Finite Group Databasebeta

← #10612336

Subgroups — #10612336

Order 640.

410 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D5×#124C211yesmaximal, characteristic, complemented
2#65914C211yesmaximal, characteristic, complemented
3C5×#464C211yesmaximal, characteristic, complemented
4#65480C211yesmaximal, characteristic, complemented
5#65475C211yesmaximal, characteristic, complemented
6#65458C211yesmaximal, characteristic, complemented
7#65457C211yesmaximal, characteristic, complemented
8#65346C211yesmaximal, characteristic, complemented
9#65317C211yesmaximal, characteristic, complemented
10#65133C211yesmaximal, characteristic, complemented
11#65122C211yesmaximal, characteristic, complemented
12#65095C211yesmaximal, characteristic, complemented
13#65089C211yesmaximal, characteristic, complemented
14#65062C211yesmaximal, characteristic
15#65057C211yesmaximal, characteristic, complemented
16C2×Q8×D5C2211yescharacteristic
17C2×D4×D5C2211yescharacteristic
18C2×C5×Q16C2211yescharacteristic, complemented
19C2×C5×SD16C2211yescharacteristic
20C5×#124C2211yescharacteristic
21C4×C5×Q8C2211yescharacteristic, complemented
22#4093C2211yescharacteristic
23#4093C2211yescharacteristic, complemented
24#4092C2211yescharacteristic
25C2×#689C2211yescharacteristic, complemented
26C2×#688C2211yescharacteristic
27C2×#688C2211yescharacteristic, complemented
28#4082C2211yescharacteristic
29C2×#686C2211yescharacteristic
30C2×D40C2211yescharacteristic
31C2×#677C2211yescharacteristic, complemented
32#4044—21no—
33#4041C2211yescharacteristic
34#4040—21no—
35#4036C2211yescharacteristic, complemented
36#4031—21no—
37#4031—21no—
38D5×C22⋊C4—21no—
39D5×C22⋊C4—21no—
40#4021C2211yescharacteristic
41C4×D20—21no—
42C4×D20C2211yescharacteristic, complemented
43#4018—21no—
44C42×D5C2211yescharacteristic
45C5×C4⋊C8C2211yescharacteristic, complemented
46C5×#103C2211yescharacteristic, complemented
47C5×#102C2211yescharacteristic
48#3967C2211yescharacteristic, complemented
49#3964C2211yescharacteristic
50#3953C2211yescharacteristic
51#3952C2211yescharacteristic
52#3952C2211yescharacteristic
53#3948C2211yescharacteristic, complemented
54C8×Dic5C2211yescharacteristic, complemented
55C8×Dic5C2211yescharacteristic, complemented
56#3942C2211yescharacteristic, complemented
57#3941C2211yescharacteristic
58C4⋊#168C2211yescharacteristic, complemented
59#3211—51nomaximal, Sylow
60C23×D5—21no—
61C2×C5×Q8C2311yescharacteristic
62C2×C5×Q8C2311yescharacteristic
63C2×C5×D4C2311yescharacteristic
64C2×C22⋊D5—21no—
65Q8×D5—41no—
66Q8×D5—41no—
67D4×D5—41no—
68D4×D5—41no—
69C2×D20—21no—
70C2×D20C2311yescharacteristic
71C2×D20C2311yescharacteristic
72C2×C4×D5—21no—
73C2×C4×D5—21no—
74C2×C4×D5D411yescharacteristic
75C2×C4×D5D411yescharacteristic
76C2×C4×D5C2311yescharacteristic
77C2×Dic10—21no—
78C2×Dic10C2311yescharacteristic
79C5×Q16—41no—
80C5×SD16—41no—
81C2×C5×C8C2311yescharacteristic
82C2×C5×C8C2311yescharacteristic
83C5×C4⋊C4—21no—
84C5×C4⋊C4C2311yescharacteristic
85C5×C22⋊C4—21no—
86C5×C22⋊C4—21no—
87C42×C5—21no—
88C42×C5C2311yescharacteristic
89C22⋊Dic5—21no—
90C22⋊Dic5—21no—
91#689—41no—
92#688—41no—
93#688—41no—
94#686—41no—
95#685—21no—
96#685—21no—
97#685—21no—
98#685—21no—
99#685—21no—
100#685—21no—
101#685—21no—
102#685—21no—
103#685—21no—
104C4⋊Dic5—21no—
105C4⋊Dic5C2311yescharacteristic
106#683—21no—
107#683—21no—
108C4×Dic5D411yescharacteristic
109C4×Dic5D411yescharacteristic
110C4×Dic5C2311yescharacteristic
111C2×#168C2311yescharacteristic
112C2×#168C2311yescharacteristic
113D40—41no—
114#677—41no—
115#529—51no—
116C2×#124—51no—
117#495—51no—
118#495—51no—
119#488—51no—
120#488—51no—
121#464—51no—
122#464D511yescharacteristic, complemented
123#463—51no—
124#463—51no—
125#451—51no—
126#450—51no—
127#448—51no—
128#447—51no—
129#431—51no—
130C23×C5—21no—
131C22×D5—21no—
132C22×D5—21no—
133C22×D5—42no—
134C22×D5—42no—
135C22×D5—42no—
136C22×D5C2×D411yescharacteristic
137C5×Q8—21no—
138C5×Q8—21no—
139C5×Q8—21no—
140C5×Q8—21no—
141C5×D4—21no—
142C5×D4—21no—
143C2×C4×C5—21no—
144C2×C4×C5—21no—
145C2×C4×C5—21no—
146C2×C4×C5C2×D411yescharacteristic
147C2×C4×C5C2×D411yescharacteristic
148C2×C4×C5C2411yescharacteristic
149C22⋊D5—42no—
150C2×Dic5—21no—
151C2×Dic5—21no—
152C2×Dic5C2×D411yescharacteristic
153C2×Dic5C2×D411yescharacteristic
154C2×Dic5C2×D411yescharacteristic
155D20—21no—
156D20—21no—
157D20—21no—
158D20—21no—
159D20—42no—
160C4×D5—22no—
161C4×D5—21no—
162C4×D5—21no—
163C4×D5—41no—
164C4×D5—42no—
165C4×D5—42no—
166Dic10—21no—
167Dic10—21no—
168Dic10—42no—
169C5×C8—21no—
170C5×C8—21no—
171#168—21no—
172#168—21no—
173C22×Q8—51no—
174C22×D4—51no—
175C2×Q16—51no—
176C2×Q16C2×D511yescharacteristic, complemented
177C2×SD16—51no—
178C2×SD16—51no—
179C2×SD16—51no—
180C2×SD16C2×D511yescharacteristic
181C2×D8—51no—
182C2×D8—51no—
183#126—101no—
184#125—51no—
185#124—51no—
186#124—51no—
187#124—51no—
188#124—51no—
189#124—101no—
190#124—101no—
191#124C2×D511yescharacteristic
192#123—101no—
193C22⋊Q8—51no—
194#121—51no—
195C4×Q8C2×D511yescharacteristic, complemented
196C4×D4—51no—
197C4×D4—101no—
198#117—101no—
199C2×C22⋊C4—101no—
200C2×C22⋊C4—101no—
201C2×C42—51no—
202C4⋊C8—51no—
203C4⋊C8C2×D511yescharacteristic, complemented
204#103—51no—
205#103—51no—
206#103—51no—
207#103C2×D511yescharacteristic, complemented
208#102—51no—
209#102—51no—
210#102—51no—
211#102C2×D511yescharacteristic
212C22⋊C8—51no—
213C22⋊C8—51no—
214C4×C8—51no—
215C4×C8—51no—
216C22×C5—41no—
217C22×C5—41no—
218C22×C5—41no—
219C22×C5C22×D411yescharacteristic
220C2×D5—21no—
221C2×D5—21no—
222C2×D5—41no—
223C2×D5—41no—
224C2×D5—41no—
225C2×D5—41no—
226C2×D5—41no—
227C2×D5—41no—
228C2×D5—84no—
229C2×D5#13512yes—
230C4×C5—21no—
231C4×C5—21no—
232C4×C5—41no—
233C4×C5—41no—
234C4×C5—41no—
235C4×C5H(5,𝔽2)11yescharacteristic
236C4×C5H(5,𝔽2)11yescharacteristic
237Dic5—21no—
238Dic5—21no—
239Dic5—21no—
240Dic5—41no—
241Dic5—41no—
242C24—101no—
243C2×Q8—51no—
244C2×Q8—101no—
245C2×Q8—201no—
246C2×Q8—201no—
247C2×Q8C22×D511yescharacteristic
248C2×Q8C22×D511yescharacteristic
249C2×D4—51no—
250C2×D4—51no—
251C2×D4—101no—
252C2×D4—101no—
253C2×D4—201no—
254C2×D4—201no—
255C2×D4C22×D511yescharacteristic
256C22×C4—51no—
257C22×C4—51no—
258C22×C4—51no—
259C22×C4—101no—
260C22×C4—101no—
261Q16—41no—
262Q16—201no—
263SD16—41no—
264SD16—201no—
265SD16—201no—
266SD16—201no—
267D8—201no—
268D8—201no—
269C2×C8—51no—
270C2×C8—51no—
271C2×C8C22×D511yescharacteristic
272C2×C8C22×D511yescharacteristic
273C4⋊C4—21no—
274C4⋊C4—51no—
275C4⋊C4—101no—
276C4⋊C4—101no—
277C4⋊C4—101no—
278C4⋊C4C22×D511yescharacteristic
279C22⋊C4—21no—
280C22⋊C4—21no—
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⋊C4—101no—
290C22⋊C4—101no—
291C22⋊C4—101no—
292C42—21no—
293C42—51no—
294C42—51no—
295C42—51no—
296C42C22×D511yescharacteristic
297C2×C5—81no—
298C2×C5#53511yescharacteristic
299C2×C5C2×#13511yescharacteristic
300C2×C5#57511yescharacteristic
301D5—22no—
302D5—81no—
303D5—81no—
304C23—21no—
305C23—51no—
306C23—101no—
307C23—101no—
308C23—202no—
309C23—202no—
310C23—202no—
311Q8—21no—
312Q8—21no—
313Q8—21no—
314Q8—21no—
315Q8—101no—
316Q8—101no—
317Q8—202no—
318D4—21no—
319D4—21no—
320D4—101no—
321D4—101no—
322D4—101no—
323D4—101no—
324D4—202no—
325D4—202no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—51no—
330C2×C4—51no—
331C2×C4—51no—
332C2×C4—102no—
333C2×C4—101no—
334C2×C4—101no—
335C2×C4—101no—
336C2×C4—101no—
337C2×C4—201no—
338C2×C4—202no—
339C2×C4—202no—
340C2×C4D4×D511yescharacteristic
341C2×C4D4×D511yescharacteristic
342C2×C4C23×D511yescharacteristic
343C8—21no—
344C8—21no—
345C8—101no—
346C8—101no—
347C5#321111yescharacteristic, complemented, Sylow
348C22—41no—
349C22—41no—
350C22—41no—
351C22—52no—
352C22—101no—
353C22—101no—
354C22—201no—
355C22—201no—
356C22—201no—
357C22—201no—
358C22—201no—
359C22—201no—
360C22—404no—
361C22C2×D4×D511yescharacteristic
362C4—21no—
363C4—21no—
364C4—41no—
365C4—41no—
366C4—41no—
367C4—101no—
368C4—101no—
369C4—101no—
370C4—201no—
371C4—201no—
372C4#414411yescharacteristic
373C4#414911yescharacteristic
374C2—81no—
375C2—102no—
376C2—401no—
377C2—401no—
378C2#6594111yescharacteristic
379C2D5×#13511yescharacteristic
380C2#6611111yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#464C2
socleC22×C5C22×D4
prime radical#65062C2
hypercenter#464D5
hypocenterC5#3211
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. D5×#124C2semidirect
2. #65914C2semidirect
3. C5×#464C2semidirect
4. #65480C2semidirect
5. #65475C2semidirect
6. #65458C2semidirect
7. #65457C2semidirect
8. #65346C2semidirect
9. #65317C2semidirect
10. #65133C2semidirect
11. #65122C2semidirect
12. #65095C2semidirect
13. #65089C2semidirect
15. #65057C2semidirect
18. C2×C5×Q16C22semidirect
21. C4×C5×Q8C22semidirect
23. #4093C22semidirect
25. C2×#689C22semidirect
27. C2×#688C22semidirect
31. C2×#677C22semidirect
35. #4036C22semidirect
42. C4×D20C22semidirect
45. C5×C4⋊C8C22semidirect
46. C5×#103C22semidirect
48. #3967C22semidirect
53. #3948C22semidirect
54. C8×Dic5C22semidirect
55. C8×Dic5C22semidirect
56. #3942C22semidirect
58. C4⋊#168C22semidirect
122. #464D5semidirect
176. C2×Q16C2×D5semidirect
195. C4×Q8C2×D5semidirect
203. C4⋊C8C2×D5semidirect
207. #103C2×D5semidirect
347. C5#3211semidirect

Maximal subgroups

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

subgroupindex
D5×#1242
#659142
C5×#4642
#654802
#654752
#654582
#654572
#653462
#653172
#651332
#651222
#650952
#650892
#650622
#650572
#32115