Finite Group Databasebeta

← #10613029

Subgroups — #10613029

Order 640.

354 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#65950C211yesmaximal, characteristic
2#65845C211yesmaximal, characteristic, complemented
3C5×#480C211yesmaximal, characteristic, complemented
4#65331C211yesmaximal, characteristic, complemented
5#65253C211yesmaximal, characteristic, complemented
6#65177C211yesmaximal, characteristic, complemented
7#65170C211yesmaximal, characteristic, complemented
8#65164C211yesmaximal, characteristic, complemented
9#65163C211yesmaximal, characteristic, complemented
10#65069C211yesmaximal, characteristic, complemented
11#65063C211yesmaximal, characteristic, complemented
12#65050C211yesmaximal, characteristic, complemented
13#65047C211yesmaximal, characteristic, complemented
14#65018C211yesmaximal, characteristic, complemented
15#65012C211yesmaximal, characteristic, complemented
16C5×#121C2211yescharacteristic
17C2×C5×C4⋊C4C2211yescharacteristic
18#4085—21no—
19#4084C2211yescharacteristic
20#4082—21no—
21D4×Dic5C2211yescharacteristic
22C2×#687C2211yescharacteristic
23C4×C22⋊D5—21no—
24C4×C22⋊D5C2211yescharacteristic
25#4072C2211yescharacteristic
26#4070—21no—
27#4070C2211yescharacteristic
28C2×#681C2211yescharacteristic
29C2×#676C2211yescharacteristic
30#4043C2211yescharacteristic
31#4042—21no—
32#4038C2211yescharacteristic
33D5×C4⋊C4C2211yescharacteristic
34#4035—21no—
35#4034C2211yescharacteristic
36#4034C2211yescharacteristic
37#4033C2211yescharacteristic
38#4031—21no—
39#4023—21no—
40C5×#107C2211yescharacteristic
41C5×#107C2211yescharacteristic
42C5×#102C2211yescharacteristic
43C5×#102C2211yescharacteristic
44C5×C22⋊C8C2211yescharacteristic
45#3964C2211yescharacteristic
46#3952C2211yescharacteristic
47#3950C2211yescharacteristic
48#3949C2211yescharacteristic
49#3948C2211yescharacteristic
50#3948C2211yescharacteristic
51#3946C2211yescharacteristic
52C8×Dic5C2211yescharacteristic
53#3942C2211yescharacteristic
54#3940C2211yescharacteristic
55#3940C2211yescharacteristic
56#3939C2211yescharacteristic
57#3939C2211yescharacteristic
58#3939C2211yescharacteristic
59#3357—51nomaximal, Sylow
60C2×C5×D4—21no—
61C2×C5×D4C2311yescharacteristic
62C22×C4×C5—21no—
63C22×C4×C5C2311yescharacteristic
64C2×C22⋊D5—21no—
65C2×C22⋊D5D411yescharacteristic
66C22×Dic5—21no—
67C2×C4×D5—21no—
68C2×C4×D5C2311yescharacteristic
69C2×Dic10—21no—
70C2×Dic10—21no—
71C2×Dic10C2311yescharacteristic
72C2×C5×C8C2311yescharacteristic
73C2×C5×C8C2311yescharacteristic
74C5×C4⋊C4—21no—
75C5×C4⋊C4C2311yescharacteristic
76C5×C4⋊C4C2311yescharacteristic
77C5×C4⋊C4C2311yescharacteristic
78C5×C22⋊C4—21no—
79C22⋊Dic5—21no—
80C22⋊Dic5—21no—
81C22⋊Dic5—21no—
82C22⋊Dic5—21no—
83C22⋊Dic5D411yescharacteristic
84#687—22no—
85#685—21no—
86#685—21no—
87#685D411yescharacteristic
88C4⋊Dic5—21no—
89C4⋊Dic5C2311yescharacteristic
90C4⋊Dic5C2311yescharacteristic
91C4⋊Dic5C2311yescharacteristic
92#683—21no—
93#683—21no—
94#683—21no—
95#683—21no—
96#683—21no—
97#683D411yescharacteristic
98C4×Dic5—21no—
99C4×Dic5—21no—
100C4×Dic5—21no—
101C4×Dic5C2311yescharacteristic
102#681—22no—
103C2×#168C2311yescharacteristic
104C2×#168C2311yescharacteristic
105#676—22no—
106#555—51no—
107#554—51no—
108#500—51no—
109#485—51no—
110#485—51no—
111#480—51no—
112#480D511yescharacteristic, complemented
113#477—51no—
114#476—51no—
115#468—51no—
116#468—51no—
117C4×SD16—51no—
118#436—51no—
119#428—51no—
120#428—51no—
121C23×C5—21no—
122C23×C5C2×D411yescharacteristic
123C22×D5C2×D411yescharacteristic
124C5×D4—21no—
125C5×D4—21no—
126C5×D4—81no—
127C2×C4×C5—21no—
128C2×C4×C5—21no—
129C2×C4×C5—21no—
130C2×C4×C5—21no—
131C2×C4×C5—21no—
132C2×C4×C5—81no—
133C2×C4×C5C2×D411yescharacteristic
134C2×C4×C5C2411yescharacteristic
135C22⋊D5—22no—
136C22⋊D5—81no—
137C2×Dic5—21no—
138C2×Dic5—21no—
139C2×Dic5—21no—
140C2×Dic5—21no—
141C2×Dic5—42no—
142C2×Dic5C2×D411yescharacteristic
143C2×Dic5C2×D411yescharacteristic
144C2×Dic5C2×D411yescharacteristic
145C4×D5—21no—
146C4×D5—21no—
147C4×D5—81no—
148Dic10—21no—
149Dic10—21no—
150Dic10—81no—
151Dic10—81no—
152C5×C8—21no—
153C5×C8—21no—
154#168—21no—
155#168#4112yes—
156C2×SD16—51no—
157C2×M16—51no—
158C2×M16—51no—
159C4⋊Q8—51no—
160C4⋊Q8—51no—
161#125—101no—
162#124—101no—
163#124—101no—
164C22⋊Q8—51no—
165C22⋊Q8—51no—
166C22⋊Q8—101no—
167C22⋊Q8—101no—
168#121—51no—
169#121—101no—
170#121C2×D511yescharacteristic
171C4×Q8—51no—
172C4×D4—51no—
173C4×D4—51no—
174C4×D4—101no—
175#117—51no—
176#117—51no—
177#117—101no—
178C2×C4⋊C4—51no—
179C2×C4⋊C4C2×D511yescharacteristic
180#107—51no—
181#107—51no—
182#107—51no—
183#107—51no—
184#107C2×D511yescharacteristic
185#107C2×D511yescharacteristic
186#106—51no—
187#106—51no—
188#106—51no—
189C4⋊C8—51no—
190#103—51no—
191#103—51no—
192#103—51no—
193#102—51no—
194#102C2×D511yescharacteristic
195#102C2×D511yescharacteristic
196C22⋊C8—51no—
197C22⋊C8C2×D511yescharacteristic
198C4×C8—51no—
199C22×C5—21no—
200C22×C5—21no—
201C22×C5—21no—
202C22×C5—41no—
203C22×C5—41no—
204C22×C5—41no—
205C22×C5C22×D411yescharacteristic
206C2×D5—21no—
207C2×D5—21no—
208C2×D5—21no—
209C4×C5—21no—
210C4×C5—41no—
211C4×C5—41no—
212C4×C5—41no—
213C4×C5C2×#4111yescharacteristic
214C4×C52−1+411yescharacteristic
215Dic5—21no—
216Dic5—21no—
217Dic5—41no—
218Dic5—41no—
219Dic5—41no—
220Dic5—41no—
221Dic5#13712yes—
222C2×Q8—51no—
223C2×Q8—101no—
224C2×Q8—101no—
225C2×D4—21no—
226C2×D4—51no—
227C2×D4—101no—
228C2×D4C22×D511yescharacteristic
229C22×C4—21no—
230C22×C4—51no—
231C22×C4—101no—
232C22×C4—101no—
233C22×C4C22×D511yescharacteristic
234SD16—102no—
235M16—102no—
236M16—102no—
237C2×C8—51no—
238C2×C8—51no—
239C2×C8C22×D511yescharacteristic
240C2×C8C22×D511yescharacteristic
241C4⋊C4—21no—
242C4⋊C4—51no—
243C4⋊C4—51no—
244C4⋊C4—51no—
245C4⋊C4—51no—
246C4⋊C4—101no—
247C4⋊C4—101no—
248C4⋊C4—101no—
249C4⋊C4—101no—
250C4⋊C4—101no—
251C4⋊C4—101no—
252C4⋊C4C22×D511yescharacteristic
253C4⋊C4C22×D511yescharacteristic
254C4⋊C4C22×D511yescharacteristic
255C22⋊C4—21no—
256C22⋊C4—51no—
257C22⋊C4—51no—
258C22⋊C4—101no—
259C22⋊C4—101no—
260C22⋊C4—101no—
261C22⋊C4—101no—
262C22⋊C4—101no—
263C22⋊C4—101no—
264C42—51no—
265C42—101no—
266C42—101no—
267C42—101no—
268C2×C5—41no—
269C2×C5—81no—
270C2×C5#54711yescharacteristic
271C2×C5C2×#13711yescharacteristic
272C2×C5#57611yescharacteristic
273D5—41no—
274C23—21no—
275C23—51no—
276C23D4×D511yescharacteristic
277Q8—101no—
278Q8—101no—
279Q8—401no—
280Q8—401no—
281D4—21no—
282D4—21no—
283D4—81no—
284D4—102no—
285D4—401no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—51no—
292C2×C4—51no—
293C2×C4—51no—
294C2×C4—81no—
295C2×C4—101no—
296C2×C4—101no—
297C2×C4—101no—
298C2×C4—101no—
299C2×C4—101no—
300C2×C4—101no—
301C2×C4—202no—
302C2×C4—401no—
303C2×C4D4×D511yescharacteristic
304C2×C4C23×D511yescharacteristic
305C8—21no—
306C8—21no—
307C8—52no—
308C8—101no—
309C5#335711yescharacteristic, complemented, Sylow
310C22—21no—
311C22—21no—
312C22—21no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—101no—
317C22—101no—
318C22—101no—
319C22C2×D4×D511yescharacteristic
320C4—21no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—52no—
325C4—101no—
326C4—101no—
327C4—201no—
328C4—201no—
329C4—201no—
330C4—201no—
331C4D5×#4111yescharacteristic
332C4#415011yescharacteristic
333C2—41no—
334C2—81no—
335C2—201no—
336C2#6598911yescharacteristic
337C2#6609111yescharacteristic
338C2D5×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#480C2
socleC22×C5C22×D4
prime radical#65950C2
hypercenter#480D5
hypocenterC5#3357
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. #65845C2semidirect
3. C5×#480C2semidirect
4. #65331C2semidirect
5. #65253C2semidirect
6. #65177C2semidirect
7. #65170C2semidirect
8. #65164C2semidirect
9. #65163C2semidirect
10. #65069C2semidirect
11. #65063C2semidirect
12. #65050C2semidirect
13. #65047C2semidirect
14. #65018C2semidirect
15. #65012C2semidirect
112. #480D5semidirect
309. C5#3357semidirect

Maximal subgroups

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

subgroupindex
#659502
#658452
C5×#4802
#653312
#652532
#651772
#651702
#651642
#651632
#650692
#650632
#650502
#650472
#650182
#650122
#33575