Finite Group Databasebeta

← #10611237

Subgroups — #10611237

Order 640.

414 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#65897C211yesmaximal, characteristic, complemented
2#65884C211yesmaximal, characteristic, complemented
3C5×#435C211yesmaximal, characteristic, complemented
4#65423C211yesmaximal, characteristic, complemented
5#65420C211yesmaximal, characteristic, complemented
6#65404C211yesmaximal, characteristic, complemented
7#65402C211yesmaximal, characteristic, complemented
8#65135C211yesmaximal, characteristic, complemented
9#65133C211yesmaximal, characteristic, complemented
10#65026C211yesmaximal, characteristic, complemented
11#65023C211yesmaximal, characteristic, complemented
12#65021C211yesmaximal, characteristic, complemented
13#65012C211yesmaximal, characteristic, complemented
14#65003C211yesmaximal, characteristic, complemented
15#65000C211yesmaximal, characteristic, complemented
16C2×#709C2211yescharacteristic
17C22×D20C2211yescharacteristic
18C2×C5×M16C2211yescharacteristic
19C22×C5×C8C2211yescharacteristic, complemented
20C4×C5×D4C2211yescharacteristic
21#4084—21no—
22#4083—21no—
23#4076C2211yescharacteristic
24#4076C2211yescharacteristic
25#4076C2211yescharacteristic, complemented
26#4072C2211yescharacteristic
27#4072C2211yescharacteristic, complemented
28#4070C2211yescharacteristic
29C2×D40C2211yescharacteristic
30C2×D40C2211yescharacteristic
31C2×#677C2211yescharacteristic, complemented
32#4040—21no—
33#4038—21no—
34#4031—21no—
35#4031—21no—
36#4030—21no—
37D5×C22⋊C4—21no—
38#4021C2211yescharacteristic
39C4×D20C2211yescharacteristic
40C4×D20C2211yescharacteristic, complemented
41#4015C2211yescharacteristic, complemented
42C5×C4⋊C8C2211yescharacteristic, complemented
43C5×C22⋊C8C2211yescharacteristic
44C5×C22⋊C8C2211yescharacteristic, complemented
45C5×#97C2211yescharacteristic, complemented
46#3953C2211yescharacteristic
47#3953C2211yescharacteristic
48#3953C2211yescharacteristic
49#3953C2211yescharacteristic
50#3953C2211yescharacteristic
51#3953C2211yescharacteristic
52#3953C2211yescharacteristic, complemented
53#3950C2211yescharacteristic, complemented
54#3950C2211yescharacteristic, complemented
55#3949C2211yescharacteristic, complemented
56#3948C2211yescharacteristic, complemented
57#3948C2211yescharacteristic, complemented
58#3948C2211yescharacteristic, complemented
59#3319—51nomaximal, Sylow
60C23×D5—21no—
61C2×C5×D4D411yescharacteristic
62C22×C4×C5C2311yescharacteristic
63C22×C4×C5C2311yescharacteristic
64C2×C22⋊D5—21no—
65C2×C22⋊D5—21no—
66C2×C22⋊D5—21no—
67C2×C22⋊D5—21no—
68#709—22no—
69#709—41no—
70C2×D20—22no—
71C2×D20—21no—
72C2×D20—41no—
73C2×D20C2311yescharacteristic
74C2×D20C2311yescharacteristic
75C2×D20C2311yescharacteristic
76C2×C4×D5—21no—
77C2×C4×D5—21no—
78C2×C4×D5—21no—
79C2×Dic10—21no—
80C2×Dic10C2311yescharacteristic
81C5×M16—41no—
82C2×C5×C8—41no—
83C2×C5×C8C2311yescharacteristic
84C2×C5×C8C2311yescharacteristic
85C2×C5×C8C2311yescharacteristic
86C2×C5×C8C2311yescharacteristic
87C5×C4⋊C4D411yescharacteristic
88C5×C22⋊C4D411yescharacteristic
89C5×C22⋊C4D411yescharacteristic
90C42×C5C2311yescharacteristic
91C22⋊Dic5—21no—
92C22⋊Dic5—21no—
93C22⋊Dic5—21no—
94#685—21no—
95#685—21no—
96#685—21no—
97#685—21no—
98#685—21no—
99#685—21no—
100#685—21no—
101C4⋊Dic5—21no—
102C4⋊Dic5C2311yescharacteristic
103C4⋊Dic5C2311yescharacteristic
104C4⋊Dic5C2311yescharacteristic
105C4⋊Dic5C2311yescharacteristic
106#683—21no—
107C4×Dic5—21no—
108C4×Dic5—21no—
109D40—22no—
110D40—41no—
111#677—41no—
112#555—51no—
113#546—51no—
114#489—51no—
115#485—51no—
116#482—51no—
117#474—51no—
118#468—51no—
119#466—51no—
120#463—51no—
121#449—51no—
122#447—51no—
123#442—51no—
124#435D511yescharacteristic, complemented
125#418—51no—
126#416—51no—
127C23×C5C2×D411yescharacteristic
128C23×C5C2×D411yescharacteristic
129C22×D5—21no—
130C22×D5—21no—
131C22×D5—21no—
132C22×D5—42no—
133C22×D5—42no—
134C22×D5—81no—
135C5×D4—41no—
136C2×C4×C5—21no—
137C2×C4×C5—21no—
138C2×C4×C5—21no—
139C2×C4×C5—21no—
140C2×C4×C5C2×D411yescharacteristic
141C2×C4×C5C2×D411yescharacteristic
142C2×C4×C5C2×D411yescharacteristic
143C2×C4×C5C2×D411yescharacteristic
144C2×C4×C5C2411yescharacteristic
145C22⋊D5—42no—
146C22⋊D5—81no—
147C22⋊D5—81no—
148C22⋊D5—81no—
149C2×Dic5—21no—
150C2×Dic5—21no—
151C2×Dic5—21no—
152C2×Dic5—21no—
153C2×Dic5—21no—
154D20—21no—
155D20—21no—
156D20—21no—
157D20—21no—
158D20—21no—
159D20—42no—
160D20#4112yes—
161C4×D5—42no—
162C4×D5—42no—
163C4×D5—81no—
164Dic10—21no—
165Dic10—21no—
166Dic10—81no—
167C5×C8—21no—
168C5×C8—21no—
169C5×C8—21no—
170C5×C8—21no—
171C2×#41—51no—
172C22×D4—51no—
173C2×SD16—51no—
174C2×D8—51no—
175C2×D8—51no—
176C2×M16C2×D511yescharacteristic
177C22×C8C2×D511yescharacteristic, complemented
178C4⋊Q8—51no—
179#124—51no—
180#124—101no—
181#124—101no—
182#123—101no—
183C22⋊Q8—51no—
184#121—51no—
185#121—51no—
186#121—51no—
187#121—101no—
188#121—101no—
189#120—101no—
190C4×D4—51no—
191C4×D4—51no—
192C4×D4C2×D511yescharacteristic
193#117—51no—
194#117—51no—
195#117—101no—
196C2×C22⋊C4—101no—
197#107—51no—
198#107—51no—
199#106—51no—
200C4⋊C8C2×D511yescharacteristic, complemented
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#102—51no—
210#102—51no—
211C22⋊C8C2×D511yescharacteristic
212C22⋊C8C2×D511yescharacteristic, complemented
213#97C2×D511yescharacteristic, complemented
214C22×C5—21no—
215C22×C5—21no—
216C22×C5—21no—
217C22×C5—21no—
218C22×C5—21no—
219C22×C5—21no—
220C22×C5C22×D411yescharacteristic
221C2×D5—22no—
222C2×D5—41no—
223C2×D5—41no—
224C2×D5—41no—
225C2×D5—41no—
226C2×D5—41no—
227C2×D5—41no—
228C2×D5—41no—
229C2×D5—41no—
230C2×D5—84no—
231C4×C5—21no—
232C4×C5—21no—
233C4×C5—21no—
234C4×C5—21no—
235C4×C5C2×#4111yescharacteristic
236C4×C5H(5,𝔽2)11yescharacteristic
237Dic5—41no—
238Dic5—41no—
239Dic5—41no—
240Dic5—41no—
241Dic5—41no—
242C24—101no—
243#41—102no—
244#41—201no—
245C2×Q8—51no—
246C2×Q8—101no—
247C2×D4—51no—
248C2×D4—51no—
249C2×D4—51no—
250C2×D4—102no—
251C2×D4—101no—
252C2×D4—101no—
253C2×D4—101no—
254C2×D4—101no—
255C2×D4—101no—
256C2×D4—201no—
257C2×D4D2011yescharacteristic
258C22×C4—101no—
259C22×C4—101no—
260C22×C4—101no—
261C22×C4C22×D511yescharacteristic
262C22×C4C22×D511yescharacteristic
263SD16—201no—
264D8—102no—
265D8—201no—
266M16—41no—
267C2×C8—41no—
268C2×C8C22×D511yescharacteristic
269C2×C8C22×D511yescharacteristic
270C2×C8C22×D511yescharacteristic
271C2×C8C22×D511yescharacteristic
272C4⋊C4—51no—
273C4⋊C4—51no—
274C4⋊C4—51no—
275C4⋊C4—51no—
276C4⋊C4—101no—
277C4⋊C4—101no—
278C4⋊C4D2011yescharacteristic
279C22⋊C4—101no—
280C22⋊C4—101no—
281C22⋊C4—101no—
282C22⋊C4—101no—
283C22⋊C4—101no—
284C22⋊C4—101no—
285C22⋊C4—101no—
286C22⋊C4—101no—
287C22⋊C4—101no—
288C22⋊C4—101no—
289C22⋊C4D2011yescharacteristic
290C22⋊C4D2011yescharacteristic
291C42—101no—
292C42—101no—
293C42C22×D511yescharacteristic
294C2×C5—41no—
295C2×C5—41no—
296C2×C5#54611yescharacteristic
297C2×C5#57511yescharacteristic
298C2×C5#57611yescharacteristic
299D5—42no—
300D5—81no—
301D5—81no—
302C23—101no—
303C23—101no—
304C23—101no—
305C23—202no—
306C23—202no—
307C23—401no—
308C23C2×D2011yescharacteristic
309C23C2×D2011yescharacteristic
310Q8—101no—
311Q8—101no—
312Q8—401no—
313D4—41no—
314D4—52no—
315D4—101no—
316D4—101no—
317D4—101no—
318D4—101no—
319D4—101no—
320D4—202no—
321D4—202no—
322D4—401no—
323D4—401no—
324D4—401no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—101no—
330C2×C4—101no—
331C2×C4—101no—
332C2×C4—101no—
333C2×C4—101no—
334C2×C4—202no—
335C2×C4—202no—
336C2×C4—401no—
337C2×C4C2×D2011yescharacteristic
338C2×C4C2×D2011yescharacteristic
339C2×C4C2×D2011yescharacteristic
340C2×C4C2×D2011yescharacteristic
341C2×C4C23×D511yescharacteristic
342C8—21no—
343C8—21no—
344C8—21no—
345C8—21no—
346C5#331911yescharacteristic, complemented, Sylow
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—102no—
354C22—201no—
355C22—201no—
356C22—201no—
357C22—201no—
358C22—201no—
359C22—201no—
360C22—201no—
361C22—201no—
362C22—404no—
363C22C22×D2011yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—201no—
369C4—201no—
370C4—201no—
371C4—201no—
372C4—201no—
373C4#414411yescharacteristic
374C4D5×#4111yescharacteristic
375C2—41no—
376C2—41no—
377C2—202no—
378C2—401no—
379C2—401no—
380C2#6588811yescharacteristic
381C2#6608211yescharacteristic
382C2#6608611yescharacteristic
subgroupG/H
centerC22C22×D20
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#435C2
socleC22×C5C22×D4
hypercenter#435D5
hypocenterC5#3319
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. #65897C2semidirect
2. #65884C2semidirect
3. C5×#435C2semidirect
4. #65423C2semidirect
5. #65420C2semidirect
6. #65404C2semidirect
7. #65402C2semidirect
8. #65135C2semidirect
9. #65133C2semidirect
10. #65026C2semidirect
11. #65023C2semidirect
12. #65021C2semidirect
13. #65012C2semidirect
14. #65003C2semidirect
15. #65000C2semidirect
19. C22×C5×C8C22semidirect
25. #4076C22semidirect
27. #4072C22semidirect
31. C2×#677C22semidirect
40. C4×D20C22semidirect
41. #4015C22semidirect
42. C5×C4⋊C8C22semidirect
44. C5×C22⋊C8C22semidirect
45. C5×#97C22semidirect
52. #3953C22semidirect
53. #3950C22semidirect
54. #3950C22semidirect
55. #3949C22semidirect
56. #3948C22semidirect
57. #3948C22semidirect
58. #3948C22semidirect
124. #435D5semidirect
177. C22×C8C2×D5semidirect
200. C4⋊C8C2×D5semidirect
212. C22⋊C8C2×D5semidirect
213. #97C2×D5semidirect
346. C5#3319semidirect

Maximal subgroups

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

subgroupindex
#658972
#658842
C5×#4352
#654232
#654202
#654042
#654022
#651352
#651332
#650262
#650232
#650212
#650122
#650032
#650002
#33195