Finite Group Databasebeta

← #10613098

Subgroups — #10613098

Order 640.

374 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#65945C211yesmaximal, characteristic
2#65867C211yesmaximal, characteristic, complemented
3C5×#482C211yesmaximal, characteristic, complemented
4#65331C211yesmaximal, characteristic, complemented
5#65284C211yesmaximal, characteristic, complemented
6#65173C211yesmaximal, characteristic, complemented
7#65163C211yesmaximal, characteristic, complemented
8#65150C211yesmaximal, characteristic, complemented
9#65143C211yesmaximal, characteristic, complemented
10#65073C211yesmaximal, characteristic, complemented
11#65069C211yesmaximal, characteristic, complemented
12#65048C211yesmaximal, characteristic, complemented
13#65047C211yesmaximal, characteristic, complemented
14#65018C211yesmaximal, characteristic, complemented
15#65011C211yesmaximal, characteristic, complemented
16C2×#711—21no—
17C5×#121C2211yescharacteristic
18C5×#117C2211yescharacteristic
19#4084C2211yescharacteristic
20#4081—21no—
21D4×Dic5C2211yescharacteristic
22C2×#687C2211yescharacteristic
23C4×C22⋊D5C2211yescharacteristic
24C2×C4⋊Dic5C2211yescharacteristic
25#4070C2211yescharacteristic
26C2×#681C2211yescharacteristic
27C2×#676C2211yescharacteristic
28#4043C2211yescharacteristic
29#4038C2211yescharacteristic
30D5×C4⋊C4C2211yescharacteristic
31#4034C2211yescharacteristic
32#4034C2211yescharacteristic
33#4033C2211yescharacteristic
34#4031—21no—
35#4029—21no—
36#4025—21no—
37#4024—21no—
38#4022—21no—
39C4×Dic10—21no—
40C5×#107C2211yescharacteristic
41C5×#106C2211yescharacteristic
42C5×#102C2211yescharacteristic
43C5×#102C2211yescharacteristic
44C5×C22⋊C8C2211yescharacteristic
45#3964C2211yescharacteristic
46#3952C2211yescharacteristic
47#3950C2211yescharacteristic
48#3950C2211yescharacteristic
49#3948C2211yescharacteristic
50#3948C2211yescharacteristic
51#3946C2211yescharacteristic
52C8×Dic5C2211yescharacteristic
53#3942C2211yescharacteristic
54#3940C2211yescharacteristic
55#3940C2211yescharacteristic
56#3939C2211yescharacteristic
57#3939C2211yescharacteristic
58#3939C2211yescharacteristic
59#3353—51nomaximal, Sylow
60C2×C5×D4—21no—
61C2×C5×D4C2311yescharacteristic
62C22×C4×C5C2311yescharacteristic
63C2×C22⋊D5—21no—
64C2×C22⋊D5D411yescharacteristic
65C22×Dic5—21no—
66C22×Dic5—21no—
67#711—44no—
68C2×C4×D5—21no—
69C2×C4×D5C2311yescharacteristic
70C2×Dic10—21no—
71C2×Dic10—21no—
72C2×Dic10C2311yescharacteristic
73C2×C5×C8C2311yescharacteristic
74C2×C5×C8C2311yescharacteristic
75C5×C4⋊C4C2311yescharacteristic
76C5×C4⋊C4C2311yescharacteristic
77C5×C4⋊C4C2311yescharacteristic
78C5×C22⋊C4—21no—
79C5×C22⋊C4—21no—
80C42×C5—21no—
81C22⋊Dic5—21no—
82C22⋊Dic5—21no—
83C22⋊Dic5—21no—
84C22⋊Dic5D411yescharacteristic
85#687—22no—
86#685—21no—
87#685—21no—
88#685D411yescharacteristic
89C4⋊Dic5—21no—
90C4⋊Dic5—21no—
91C4⋊Dic5C2311yescharacteristic
92C4⋊Dic5C2311yescharacteristic
93C4⋊Dic5C2311yescharacteristic
94#683—21no—
95#683—21no—
96#683—21no—
97#683—21no—
98#683—21no—
99#683D411yescharacteristic
100C4×Dic5—21no—
101C4×Dic5—21no—
102C4×Dic5C2311yescharacteristic
103#681—22no—
104C2×#168C2311yescharacteristic
105C2×#168C2311yescharacteristic
106#676—22no—
107#556—51no—
108#547—51no—
109#500—51no—
110#484—51no—
111#484—51no—
112#482—51no—
113#482D511yescharacteristic, complemented
114#475—51no—
115#474—51no—
116#468—51no—
117#468—51no—
118C4×SD16—51no—
119#436—51no—
120#428—51no—
121#428—51no—
122C23×C5—21no—
123C23×C5C2×D411yescharacteristic
124C22×D5C2×D411yescharacteristic
125C5×D4—21no—
126C5×D4—21no—
127C5×D4—42no—
128C2×C4×C5—21no—
129C2×C4×C5—21no—
130C2×C4×C5—21no—
131C2×C4×C5—21no—
132C2×C4×C5—21no—
133C2×C4×C5C2×D411yescharacteristic
134C2×C4×C5C2411yescharacteristic
135C22⋊D5—22no—
136C22⋊D5—42no—
137C2×Dic5—21no—
138C2×Dic5—21no—
139C2×Dic5—21no—
140C2×Dic5—21no—
141C2×Dic5—42no—
142C2×Dic5—42no—
143C2×Dic5C2×D411yescharacteristic
144C2×Dic5C2×D411yescharacteristic
145C2×Dic5C2×D411yescharacteristic
146C4×D5—21no—
147C4×D5—21no—
148C4×D5—42no—
149Dic10—21no—
150Dic10—21no—
151Dic10—42no—
152Dic10—42no—
153C5×C8—21no—
154C5×C8—21no—
155#168—21no—
156#168#4112yescomplemented
157C2×#41—101no—
158C2×SD16—51no—
159C2×M16—51no—
160C2×M16—51no—
161C4⋊Q8—51no—
162C4⋊Q8—51no—
163#126—101no—
164#126—101no—
165#124—101no—
166#123—101no—
167#123—101no—
168C22⋊Q8—51no—
169C22⋊Q8—51no—
170C22⋊Q8—101no—
171#121—51no—
172#121C2×D511yescharacteristic
173C4×Q8—51no—
174C4×Q8—101no—
175C4×D4—51no—
176C4×D4—51no—
177#117—51no—
178#117C2×D511yescharacteristic
179C2×C4⋊C4—51no—
180C2×C4⋊C4—51no—
181#107—51no—
182#107—51no—
183#107—51no—
184#107—51no—
185#107—51no—
186#107C2×D511yescharacteristic
187#106—51no—
188#106—51no—
189#106C2×D511yescharacteristic
190C4⋊C8—51no—
191#103—51no—
192#103—51no—
193#103—51no—
194#102—51no—
195#102C2×D511yescharacteristic
196#102C2×D511yescharacteristic
197C22⋊C8—51no—
198C22⋊C8C2×D511yescharacteristic
199C4×C8—51no—
200C22×C5—21no—
201C22×C5—21no—
202C22×C5—21no—
203C22×C5—41no—
204C22×C5—41no—
205C22×C5—41no—
206C22×C5C22×D411yescharacteristic
207C2×D5—21no—
208C2×D5—21no—
209C2×D5—21no—
210C4×C5—21no—
211C4×C5—41no—
212C4×C5—41no—
213C4×C5—41no—
214C4×C5C2×#4111yescharacteristic
215C4×C52−1+411yescharacteristic
216Dic5—21no—
217Dic5—21no—
218Dic5—41no—
219Dic5—41no—
220Dic5—41no—
221Dic5—41no—
222Dic5AGL(1,ℤ/8)12yes—
223#41—204no—
224C2×Q8—51no—
225C2×Q8—101no—
226C2×Q8—101no—
227C2×D4—21no—
228C2×D4—51no—
229C2×D4—101no—
230C2×D4C22×D511yescharacteristic
231C22×C4—51no—
232C22×C4—101no—
233C22×C4—101no—
234C22×C4—101no—
235C22×C4C22×D511yescharacteristic
236SD16—102no—
237M16—102no—
238M16—102no—
239C2×C8—51no—
240C2×C8—51no—
241C2×C8C22×D511yescharacteristic
242C2×C8C22×D511yescharacteristic
243C4⋊C4—51no—
244C4⋊C4—51no—
245C4⋊C4—51no—
246C4⋊C4—51no—
247C4⋊C4—101no—
248C4⋊C4—101no—
249C4⋊C4—101no—
250C4⋊C4—101no—
251C4⋊C4—101no—
252C4⋊C4—101no—
253C4⋊C4—101no—
254C4⋊C4C22×D511yescharacteristic
255C4⋊C4C22×D511yescharacteristic
256C4⋊C4C22×D511yescharacteristic
257C22⋊C4—21no—
258C22⋊C4—21no—
259C22⋊C4—51no—
260C22⋊C4—51no—
261C22⋊C4—101no—
262C22⋊C4—101no—
263C22⋊C4—101no—
264C22⋊C4—101no—
265C22⋊C4—101no—
266C42—21no—
267C42—51no—
268C42—101no—
269C42—101no—
270C2×C5—41no—
271C2×C5—81no—
272C2×C5#54711yescharacteristic
273C2×C5C2×AGL(1,ℤ/8)11yescharacteristic
274C2×C5#57811yescharacteristic
275D5—41no—
276C23—21no—
277C23—51no—
278C23D4×D511yescharacteristic
279Q8—101no—
280Q8—101no—
281Q8—202no—
282Q8—202no—
283D4—21no—
284D4—21no—
285D4—42no—
286D4—102no—
287D4—202no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—51no—
294C2×C4—51no—
295C2×C4—51no—
296C2×C4—101no—
297C2×C4—101no—
298C2×C4—101no—
299C2×C4—101no—
300C2×C4—101no—
301C2×C4—101no—
302C2×C4—202no—
303C2×C4—202no—
304C2×C4—202no—
305C2×C4D4×D511yescharacteristic
306C2×C4C23×D511yescharacteristic
307C8—21no—
308C8—21no—
309C8—52no—
310C8—101no—
311C5#335311yescharacteristic, complemented, Sylow
312C22—21no—
313C22—21no—
314C22—21no—
315C22—41no—
316C22—41no—
317C22—41no—
318C22—101no—
319C22—101no—
320C22—101no—
321C22C2×D4×D511yescharacteristic
322C4—21no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—52no—
327C4—101no—
328C4—101no—
329C4—201no—
330C4—201no—
331C4—201no—
332C4—201no—
333C4D5×#4111yescharacteristic
334C4#415011yescharacteristic
335C2—41no—
336C2—81no—
337C2—201no—
338C2#6598911yescharacteristic
339C2#6610511yescharacteristic
340C2D5×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#482C2
socleC22×C5C22×D4
prime radical#65945C2
hypercenter#482D5
hypocenterC5#3353
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
2. #65867C2semidirect
3. C5×#482C2semidirect
4. #65331C2semidirect
5. #65284C2semidirect
6. #65173C2semidirect
7. #65163C2semidirect
8. #65150C2semidirect
9. #65143C2semidirect
10. #65073C2semidirect
11. #65069C2semidirect
12. #65048C2semidirect
13. #65047C2semidirect
14. #65018C2semidirect
15. #65011C2semidirect
113. #482D5semidirect
156. #168#41semidirect
311. C5#3353semidirect

Maximal subgroups

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

subgroupindex
#659452
#658672
C5×#4822
#653312
#652842
#651732
#651632
#651502
#651432
#650732
#650692
#650482
#650472
#650182
#650112
#33535