Finite Group Databasebeta

← #10613087

Subgroups — #10613087

Order 640.

370 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#65930C211yesmaximal, characteristic
2#65851C211yesmaximal, characteristic, complemented
3C5×#482C211yesmaximal, characteristic, complemented
4#65324C211yesmaximal, characteristic, complemented
5#65292C211yesmaximal, characteristic, complemented
6#65180C211yesmaximal, characteristic, complemented
7#65163C211yesmaximal, characteristic, complemented
8#65160C211yesmaximal, characteristic, complemented
9#65144C211yesmaximal, characteristic, complemented
10#65055C211yesmaximal, characteristic, complemented
11#65052C211yesmaximal, characteristic, complemented
12#65049C211yesmaximal, characteristic, complemented
13#65049C211yesmaximal, characteristic, complemented
14#65026C211yesmaximal, characteristic, complemented
15#65007C211yesmaximal, characteristic, complemented
16C2×#711—21no—
17C5×#121C2211yescharacteristic
18C5×#117C2211yescharacteristic
19#4086C2211yescharacteristic
20#4085—21no—
21D4×Dic5C2211yescharacteristic
22#4076C2211yescharacteristic
23#4072C2211yescharacteristic
24C2×C4×Dic5C2211yescharacteristic
25#4039C2211yescharacteristic
26#4036C2211yescharacteristic
27#4036C2211yescharacteristic
28#4034C2211yescharacteristic
29#4034C2211yescharacteristic
30#4034C2211yescharacteristic
31#4031—21no—
32#4027—21no—
33#4024—21no—
34#4024—21no—
35C4×Dic10—21no—
36C4×Dic10—21no—
37C5×#107C2211yescharacteristic
38C5×#106C2211yescharacteristic
39C5×#102C2211yescharacteristic
40C5×#102C2211yescharacteristic
41C5×C22⋊C8C2211yescharacteristic
42#3964C2211yescharacteristic
43#3964C2211yescharacteristic
44C22⋊#168C2211yescharacteristic
45#3953C2211yescharacteristic
46#3953C2211yescharacteristic
47#3950C2211yescharacteristic
48#3949C2211yescharacteristic
49#3946C2211yescharacteristic
50#3946C2211yescharacteristic
51C8×Dic5C2211yescharacteristic
52C8×Dic5C2211yescharacteristic
53#3941C2211yescharacteristic
54#3941C2211yescharacteristic
55#3940C2211yescharacteristic
56#3940C2211yescharacteristic
57#3939C2211yescharacteristic
58#3939C2211yescharacteristic
59#3228—51nomaximal, Sylow
60C2×C5×D4—21no—
61C2×C5×D4C2311yescharacteristic
62C22×C4×C5C2311yescharacteristic
63C2×C22⋊D5—21no—
64C2×C22⋊D5—21no—
65C2×C22⋊D5—21no—
66C22×Dic5—21no—
67C22×Dic5D411yescharacteristic
68C22×Dic5D411yescharacteristic
69#711—44no—
70C2×D20C2311yescharacteristic
71C2×C4×D5—21no—
72C2×Dic10—21no—
73C2×Dic10—21no—
74C2×Dic10—21no—
75C2×C5×C8C2311yescharacteristic
76C2×C5×C8C2311yescharacteristic
77C5×C4⋊C4C2311yescharacteristic
78C5×C4⋊C4C2311yescharacteristic
79C5×C4⋊C4C2311yescharacteristic
80C5×C22⋊C4—21no—
81C5×C22⋊C4—21no—
82C42×C5—21no—
83C22⋊Dic5—21no—
84C22⋊Dic5—21no—
85#685—21no—
86#685—21no—
87C4⋊Dic5—21no—
88C4⋊Dic5—21no—
89C4⋊Dic5C2311yescharacteristic
90C4⋊Dic5C2311yescharacteristic
91C4⋊Dic5C2311yescharacteristic
92#683—21no—
93#683—21no—
94#683—21no—
95#683—21no—
96#683—21no—
97C4×Dic5—21no—
98C4×Dic5D411yescharacteristic
99C4×Dic5D411yescharacteristic
100C4×Dic5C2311yescharacteristic
101C4×Dic5C2311yescharacteristic
102C2×#168C2311yescharacteristic
103C2×#168C2311yescharacteristic
104#533—51no—
105#532—51no—
106#500—51no—
107#498—51no—
108#486—51no—
109#486—51no—
110#482—51no—
111#482—51no—
112#482—51no—
113#482D511yescharacteristic, complemented
114#478—51no—
115#478—51no—
116#478—51no—
117#478—51no—
118#431—51no—
119C23×C5—21no—
120C23×C5C2×D411yescharacteristic
121C22×D5—21no—
122C5×D4—21no—
123C5×D4—21no—
124C5×D4—42no—
125C2×C4×C5—21no—
126C2×C4×C5—21no—
127C2×C4×C5—21no—
128C2×C4×C5—21no—
129C2×C4×C5—21no—
130C2×C4×C5C2×D411yescharacteristic
131C2×C4×C5C2411yescharacteristic
132C22⋊D5—42no—
133C22⋊D5—42no—
134C22⋊D5—42no—
135C2×Dic5—22no—
136C2×Dic5—22no—
137C2×Dic5—21no—
138C2×Dic5—21no—
139C2×Dic5—21no—
140C2×Dic5—42no—
141C2×Dic5C2×D411yescharacteristic
142C2×Dic5C2×D411yescharacteristic
143C2×Dic5C2×D411yescharacteristic
144C2×Dic5C2×D411yescharacteristic
145D20—21no—
146D20—21no—
147C4×D5—42no—
148Dic10—42no—
149Dic10—42no—
150Dic10—42no—
151C5×C8—21no—
152C5×C8—21no—
153#168—21no—
154#168—21no—
155C2×#41—101no—
156C4⋊Q8—51no—
157C4⋊Q8—51no—
158C4⋊Q8—51no—
159#127—51no—
160#125—51no—
161#125—51no—
162#124—101no—
163C22⋊Q8—101no—
164C22⋊Q8—101no—
165#121—51no—
166#121—101no—
167#121C2×D511yescharacteristic
168C4×Q8—101no—
169C4×Q8—101no—
170C4×D4—51no—
171C4×D4—51no—
172C4×D4—101no—
173#117—51no—
174#117C2×D511yescharacteristic
175C2×C42—51no—
176#107—51no—
177#107—51no—
178#107—51no—
179#107C2×D511yescharacteristic
180#106—51no—
181#106—51no—
182#106—51no—
183#106C2×D511yescharacteristic
184C4⋊C8—51no—
185C4⋊C8—51no—
186#102—51no—
187#102—51no—
188#102—51no—
189#102—51no—
190#102—51no—
191#102—51no—
192#102C2×D511yescharacteristic
193#102C2×D511yescharacteristic
194C22⋊C8—51no—
195C22⋊C8C2×D511yescharacteristic
196C4×C8—51no—
197C4×C8—51no—
198C22×C5—21no—
199C22×C5—21no—
200C22×C5—21no—
201C22×C5—41no—
202C22×C5—41no—
203C22×C5—41no—
204C22×C5C22×D411yescharacteristic
205C2×D5—41no—
206C2×D5—41no—
207C2×D5—41no—
208C4×C5—21no—
209C4×C5—41no—
210C4×C5—41no—
211C4×C5—41no—
212C4×C52−1+411yescharacteristic
213C4×C52−1+411yescharacteristic
214Dic5—21no—
215Dic5—21no—
216Dic5—41no—
217Dic5—41no—
218Dic5—41no—
219Dic5#13512yes—
220Dic5AGL(1,ℤ/8)12yes—
221#41—204no—
222C2×Q8—101no—
223C2×Q8—101no—
224C2×Q8—101no—
225C2×D4—21no—
226C2×D4—51no—
227C2×D4—101no—
228C2×D4—101no—
229C2×D4—101no—
230C2×D4C22×D511yescharacteristic
231C22×C4—51no—
232C22×C4—51no—
233C22×C4—101no—
234C22×C4—101no—
235C22×C4C22×D511yescharacteristic
236C2×C8—51no—
237C2×C8—51no—
238C2×C8C22×D511yescharacteristic
239C2×C8C22×D511yescharacteristic
240C4⋊C4—51no—
241C4⋊C4—51no—
242C4⋊C4—51no—
243C4⋊C4—101no—
244C4⋊C4—101no—
245C4⋊C4—101no—
246C4⋊C4—101no—
247C4⋊C4—101no—
248C4⋊C4—101no—
249C4⋊C4—101no—
250C4⋊C4C22×D511yescharacteristic
251C4⋊C4C22×D511yescharacteristic
252C4⋊C4C22×D511yescharacteristic
253C22⋊C4—21no—
254C22⋊C4—21no—
255C22⋊C4—101no—
256C22⋊C4—101no—
257C22⋊C4—101no—
258C22⋊C4—101no—
259C42—21no—
260C42—51no—
261C42—51no—
262C42—51no—
263C42—51no—
264C42—101no—
265C2×C5—41no—
266C2×C5—81no—
267C2×C5#53611yescharacteristic
268C2×C5C2×#13511yescharacteristic
269C2×C5C2×AGL(1,ℤ/8)11yescharacteristic
270D5—81no—
271C23—21no—
272C23—101no—
273C23D4×D511yescharacteristic
274Q8—202no—
275Q8—202no—
276Q8—202no—
277D4—21no—
278D4—21no—
279D4—42no—
280D4—101no—
281D4—101no—
282D4—202no—
283D4—202no—
284D4—202no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—51no—
291C2×C4—51no—
292C2×C4—51no—
293C2×C4—51no—
294C2×C4—102no—
295C2×C4—102no—
296C2×C4—101no—
297C2×C4—101no—
298C2×C4—101no—
299C2×C4—202no—
300C2×C4—202no—
301C2×C4D4×D511yescharacteristic
302C2×C4C23×D511yescharacteristic
303C8—21no—
304C8—21no—
305C8—101no—
306C8—101no—
307C5#322811yescharacteristic, complemented, Sylow
308C22—21no—
309C22—21no—
310C22—21no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—201no—
315C22—201no—
316C22—201no—
317C22C2×D4×D511yescharacteristic
318C4—21no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—52no—
323C4—52no—
324C4—101no—
325C4—101no—
326C4—201no—
327C4—201no—
328C4—201no—
329C4#415011yescharacteristic
330C4#415011yescharacteristic
331C2—41no—
332C2—81no—
333C2—401no—
334C2#6598211yescharacteristic
335C2D5×#13511yescharacteristic
336C2D5×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#482C2
socleC22×C5C22×D4
prime radical#65930C2
hypercenter#482D5
hypocenterC5#3228
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. #65851C2semidirect
3. C5×#482C2semidirect
4. #65324C2semidirect
5. #65292C2semidirect
6. #65180C2semidirect
7. #65163C2semidirect
8. #65160C2semidirect
9. #65144C2semidirect
10. #65055C2semidirect
11. #65052C2semidirect
12. #65049C2semidirect
13. #65049C2semidirect
14. #65026C2semidirect
15. #65007C2semidirect
113. #482D5semidirect
307. C5#3228semidirect

Maximal subgroups

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

subgroupindex
#659302
#658512
C5×#4822
#653242
#652922
#651802
#651632
#651602
#651442
#650552
#650522
#650492
#650262
#650072
#32285