Finite Group Databasebeta

← #10612458

Subgroups — #10612458

Order 640.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4057C211yesmaximal, characteristic, complemented
2C22×#677C211yesmaximal, characteristic, complemented
3#65943C211yesmaximal, characteristic, complemented
4#65932C211yesmaximal, characteristic, complemented
5C5×#466C211yesmaximal, characteristic, complemented
6#65443C211yesmaximal, characteristic, complemented
7#65400C211yesmaximal, characteristic, complemented
8#65329C211yesmaximal, characteristic, complemented
9#65325C211yesmaximal, characteristic, complemented
10#65178C211yesmaximal, characteristic, complemented
11#65175C211yesmaximal, characteristic, complemented
12#65076C211yesmaximal, characteristic, complemented
13#65070C211yesmaximal, characteristic, complemented
14#65065C211yesmaximal, characteristic, complemented
15#65051C211yesmaximal, characteristic, complemented
16C2×#711C2211yescharacteristic, complemented
17C2×D4×D5—21no—
18C2×D4×D5C2211yescharacteristic
19C22×D20C2211yescharacteristic
20C22×Dic10C2211yescharacteristic, complemented
21C2×C5×D8C2211yescharacteristic, complemented
22C22×C5×C8C2211yescharacteristic, complemented
23C5×#121C2211yescharacteristic
24C5×#121C2211yescharacteristic, complemented
25#4086C2211yescharacteristic
26#4085—21no—
27#4083—21no—
28#4082C2211yescharacteristic, complemented
29D4×Dic5—21no—
30C2×#687C2211yescharacteristic, complemented
31C2×#687C2211yescharacteristic, complemented
32C2×#686C2211yescharacteristic
33C2×#686C2211yescharacteristic, complemented
34C4×C22⋊D5C2211yescharacteristic
35#4057—24no—
36C2×#677—24no—
37C2×#677C2211yescharacteristic
38C2×#677C2211yescharacteristic
39C2×#677C2211yescharacteristic, complemented
40C2×#677C2211yescharacteristic, complemented
41C2×#676C2211yescharacteristic, complemented
42#4043C2211yescharacteristic, complemented
43#4041C2211yescharacteristic
44#4039C2211yescharacteristic
45#4033C2211yescharacteristic, complemented
46#4031—21no—
47#4030—21no—
48#4028—21no—
49#4024—21no—
50C5×#107C2211yescharacteristic, complemented
51C5×#102C2211yescharacteristic, complemented
52C5×#102C2211yescharacteristic, complemented
53C22⋊#168C2211yescharacteristic, complemented
54#3952C2211yescharacteristic, complemented
55#3947C2211yescharacteristic, complemented
56#3946C2211yescharacteristic, complemented
57#3942C2211yescharacteristic, complemented
58#3942C2211yescharacteristic, complemented
59#3941C2211yescharacteristic
60#3941C2211yescharacteristic, complemented
61#3265—51nomaximal, Sylow
62C23×D5—21no—
63C23×D5—21no—
64C2×C5×D4—21no—
65C2×C5×D4—21no—
66C2×C5×D4C2311yescharacteristic
67C2×C5×D4C2311yescharacteristic
68C22×C4×C5C2311yescharacteristic
69C2×C22⋊D5—21no—
70C2×C22⋊D5—21no—
71C2×C22⋊D5—21no—
72C2×C22⋊D5—21no—
73C2×C22⋊D5D411yescharacteristic
74C22×Dic5—21no—
75C22×Dic5—21no—
76#711—22no—
77#711—41no—
78D4×D5—22no—
79D4×D5—41no—
80D4×D5—44no—
81C2×D20—22no—
82C2×D20—21no—
83C2×D20—21no—
84C2×D20—41no—
85C2×D20C2311yescharacteristic
86C2×D20C2311yescharacteristic
87C2×C4×D5—21no—
88C2×C4×D5C2311yescharacteristic
89C2×Dic10—22no—
90C2×Dic10—21no—
91C2×Dic10—41no—
92C2×Dic10C2311yescharacteristic
93C2×Dic10C2311yescharacteristic
94C5×D8—22no—
95C2×C5×C8—22no—
96C2×C5×C8C2311yescharacteristic
97C2×C5×C8C2311yescharacteristic
98C5×C4⋊C4C2311yescharacteristic
99C5×C4⋊C4C2311yescharacteristic
100C5×C22⋊C4—21no—
101C5×C22⋊C4—21no—
102C22⋊Dic5—21no—
103C22⋊Dic5—21no—
104C22⋊Dic5D411yescharacteristic
105#687—22no—
106#687—41no—
107#686—22no—
108#686—41no—
109#685—21no—
110#685—21no—
111#685—21no—
112#685D411yescharacteristic
113C4⋊Dic5—21no—
114#683—21no—
115#683D411yescharacteristic
116C4×Dic5—21no—
117C4×Dic5C2311yescharacteristic
118C2×#168C2311yescharacteristic
119C2×#168C2311yescharacteristic
120#677—22no—
121#677—22no—
122#677—22no—
123#677D414yescomplemented
124#676—22no—
125C2×AGL(1,ℤ/8)—51no—
126C22×SD16—51no—
127#548—51no—
128D42—51no—
129#496—51no—
130#468—51no—
131#466D511yescharacteristic, complemented
132#463—51no—
133#460—51no—
134#450—51no—
135#449—51no—
136#448—51no—
137#447—51no—
138#440—51no—
139#435—51no—
140C23×C5—21no—
141C23×C5—21no—
142C23×C5C2×D411yescharacteristic
143C22×D5—21no—
144C22×D5—21no—
145C22×D5—42no—
146C22×D5—42no—
147C22×D5—42no—
148C22×D5—42no—
149C22×D5—81no—
150C22×D5—81no—
151C22×D5C2×D411yescharacteristic
152C5×D4—21no—
153C5×D4—21no—
154C5×D4—21no—
155C5×D4—21no—
156C5×D4—42no—
157C5×D4—42no—
158C2×C4×C5—21no—
159C2×C4×C5—21no—
160C2×C4×C5—21no—
161C2×C4×C5—21no—
162C2×C4×C5C2×D411yescharacteristic
163C2×C4×C5C2411yescharacteristic
164C22⋊D5—22no—
165C22⋊D5—42no—
166C22⋊D5—42no—
167C22⋊D5—42no—
168C22⋊D5—81no—
169C2×Dic5—21no—
170C2×Dic5—21no—
171C2×Dic5—42no—
172C2×Dic5—42no—
173C2×Dic5C2×D411yescharacteristic
174C2×Dic5C2×D411yescharacteristic
175C2×Dic5C2×D411yescharacteristic
176D20—21no—
177D20—21no—
178D20—21no—
179D20—42no—
180D20—42no—
181D20C2×D412yes—
182C4×D5—21no—
183C4×D5—21no—
184C4×D5—42no—
185Dic10—21no—
186Dic10—21no—
187Dic10—21no—
188Dic10—42no—
189Dic10C2×D412yescomplemented
190C5×C8—21no—
191C5×C8C2×D412yescomplemented
192#168—21no—
193#168—21no—
194C2×#41—51no—
195C22×Q8—51no—
196C22×D4—51no—
197C22×D4—51no—
198C22×D4—101no—
199AGL(1,ℤ/8)—104no—
200C2×SD16—51no—
201C2×SD16—51no—
202C2×SD16—51no—
203C2×SD16—51no—
204C2×SD16—51no—
205C2×SD16—51no—
206C2×SD16—104no—
207C2×D8—51no—
208C2×D8—51no—
209C2×D8C2×D511yescharacteristic, complemented
210C2×M16—51no—
211C22×C8C2×D511yescharacteristic, complemented
212#127—51no—
213#124—51no—
214#124—101no—
215C22⋊Q8—51no—
216C22⋊Q8—101no—
217#121—51no—
218#121—101no—
219#121—101no—
220#121C2×D511yescharacteristic
221#121C2×D511yescharacteristic, complemented
222#120—101no—
223#120—101no—
224C4×Q8—51no—
225C4×D4—51no—
226C4×D4—51no—
227C4×D4—101no—
228#107C2×D511yescharacteristic, complemented
229C4⋊C8—51no—
230#103—51no—
231#103—51no—
232#102—51no—
233#102—51no—
234#102C2×D511yescharacteristic, complemented
235#102C2×D511yescharacteristic, complemented
236C22⋊C8—51no—
237C22⋊C8—51no—
238#97—51no—
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×C5AGL(1,ℤ/8)12yescomplemented
248C22×C5C22×D411yescharacteristic
249C2×D5—21no—
250C2×D5—22no—
251C2×D5—21no—
252C2×D5—21no—
253C2×D5—41no—
254C2×D5—41no—
255C2×D5—41no—
256C2×D5—41no—
257C2×D5—41no—
258C2×D5—84no—
259C2×D5—84no—
260C4×C5—21no—
261C4×C5—41no—
262C4×C5—41no—
263C4×C5C22×D411yescharacteristic
264C4×C5H(5,𝔽2)11yescharacteristic
265Dic5—22no—
266Dic5—21no—
267Dic5—21no—
268Dic5—21no—
269Dic5—41no—
270C24—101no—
271C24—101no—
272#41—102no—
273#41—201no—
274C2×Q8—51no—
275C2×Q8—51no—
276C2×Q8—102no—
277C2×Q8—101no—
278C2×Q8—201no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—51no—
282C2×D4—51no—
283C2×D4—51no—
284C2×D4—102no—
285C2×D4—101no—
286C2×D4—101no—
287C2×D4—101no—
288C2×D4—101no—
289C2×D4—102no—
290C2×D4—101no—
291C2×D4—101no—
292C2×D4—201no—
293C2×D4—201no—
294C2×D4—204no—
295C2×D4C22×D511yescharacteristic
296C2×D4C22×D511yescharacteristic
297C22×C4—51no—
298C22×C4—101no—
299C22×C4—101no—
300C22×C4—101no—
301C22×C4C22×D511yescharacteristic
302SD16—54no—
303SD16—102no—
304SD16—102no—
305SD16—102no—
306SD16—102no—
307SD16—201no—
308D8—22no—
309D8—102no—
310D8—201no—
311M16—102no—
312C2×C8—22no—
313C2×C8—51no—
314C2×C8—51no—
315C2×C8C22×D511yescharacteristic
316C2×C8C22×D511yescharacteristic
317C4⋊C4—51no—
318C4⋊C4—101no—
319C4⋊C4—101no—
320C4⋊C4C22×D511yescharacteristic
321C4⋊C4C22×D511yescharacteristic
322C22⋊C4—21no—
323C22⋊C4—21no—
324C22⋊C4—51no—
325C22⋊C4—51no—
326C22⋊C4—101no—
327C22⋊C4—101no—
328C22⋊C4—101no—
329C22⋊C4—101no—
330C22⋊C4—101no—
331C42—51no—
332C42—101no—
333C2×C5—22no—
334C2×C5—81no—
335C2×C5—81no—
336C2×C5D4211yescharacteristic
337C2×C5C2×AGL(1,ℤ/8)11yescharacteristic
338C2×C5#57711yescharacteristic
339D5—41no—
340D5—42no—
341D5—81no—
342C23—21no—
343C23—21no—
344C23—51no—
345C23—101no—
346C23—101no—
347C23—202no—
348C23—202no—
349C23—202no—
350C23—202no—
351C23—401no—
352C23—401no—
353C23D4×D511yescharacteristic
354Q8—52no—
355Q8—101no—
356Q8—101no—
357Q8—101no—
358Q8—202no—
359D4—21no—
360D4—21no—
361D4—21no—
362D4—21no—
363D4—42no—
364D4—42no—
365D4—52no—
366D4—102no—
367D4—101no—
368D4—101no—
369D4—101no—
370D4—202no—
371D4—202no—
372D4—202no—
373D4—202no—
374D4—202no—
375D4—401no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—51no—
381C2×C4—51no—
382C2×C4—51no—
383C2×C4—101no—
384C2×C4—101no—
385C2×C4—101no—
386C2×C4—101no—
387C2×C4—202no—
388C2×C4—202no—
389C2×C4—202no—
390C2×C4D4×D511yescharacteristic
391C2×C4C23×D511yescharacteristic
392C8—21no—
393C8—101no—
394C8—101no—
395C8D4×D512yescomplemented
396C5#326511yescharacteristic, complemented, Sylow
397C22—21no—
398C22—21no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—101no—
406C22—101no—
407C22—101no—
408C22—102no—
409C22—201no—
410C22—201no—
411C22—201no—
412C22—201no—
413C22—201no—
414C22—404no—
415C22—404no—
416C22#405712yescomplemented
417C22C2×D4×D511yescharacteristic
418C4—21no—
419C4—41no—
420C4—41no—
421C4—101no—
422C4—101no—
423C4—101no—
424C4—102no—
425C4—201no—
426C4C2×D4×D511yescharacteristic
427C4#414911yescharacteristic
428C2—22no—
429C2—81no—
430C2—81no—
431C2—201no—
432C2—202no—
433C2—401no—
434C2#6594311yescharacteristic
435C2C2×#405711yescharacteristic
436C2#6610411yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#466C2
socleC22×C5C22×D4
hypercenter#466D5
hypocenterC5#3265
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. C2×#4057C2semidirect
2. C22×#677C2semidirect
3. #65943C2semidirect
4. #65932C2semidirect
5. C5×#466C2semidirect
6. #65443C2semidirect
7. #65400C2semidirect
8. #65329C2semidirect
9. #65325C2semidirect
10. #65178C2semidirect
11. #65175C2semidirect
12. #65076C2semidirect
13. #65070C2semidirect
14. #65065C2semidirect
15. #65051C2semidirect
16. C2×#711C22semidirect
20. C22×Dic10C22semidirect
21. C2×C5×D8C22semidirect
22. C22×C5×C8C22semidirect
24. C5×#121C22semidirect
28. #4082C22semidirect
30. C2×#687C22semidirect
31. C2×#687C22semidirect
33. C2×#686C22semidirect
39. C2×#677C22semidirect
40. C2×#677C22semidirect
41. C2×#676C22semidirect
42. #4043C22semidirect
45. #4033C22semidirect
50. C5×#107C22semidirect
51. C5×#102C22semidirect
52. C5×#102C22semidirect
53. C22⋊#168C22semidirect
54. #3952C22semidirect
55. #3947C22semidirect
56. #3946C22semidirect
57. #3942C22semidirect
58. #3942C22semidirect
60. #3941C22semidirect
123. #677D4semidirect
131. #466D5semidirect
189. Dic10C2×D4semidirect
191. C5×C8C2×D4semidirect
209. C2×D8C2×D5semidirect
211. C22×C8C2×D5semidirect
221. #121C2×D5semidirect
228. #107C2×D5semidirect
234. #102C2×D5semidirect
235. #102C2×D5semidirect
247. C22×C5AGL(1,ℤ/8)semidirect
395. C8D4×D5semidirect
396. C5#3265semidirect
416. C22#4057semidirect

Maximal subgroups

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

subgroupindex
C2×#40572
C22×#6772
#659432
#659322
C5×#4662
#654432
#654002
#653292
#653252
#651782
#651752
#650762
#650702
#650652
#650512
#32655