Finite Group Databasebeta

← #10613113

Subgroups — #10613113

Order 640.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#65946C211yesmaximal, characteristic, complemented
2#65864C211yesmaximal, characteristic, complemented
3C5×#482C211yesmaximal, characteristic, complemented
4#65327C211yesmaximal, characteristic, complemented
5#65291C211yesmaximal, characteristic, complemented
6#65173C211yesmaximal, characteristic, complemented
7#65161C211yesmaximal, characteristic, complemented
8#65160C211yesmaximal, characteristic, complemented
9#65153C211yesmaximal, characteristic, complemented
10#65076C211yesmaximal, characteristic, complemented
11#65062C211yesmaximal, characteristic, complemented
12D5×#102C211yesmaximal, characteristic, complemented
13#65057C211yesmaximal, characteristic, complemented
14#65023C211yesmaximal, characteristic, complemented
15#65014C211yesmaximal, characteristic, complemented
16C2×D4×D5C2211yescharacteristic
17C2×#709C2211yescharacteristic
18C5×#121C2211yescharacteristic
19C5×#117C2211yescharacteristic, complemented
20#4084—21no—
21#4083—21no—
22#4082C2211yescharacteristic
23C2×#686C2211yescharacteristic
24#4076C2211yescharacteristic, complemented
25C4×C22⋊D5C2211yescharacteristic
26C22×#168C2211yescharacteristic, complemented
27C2×D40C2211yescharacteristic
28C2×#677C2211yescharacteristic, complemented
29C2×C8×D5C2211yescharacteristic
30#4043C2211yescharacteristic
31#4041C2211yescharacteristic
32#4039C2211yescharacteristic
33#4039C2211yescharacteristic, complemented
34D5×C4⋊C4C2211yescharacteristic
35#4036C2211yescharacteristic, complemented
36#4030—21no—
37#4029—21no—
38#4029—21no—
39D5×C22⋊C4—21no—
40#4022—21no—
41C4×D20—21no—
42C5×#107C2211yescharacteristic, complemented
43C5×#106C2211yescharacteristic, complemented
44C5×#102C2211yescharacteristic
45C5×#102C2211yescharacteristic
46C5×C22⋊C8C2211yescharacteristic, complemented
47#3964C2211yescharacteristic
48#3953C2211yescharacteristic
49#3952C2211yescharacteristic
50#3948C2211yescharacteristic, complemented
51#3946C2211yescharacteristic, complemented
52C8×Dic5C2211yescharacteristic, complemented
53#3942C2211yescharacteristic, complemented
54#3942C2211yescharacteristic, complemented
55#3941C2211yescharacteristic
56#3941C2211yescharacteristic
57#3941C2211yescharacteristic, complemented
58#3939C2211yescharacteristic, complemented
59#3297—51nomaximal, Sylow
60C23×D5—21no—
61C2×C5×D4—21no—
62C2×C5×D4C2311yescharacteristic
63C22×C4×C5C2311yescharacteristic
64C2×C22⋊D5—21no—
65C2×C22⋊D5—21no—
66C2×C22⋊D5—21no—
67C2×C22⋊D5D411yescharacteristic
68D4×D5—22no—
69D4×D5—41no—
70#709—22no—
71#709—41no—
72C2×D20—21no—
73C2×D20C2311yescharacteristic
74C2×D20C2311yescharacteristic
75C2×C4×D5—21no—
76C2×C4×D5—21no—
77C2×C4×D5—21no—
78C2×C4×D5—21no—
79C2×C4×D5C2311yescharacteristic
80C2×Dic10C2311yescharacteristic
81C2×C5×C8C2311yescharacteristic
82C2×C5×C8C2311yescharacteristic
83C5×C4⋊C4C2311yescharacteristic
84C5×C4⋊C4C2311yescharacteristic
85C5×C4⋊C4C2311yescharacteristic
86C5×C22⋊C4—21no—
87C5×C22⋊C4—21no—
88C42×C5—21no—
89C22⋊Dic5—21no—
90C22⋊Dic5—21no—
91C22⋊Dic5D411yescharacteristic
92#686—41no—
93#685—21no—
94#685—21no—
95#685—21no—
96#685—21no—
97#685—21no—
98#685D411yescharacteristic
99C4⋊Dic5—21no—
100C4⋊Dic5—21no—
101C4⋊Dic5C2311yescharacteristic
102#683—21no—
103#683—21no—
104#683D411yescharacteristic
105C4×Dic5C2311yescharacteristic
106C2×#168—41no—
107C2×#168C2311yescharacteristic
108C2×#168C2311yescharacteristic
109D40—22no—
110#677—41no—
111C8×D5—41no—
112#553—51no—
113#546—51no—
114#488—51no—
115#484—51no—
116#482D511yescharacteristic, complemented
117#478—51no—
118#466—51no—
119#465—51no—
120#463—51no—
121#449—51no—
122#447—51no—
123C4×D8—51no—
124C8×D4—51no—
125#416—51no—
126C2×#102—51no—
127C23×C5—21no—
128C23×C5C2×D411yescharacteristic
129C22×D5—21no—
130C22×D5—21no—
131C22×D5—42no—
132C22×D5—42no—
133C22×D5—81no—
134C22×D5C2×D411yescharacteristic
135C5×D4—21no—
136C5×D4—21no—
137C5×D4—81no—
138C2×C4×C5—21no—
139C2×C4×C5—21no—
140C2×C4×C5—21no—
141C2×C4×C5—21no—
142C2×C4×C5—21no—
143C2×C4×C5C2×D411yescharacteristic
144C2×C4×C5C2411yescharacteristic
145C22⋊D5—22no—
146C22⋊D5—42no—
147C22⋊D5—42no—
148C22⋊D5—81no—
149C2×Dic5—21no—
150C2×Dic5—21no—
151C2×Dic5C2×D411yescharacteristic
152C2×Dic5C2×D411yescharacteristic
153C2×Dic5C2×D411yescharacteristic
154D20—21no—
155D20—21no—
156D20—21no—
157D20—42no—
158D20#4112yes—
159C4×D5—21no—
160C4×D5—21no—
161C4×D5—42no—
162C4×D5—81no—
163C4×D5—81no—
164C4×D5—81no—
165Dic10—21no—
166Dic10—21no—
167C5×C8—21no—
168C5×C8—21no—
169#168—21no—
170#168—21no—
171C2×#41—51no—
172C22×D4—51no—
173C2×SD16—51no—
174C2×D8—51no—
175C2×D8—51no—
176C22×C8—51no—
177C22×C8—51no—
178#126—101no—
179#125—51no—
180#124—51no—
181#123—101no—
182#123—101no—
183C22⋊Q8—51no—
184#121—51no—
185#121—51no—
186#121—101no—
187#121—101no—
188#121C2×D511yescharacteristic
189#120—101no—
190C4×D4—51no—
191C4×D4—51no—
192C4×D4—51no—
193C4×D4—101no—
194#117C2×D511yescharacteristic, complemented
195C2×C4⋊C4—51no—
196C2×C22⋊C4—101no—
197#107—51no—
198#107C2×D511yescharacteristic, complemented
199#106C2×D511yescharacteristic, complemented
200C4⋊C8—51no—
201#103—51no—
202#103—51no—
203#103—51no—
204#102—51no—
205#102—51no—
206#102—51no—
207#102—51no—
208#102—51no—
209#102C2×D511yescharacteristic
210#102C2×D511yescharacteristic
211C22⋊C8—51no—
212C22⋊C8C2×D511yescharacteristic, complemented
213C4×C8—51no—
214C22×C5—21no—
215C22×C5—21no—
216C22×C5—21no—
217C22×C5—41no—
218C22×C5—41no—
219C22×C5—41no—
220C22×C5C22×D411yescharacteristic
221C2×D5—21no—
222C2×D5—21no—
223C2×D5—22no—
224C2×D5—41no—
225C2×D5—41no—
226C2×D5—41no—
227C2×D5—41no—
228C2×D5—41no—
229C2×D5—84no—
230C2×D5#13512yes—
231C4×C5—21no—
232C4×C5—41no—
233C4×C5—41no—
234C4×C5—41no—
235C4×C5C2×#4111yescharacteristic
236C4×C5H(5,𝔽2)11yescharacteristic
237Dic5—21no—
238Dic5—21no—
239Dic5—21no—
240Dic5—41no—
241Dic5—41no—
242C24—101no—
243#41—102no—
244#41—201no—
245C2×Q8—51no—
246C2×D4—21no—
247C2×D4—51no—
248C2×D4—51no—
249C2×D4—51no—
250C2×D4—101no—
251C2×D4—101no—
252C2×D4—102no—
253C2×D4—101no—
254C2×D4—101no—
255C2×D4—201no—
256C2×D4C22×D511yescharacteristic
257C22×C4—51no—
258C22×C4—101no—
259C22×C4—101no—
260C22×C4—101no—
261C22×C4—101no—
262C22×C4C22×D511yescharacteristic
263SD16—201no—
264D8—102no—
265D8—201no—
266C2×C8—51no—
267C2×C8—51no—
268C2×C8—201no—
269C2×C8—201no—
270C2×C8C22×D511yescharacteristic
271C2×C8C22×D511yescharacteristic
272C4⋊C4—51no—
273C4⋊C4—51no—
274C4⋊C4—101no—
275C4⋊C4—101no—
276C4⋊C4—101no—
277C4⋊C4—101no—
278C4⋊C4C22×D511yescharacteristic
279C4⋊C4C22×D511yescharacteristic
280C4⋊C4C22×D511yescharacteristic
281C22⋊C4—21no—
282C22⋊C4—21no—
283C22⋊C4—51no—
284C22⋊C4—51no—
285C22⋊C4—101no—
286C22⋊C4—101no—
287C22⋊C4—101no—
288C22⋊C4—101no—
289C22⋊C4—101no—
290C22⋊C4—101no—
291C22⋊C4—101no—
292C42—21no—
293C42—51no—
294C2×C5—41no—
295C2×C5—81no—
296C2×C5#54611yescharacteristic
297C2×C5C2×#13511yescharacteristic
298C2×C5#57611yescharacteristic
299D5—22no—
300D5—42no—
301D5—81no—
302C23—21no—
303C23—51no—
304C23—101no—
305C23—101no—
306C23—202no—
307C23—202no—
308C23—401no—
309C23D4×D511yescharacteristic
310Q8—101no—
311Q8—101no—
312D4—21no—
313D4—21no—
314D4—52no—
315D4—81no—
316D4—102no—
317D4—101no—
318D4—101no—
319D4—101no—
320D4—202no—
321D4—202no—
322D4—202no—
323D4—401no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—51no—
330C2×C4—51no—
331C2×C4—51no—
332C2×C4—101no—
333C2×C4—101no—
334C2×C4—101no—
335C2×C4—101no—
336C2×C4—202no—
337C2×C4—401no—
338C2×C4—401no—
339C2×C4—401no—
340C2×C4D4×D511yescharacteristic
341C2×C4C23×D511yescharacteristic
342C8—21no—
343C8—21no—
344C8—101no—
345C8—101no—
346C5#329711yescharacteristic, complemented, Sylow
347C22—21no—
348C22—21no—
349C22—21no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—52no—
354C22—101no—
355C22—101no—
356C22—102no—
357C22—201no—
358C22—201no—
359C22—201no—
360C22—201no—
361C22—201no—
362C22—404no—
363C22C2×D4×D511yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—101no—
369C4—101no—
370C4—101no—
371C4—201no—
372C4—201no—
373C4D5×#4111yescharacteristic
374C4#414911yescharacteristic
375C2—41no—
376C2—81no—
377C2—102no—
378C2—202no—
379C2—401no—
380C2#6598811yescharacteristic
381C2D5×#13511yescharacteristic
382C2#6610811yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#482C2
socleC22×C5C22×D4
hypercenter#482D5
hypocenterC5#3297
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #482

Complements

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

NH ≅ G/Ntype
1. #65946C2semidirect
2. #65864C2semidirect
3. C5×#482C2semidirect
4. #65327C2semidirect
5. #65291C2semidirect
6. #65173C2semidirect
7. #65161C2semidirect
8. #65160C2semidirect
9. #65153C2semidirect
10. #65076C2semidirect
11. #65062C2semidirect
12. D5×#102C2semidirect
13. #65057C2semidirect
14. #65023C2semidirect
15. #65014C2semidirect
19. C5×#117C22semidirect
24. #4076C22semidirect
26. C22×#168C22semidirect
28. C2×#677C22semidirect
33. #4039C22semidirect
35. #4036C22semidirect
42. C5×#107C22semidirect
43. C5×#106C22semidirect
46. C5×C22⋊C8C22semidirect
50. #3948C22semidirect
51. #3946C22semidirect
52. C8×Dic5C22semidirect
53. #3942C22semidirect
54. #3942C22semidirect
57. #3941C22semidirect
58. #3939C22semidirect
116. #482D5semidirect
194. #117C2×D5semidirect
198. #107C2×D5semidirect
199. #106C2×D5semidirect
212. C22⋊C8C2×D5semidirect
346. C5#3297semidirect

Maximal subgroups

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

subgroupindex
#659462
#658642
C5×#4822
#653272
#652912
#651732
#651612
#651602
#651532
#650762
#650622
D5×#1022
#650572
#650232
#650142
#32975