Finite Group Databasebeta

← #10612337

Subgroups — #10612337

Order 640.

400 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D5×#124C211yesmaximal, characteristic
2#65912C211yesmaximal, characteristic, complemented
3C5×#464C211yesmaximal, characteristic, complemented
4#65479C211yesmaximal, characteristic, complemented
5#65475C211yesmaximal, characteristic, complemented
6#65458C211yesmaximal, characteristic
7#65456C211yesmaximal, characteristic, complemented
8#65343C211yesmaximal, characteristic, complemented
9#65317C211yesmaximal, characteristic, complemented
10#65134C211yesmaximal, characteristic, complemented
11#65124C211yesmaximal, characteristic, complemented
12#65097C211yesmaximal, characteristic, complemented
13#65084C211yesmaximal, characteristic, complemented
14#65065C211yesmaximal, characteristic
15#65054C211yesmaximal, characteristic, complemented
16C2×Q8×D5C2211yescharacteristic
17C2×D4×D5C2211yescharacteristic
18C2×C5×Q16C2211yescharacteristic, complemented
19C2×C5×SD16C2211yescharacteristic
20C5×#124C2211yescharacteristic
21C4×C5×Q8C2211yescharacteristic, complemented
22#4093C2211yescharacteristic
23#4092C2211yescharacteristic
24#4090C2211yescharacteristic, complemented
25C2×#689C2211yescharacteristic, complemented
26C2×#689C2211yescharacteristic, complemented
27C2×#688C2211yescharacteristic
28#4082C2211yescharacteristic
29C2×#687C2211yescharacteristic
30C2×Dic20C2211yescharacteristic, complemented
31C2×#677C2211yescharacteristic
32#4043C2211yescharacteristic
33#4042—21no—
34#4035—21no—
35#4034C2211yescharacteristic, complemented
36#4031—21no—
37#4031—21no—
38D5×C22⋊C4—21no—
39D5×C22⋊C4—21no—
40#4021C2211yescharacteristic
41#4018—21no—
42C42×D5C2211yescharacteristic
43C4×Dic10—21no—
44C4×Dic10C2211yescharacteristic, complemented
45C5×C4⋊C8C2211yescharacteristic, complemented
46C5×#103C2211yescharacteristic, complemented
47C5×#102C2211yescharacteristic
48#3967C2211yescharacteristic, complemented
49#3964C2211yescharacteristic
50#3953C2211yescharacteristic
51#3952C2211yescharacteristic
52#3952C2211yescharacteristic
53#3948C2211yescharacteristic, complemented
54#3947C2211yescharacteristic, complemented
55#3947C2211yescharacteristic, complemented
56#3942C2211yescharacteristic, complemented
57#3941C2211yescharacteristic
58C4⋊#168C2211yescharacteristic, complemented
59#3215—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×D20C2311yescharacteristic
70C2×C4×D5—21no—
71C2×C4×D5D411yescharacteristic
72C2×C4×D5D411yescharacteristic
73C2×C4×D5C2311yescharacteristic
74C2×Dic10—21no—
75C2×Dic10—21no—
76C2×Dic10C2311yescharacteristic
77C2×Dic10C2311yescharacteristic
78C5×Q16—41no—
79C5×SD16—41no—
80C2×C5×C8C2311yescharacteristic
81C2×C5×C8C2311yescharacteristic
82C5×C4⋊C4—21no—
83C5×C4⋊C4C2311yescharacteristic
84C5×C22⋊C4—21no—
85C5×C22⋊C4—21no—
86C42×C5—21no—
87C42×C5C2311yescharacteristic
88C22⋊Dic5—21no—
89C22⋊Dic5—21no—
90#689—41no—
91#689—41no—
92#688—41no—
93#687—41no—
94#685—21no—
95#685—21no—
96#685—21no—
97#685—21no—
98#685—21no—
99#685—21no—
100C4⋊Dic5—21no—
101C4⋊Dic5C2311yescharacteristic
102#683—21no—
103#683—21no—
104#683—21no—
105#683—21no—
106#683—21no—
107C4×Dic5—21no—
108C4×Dic5D411yescharacteristic
109C4×Dic5D411yescharacteristic
110C4×Dic5C2311yescharacteristic
111C2×#168C2311yescharacteristic
112C2×#168C2311yescharacteristic
113Dic20—41no—
114#677—41no—
115#533—51no—
116C2×#124—51no—
117#497—51no—
118#497—51no—
119#489—51no—
120#489—51no—
121#464—51no—
122#464—51no—
123#464—51no—
124#464D511yescharacteristic, complemented
125#451—51no—
126#451—51no—
127#450—51no—
128#450—51no—
129#432—51no—
130C23×C5—21no—
131C22×D5—21no—
132C22×D5—42no—
133C22×D5—42no—
134C22×D5—42no—
135C22×D5C2×D411yescharacteristic
136C5×Q8—21no—
137C5×Q8—21no—
138C5×Q8—21no—
139C5×Q8—21no—
140C5×D4—21no—
141C5×D4—21no—
142C2×C4×C5—21no—
143C2×C4×C5—21no—
144C2×C4×C5—21no—
145C2×C4×C5C2×D411yescharacteristic
146C2×C4×C5C2×D411yescharacteristic
147C2×C4×C5C2411yescharacteristic
148C22⋊D5—42no—
149C2×Dic5—21no—
150C2×Dic5—21no—
151C2×Dic5—21no—
152C2×Dic5C2×D411yescharacteristic
153C2×Dic5C2×D411yescharacteristic
154C2×Dic5C2×D411yescharacteristic
155D20—21no—
156D20—21no—
157C4×D5—22no—
158C4×D5—21no—
159C4×D5—21no—
160C4×D5—41no—
161C4×D5—42no—
162Dic10—21no—
163Dic10—21no—
164Dic10—21no—
165Dic10—21no—
166Dic10—42no—
167Dic10—42no—
168C5×C8—21no—
169C5×C8—21no—
170#168—21no—
171#168—21no—
172C22×Q8—51no—
173C22×D4—51no—
174C2×Q16—51no—
175C2×Q16—51no—
176C2×Q16—51no—
177C2×Q16C2×D511yescharacteristic, complemented
178C2×SD16—51no—
179C2×SD16—51no—
180C2×SD16—51no—
181C2×SD16C2×D511yescharacteristic
182C4⋊Q8—51no—
183C4⋊Q8—51no—
184#125—101no—
185#124—51no—
186#124—51no—
187#124—51no—
188#124—101no—
189#124—101no—
190#124C2×D511yescharacteristic
191C22⋊Q8—51no—
192C22⋊Q8—51no—
193C22⋊Q8—101no—
194C4×Q8—51no—
195C4×Q8—101no—
196C4×Q8C2×D511yescharacteristic, complemented
197#117—101no—
198C2×C22⋊C4—101no—
199C2×C22⋊C4—101no—
200C2×C42—51no—
201C4⋊C8—51no—
202C4⋊C8C2×D511yescharacteristic, complemented
203#103—51no—
204#103—51no—
205#103—51no—
206#103C2×D511yescharacteristic, complemented
207#102—51no—
208#102—51no—
209#102—51no—
210#102C2×D511yescharacteristic
211C22⋊C8—51no—
212C22⋊C8—51no—
213#97—51no—
214#97—51no—
215C22×C5—41no—
216C22×C5—41no—
217C22×C5—41no—
218C22×C5C22×D411yescharacteristic
219C2×D5—21no—
220C2×D5—21no—
221C2×D5—41no—
222C2×D5—41no—
223C2×D5—41no—
224C2×D5—84no—
225C2×D5#13712yes—
226C4×C5—21no—
227C4×C5—21no—
228C4×C5—41no—
229C4×C5—41no—
230C4×C5—41no—
231C4×C5H(5,𝔽2)11yescharacteristic
232C4×C5H(5,𝔽2)11yescharacteristic
233Dic5—21no—
234Dic5—21no—
235Dic5—21no—
236Dic5—41no—
237Dic5—41no—
238Dic5—41no—
239C24—101no—
240C2×Q8—51no—
241C2×Q8—51no—
242C2×Q8—101no—
243C2×Q8—101no—
244C2×Q8—201no—
245C2×Q8—201no—
246C2×Q8C22×D511yescharacteristic
247C2×Q8C22×D511yescharacteristic
248C2×D4—51no—
249C2×D4—101no—
250C2×D4—201no—
251C2×D4—201no—
252C2×D4C22×D511yescharacteristic
253C22×C4—51no—
254C22×C4—51no—
255C22×C4—51no—
256C22×C4—101no—
257Q16—41no—
258Q16—201no—
259Q16—201no—
260Q16—201no—
261SD16—41no—
262SD16—201no—
263SD16—201no—
264SD16—201no—
265C2×C8—51no—
266C2×C8—51no—
267C2×C8C22×D511yescharacteristic
268C2×C8C22×D511yescharacteristic
269C4⋊C4—21no—
270C4⋊C4—51no—
271C4⋊C4—101no—
272C4⋊C4—101no—
273C4⋊C4—101no—
274C4⋊C4—101no—
275C4⋊C4—101no—
276C4⋊C4—101no—
277C4⋊C4C22×D511yescharacteristic
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—101no—
281C22⋊C4—101no—
282C22⋊C4—101no—
283C22⋊C4—101no—
284C22⋊C4—101no—
285C22⋊C4—101no—
286C22⋊C4—101no—
287C22⋊C4—101no—
288C42—21no—
289C42—51no—
290C42—51no—
291C42—51no—
292C42—101no—
293C42C22×D511yescharacteristic
294C2×C5—81no—
295C2×C5#53511yescharacteristic
296C2×C5C2×#13711yescharacteristic
297C2×C5#57511yescharacteristic
298D5—22no—
299D5—81no—
300C23—21no—
301C23—51no—
302C23—101no—
303C23—202no—
304C23—202no—
305C23—202no—
306Q8—21no—
307Q8—21no—
308Q8—21no—
309Q8—21no—
310Q8—101no—
311Q8—101no—
312Q8—101no—
313Q8—101no—
314Q8—202no—
315Q8—202no—
316D4—21no—
317D4—21no—
318D4—101no—
319D4—101no—
320D4—202no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—51no—
325C2×C4—51no—
326C2×C4—51no—
327C2×C4—102no—
328C2×C4—101no—
329C2×C4—101no—
330C2×C4—101no—
331C2×C4—101no—
332C2×C4—101no—
333C2×C4—201no—
334C2×C4—202no—
335C2×C4D4×D511yescharacteristic
336C2×C4D4×D511yescharacteristic
337C2×C4C23×D511yescharacteristic
338C8—21no—
339C8—21no—
340C8—101no—
341C8—101no—
342C5#321511yescharacteristic, complemented, Sylow
343C22—41no—
344C22—41no—
345C22—41no—
346C22—52no—
347C22—101no—
348C22—101no—
349C22—201no—
350C22—201no—
351C22—201no—
352C22—404no—
353C22C2×D4×D511yescharacteristic
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—101no—
360C4—101no—
361C4—101no—
362C4—201no—
363C4—201no—
364C4—201no—
365C4#414411yescharacteristic
366C4#414911yescharacteristic
367C2—81no—
368C2—102no—
369C2—401no—
370C2#6594111yescharacteristic
371C2#6610211yescharacteristic
372C2D5×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#464C2
socleC22×C5C22×D4
prime radicalC2×D4×D5C22
hypercenter#464D5
hypocenterC5#3215
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
2. #65912C2semidirect
3. C5×#464C2semidirect
4. #65479C2semidirect
5. #65475C2semidirect
7. #65456C2semidirect
8. #65343C2semidirect
9. #65317C2semidirect
10. #65134C2semidirect
11. #65124C2semidirect
12. #65097C2semidirect
13. #65084C2semidirect
15. #65054C2semidirect
18. C2×C5×Q16C22semidirect
21. C4×C5×Q8C22semidirect
24. #4090C22semidirect
25. C2×#689C22semidirect
26. C2×#689C22semidirect
30. C2×Dic20C22semidirect
35. #4034C22semidirect
44. C4×Dic10C22semidirect
45. C5×C4⋊C8C22semidirect
46. C5×#103C22semidirect
48. #3967C22semidirect
53. #3948C22semidirect
54. #3947C22semidirect
55. #3947C22semidirect
56. #3942C22semidirect
58. C4⋊#168C22semidirect
124. #464D5semidirect
177. C2×Q16C2×D5semidirect
196. C4×Q8C2×D5semidirect
202. C4⋊C8C2×D5semidirect
206. #103C2×D5semidirect
342. C5#3215semidirect

Maximal subgroups

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

subgroupindex
D5×#1242
#659122
C5×#4642
#654792
#654752
#654582
#654562
#653432
#653172
#651342
#651242
#650972
#650842
#650652
#650542
#32155