no—
248C22×C5—84no—
249C22×C5C22×D411yescharacteristic
250C2×D5—21no—
251C2×D5—21no—
252C2×D5—21no—
253C2×D5—41no—
254C2×D5—41no—
255C2×D5—41no—
256C2×D5—84no—
257C4×C5—21no—
258C4×C5—41no—
259C4×C5C22×D411yescharacteristic
260C4×C5H(5,𝔽2)11yescharacteristic
261Dic5—21no—
262Dic5—21no—
263Dic5—21no—
264Dic5—41no—
265Dic5—41no—
266Dic5—41no—
267C24—21no—
268C24—101no—
269#41—22no—
270#41—41no—
271#41—102no—
272#41—201no—
273C2×Q8—51no—
274C2×Q8C22×D511yescharacteristic
275C2×D4—22no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—41no—
279C2×D4—51no—
280C2×D4—51no—
281C2×D4—101no—
282C2×D4—102no—
283C2×D4—101no—
284C2×D4—101no—
285C2×D4—101no—
286C2×D4—201no—
287C2×D4C22×D511yescharacteristic
288C2×D4C22×D511yescharacteristic
289C2×D4C22×D511yescharacteristic
290C22×C4—21no—
291C22×C4—51no—
292C22×C4—101no—
293C22×C4—101no—
294C22×C4—101no—
295C22×C4C22×D511yescharacteristic
296Q16—201no—
297SD16—22no—
298SD16—22no—
299SD16—102no—
300SD16—102no—
301D8—22no—
302D8—102no—
303D8—201no—
304D8C22⋊D514yescomplemented
305M16—22no—
306M16—102no—
307C2×C8—51no—
308C2×C8—51no—
309C2×C8C22×D511yescharacteristic
310C2×C8C22×D511yescharacteristic
311C4⋊C4—51no—
312C4⋊C4—51no—
313C4⋊C4—51no—
314C4⋊C4—101no—
315C4⋊C4—101no—
316C22⋊C4—51no—
317C22⋊C4—51no—
318C22⋊C4—101no—
319C22⋊C4—101no—
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—42no—
330C2×C5—41no—
331C2×C5—42no—
332C2×C5—81no—
333C2×C5D4211yescharacteristic
334C2×C5#57511yescharacteristic
335C2×C5#57611yescharacteristic
336D5—41no—
337D5—81no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—42no—
342C23—42no—
343C23—51no—
344C23—81no—
345C23—101no—
346C23—202no—
347C23—202no—
348C23—401no—
349C23D4×D511yescharacteristic
350Q8—21no—
351Q8—21no—
352Q8—101no—
353Q8—101no—
354D4—21no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—42no—
359D4—42no—
360D4—101no—
361D4—101no—
362D4—201no—
363D4—202no—
364D4—202no—
365D4—401no—
366D4—401no—
367D4C2×C22⋊D512yes—
368D4C2×C22⋊D512yes—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—42no—
373C2×C4—51no—
374C2×C4—51no—
375C2×C4—51no—
376C2×C4—101no—
377C2×C4—101no—
378C2×C4—101no—
379C2×C4—101no—
380C2×C4—101no—
381C2×C4—202no—
382C2×C4—401no—
383C2×C4—401no—
384C2×C4D4×D511yescharacteristic
385C2×C4C23×D511yescharacteristic
386C8—21no—
387C8—101no—
388C8—101no—
389C8C2×C22⋊D512yes—
390C5#327311yescharacteristic, complemented, Sylow
391C22—21no—
392C22—21no—
393C22—21no—
394C22—22no—
395C22—22no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—84no—
404C22—101no—
405C22—101no—
406C22—101no—
407C22—201no—
408C22—201no—
409C22—201no—
410C22—404no—
411C22C2×D4×D511yescharacteristic
412C4—21no—
413C4—41no—
414C4—101no—
415C4—101no—
416C4—101no—
417C4—201no—
418C4—201no—
419C4—201no—
420C4#414411yescharacteristic
421C4C22×C22⋊D511yescharacteristic
422C2—41no—
423C2—42no—
424C2—42no—
425C2—81no—
426C2—201no—
427C2—401no—
428C2#6610711yescharacteristic
429C2#6610811yescharacteristic
430C2D4×C22⋊D511yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC2×C5×AGL(1,ℤ/8)C2
socleC22×C5C22×D4
hypercenterC2×AGL(1,ℤ/8)D5
hypocenterC5#3273
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×C5×AGL(1,ℤ/8)C2semidirect
2. #66163C2semidirect
3. #66140C2semidirect
4. C2×#4057C2semidirect
5. #65528C2semidirect
6. #65506C2semidirect
7. #65463C2semidirect
8. #65462C2semidirect
9. #65457C2semidirect
10. #65454C2semidirect
11. #65445C2semidirect
12. #65442C2semidirect
13. D8×Dic5C2semidirect
14. #65423C2semidirect
15. #65417C2semidirect
17. C22×C5×D4C22semidirect
19. C2×D4×D5C22semidirect
21. C2×C5×SD16C22semidirect
22. C2×C5×SD16C22semidirect
25. C2×C5×M16C22semidirect
28. #4093C22semidirect
29. #4092C22semidirect
30. C2×#689C22semidirect
35. #4082C22semidirect
37. D4×Dic5C22semidirect
38. D4×Dic5C22semidirect
39. C2×#687C22semidirect
46. #4070C22semidirect
48. C2×#677C22semidirect
49. C2×#676C22semidirect
50. #3967C22semidirect
51. #3964C22semidirect
52. #3964C22semidirect
53. #3964C22semidirect
54. C22⋊#168C22semidirect
55. #3953C22semidirect
56. #3952C22semidirect
57. #3950C22semidirect
58. #3948C22semidirect
59. #3946C22semidirect
60. C8×Dic5C22semidirect
94. C5×D8D4semidirect
125. C2×AGL(1,ℤ/8)D5semidirect
194. C22×D4C2×D5semidirect
200. C2×SD16C2×D5semidirect
201. C2×SD16C2×D5semidirect
207. C2×M16C2×D5semidirect
304. D8C22⋊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×C5×AGL(1,ℤ/8)2
#661632
#661402
C2×#40572
#655282
#655062
#654632
#654622
#654572
#654542
#654452
#654422
D8×Dic52
#654232
#654172
#32735