Finite Group Databasebeta

← #10613402

Subgroups — #10613402

Order 640.

338 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#66056C211yesmaximal, characteristic, complemented
2#66005C211yesmaximal, characteristic
3C5×#489C211yesmaximal, characteristic
4#65383C211yesmaximal, characteristic, complemented
5#65343C211yesmaximal, characteristic
6#65082C211yesmaximal, characteristic, complemented
7#65081C211yesmaximal, characteristic, complemented
8#65080C211yesmaximal, characteristic, complemented
9#65078C211yesmaximal, characteristic, complemented
10#65056C211yesmaximal, characteristic
11#65054C211yesmaximal, characteristic
12#65050C211yesmaximal, characteristic
13#65047C211yesmaximal, characteristic
14#65005C211yesmaximal, characteristic, complemented
15#64990C211yesmaximal, characteristic, complemented
16C5×C4⋊Q8C2211yescharacteristic
17C5×#124C2211yescharacteristic
18Q8×Dic5C2211yescharacteristic
19#4090—21no—
20#4090—21no—
21#4090C2211yescharacteristic
22C2×#689C2211yescharacteristic
23#4082C2211yescharacteristic
24D4×Dic5C2211yescharacteristic
25C2×#687C2211yescharacteristic
26C2×Dic20C2211yescharacteristic
27#4034—21no—
28#4034C2211yescharacteristic
29#4034C2211yescharacteristic
30#4033—21no—
31#4033C2211yescharacteristic
32#4033C2211yescharacteristic
33#4025—21no—
34#4025—21no—
35#4024—21no—
36#4023—21no—
37C4×Dic10C2211yescharacteristic
38C4×Dic10C2211yescharacteristic
39C4×Dic10C2211yescharacteristic
40C5×#103C2211yescharacteristic
41C5×#103C2211yescharacteristic
42C5×#102C2211yescharacteristic
43C5×#102C2211yescharacteristic
44C5×#97C2211yescharacteristic
45#3967C2211yescharacteristic
46#3964C2211yescharacteristic
47#3949C2211yescharacteristic
48#3948C2211yescharacteristic
49#3948C2211yescharacteristic
50#3947C2211yescharacteristic
51#3946C2211yescharacteristic
52#3946C2211yescharacteristic
53C8×Dic5C2211yescharacteristic
54#3942C2211yescharacteristic
55#3942C2211yescharacteristic
56#3940C2211yescharacteristic
57#3940C2211yescharacteristic
58C4⋊#168C2211yescharacteristic
59#3369—51nomaximal, Sylow
60C2×C5×Q8—21no—
61C2×C5×Q8C2311yescharacteristic
62C2×C5×D4C2311yescharacteristic
63C22×Dic5—21no—
64C2×Dic10—21no—
65C2×Dic10—21no—
66C2×Dic10D411yescharacteristic
67C2×Dic10C2311yescharacteristic
68C2×Dic10C2311yescharacteristic
69C2×C5×C8C2311yescharacteristic
70C2×C5×C8C2311yescharacteristic
71C5×C4⋊C4—21no—
72C5×C4⋊C4C2311yescharacteristic
73C5×C4⋊C4C2311yescharacteristic
74C5×C22⋊C4—21no—
75C5×C22⋊C4—21no—
76C42×C5C2311yescharacteristic
77C22⋊Dic5—21no—
78C22⋊Dic5—21no—
79C22⋊Dic5—21no—
80#689—22no—
81#687—22no—
82C4⋊Dic5—21no—
83C4⋊Dic5D411yescharacteristic
84C4⋊Dic5C2311yescharacteristic
85C4⋊Dic5C2311yescharacteristic
86#683—21no—
87#683—21no—
88#683—21no—
89#683—21no—
90#683—21no—
91#683—21no—
92#683—21no—
93#683—21no—
94#683D411yescharacteristic
95#683D411yescharacteristic
96C4×Dic5—21no—
97C4×Dic5—21no—
98C4×Dic5—21no—
99C4×Dic5—21no—
100C4×Dic5—21no—
101C4×Dic5C2311yescharacteristic
102C4×Dic5C2311yescharacteristic
103C2×#168C2311yescharacteristic
104C2×#168C2311yescharacteristic
105Dic20—22no—
106Q82—51no—
107#556—51no—
108#489—51no—
109#489D511yescharacteristic
110#487—51no—
111#476—51no—
112#475—51no—
113#475—51no—
114#464—51no—
115#464—51no—
116#462—51no—
117#446—51no—
118#441—51no—
119#441—51no—
120C4×SD16—51no—
121C23×C5—21no—
122C5×Q8—21no—
123C5×Q8—21no—
124C5×Q8—81no—
125C5×D4—21no—
126C5×D4—21no—
127C2×C4×C5—21no—
128C2×C4×C5—21no—
129C2×C4×C5—21no—
130C2×C4×C5C2×D411yescharacteristic
131C2×C4×C5C2×D411yescharacteristic
132C2×C4×C5C2411yescharacteristic
133C2×Dic5—21no—
134C2×Dic5—21no—
135C2×Dic5—21no—
136C2×Dic5—21no—
137C2×Dic5—42no—
138C2×Dic5C2×D411yescharacteristic
139C2×Dic5C2×D411yescharacteristic
140C2×Dic5C2×D411yescharacteristic
141C2×Dic5C2×D411yescharacteristic
142Dic10—21no—
143Dic10—22no—
144Dic10—21no—
145Dic10—21no—
146Dic10—81no—
147Dic10—81no—
148Dic10#4112yes—
149C5×C8—21no—
150C5×C8—21no—
151#168—21no—
152#168—21no—
153C2×Q16—51no—
154C2×Q16—51no—
155C2×SD16—51no—
156C4⋊Q8—51no—
157C4⋊Q8—51no—
158C4⋊Q8—51no—
159C4⋊Q8—101no—
160C4⋊Q8—101no—
161C4⋊Q8—101no—
162C4⋊Q8C2×D511yescharacteristic
163#126—101no—
164#126—101no—
165#124—51no—
166#124C2×D511yescharacteristic
167C22⋊Q8—101no—
168C4×Q8—51no—
169C4×Q8—51no—
170C4×Q8—51no—
171C4×Q8—51no—
172C4×Q8—51no—
173C4×Q8—51no—
174C4×Q8—101no—
175C4×D4—51no—
176#117—101no—
177#106—51no—
178#106—51no—
179#106—51no—
180C4⋊C8—51no—
181C4⋊C8—51no—
182C4⋊C8—51no—
183#103—51no—
184#103—51no—
185#103—51no—
186#103—51no—
187#103—51no—
188#103C2×D511yescharacteristic
189#103C2×D511yescharacteristic
190#102—51no—
191#102C2×D511yescharacteristic
192#102C2×D511yescharacteristic
193#97—51no—
194#97C2×D511yescharacteristic
195C4×C8—51no—
196C22×C5—41no—
197C22×C5—41no—
198C22×C5—41no—
199C22×C5C22×D411yescharacteristic
200C4×C5—21no—
201C4×C5—21no—
202C4×C5—41no—
203C4×C5—41no—
204C4×C5—41no—
205C4×C5C2×#4111yescharacteristic
206C4×C52−1+411yescharacteristic
207Dic5—22no—
208Dic5—21no—
209Dic5—21no—
210Dic5—21no—
211Dic5—41no—
212Dic5—41no—
213Dic5—41no—
214Dic5#13712yes—
215C2×Q8—21no—
216C2×Q8—51no—
217C2×Q8—51no—
218C2×Q8—51no—
219C2×Q8—101no—
220C2×Q8—101no—
221C2×Q8C22×D511yescharacteristic
222C2×D4C22×D511yescharacteristic
223C22×C4—101no—
224Q16—102no—
225Q16—102no—
226SD16—102no—
227C2×C8—51no—
228C2×C8—51no—
229C2×C8C22×D511yescharacteristic
230C2×C8C22×D511yescharacteristic
231C4⋊C4—21no—
232C4⋊C4—51no—
233C4⋊C4—51no—
234C4⋊C4—51no—
235C4⋊C4—51no—
236C4⋊C4—51no—
237C4⋊C4—101no—
238C4⋊C4—101no—
239C4⋊C4—101no—
240C4⋊C4—101no—
241C4⋊C4—101no—
242C4⋊C4—101no—
243C4⋊C4—101no—
244C4⋊C4—101no—
245C4⋊C4—101no—
246C4⋊C4C22×D511yescharacteristic
247C4⋊C4C22×D511yescharacteristic
248C22⋊C4—21no—
249C22⋊C4—21no—
250C22⋊C4—101no—
251C22⋊C4—101no—
252C22⋊C4—101no—
253C42—51no—
254C42—51no—
255C42—101no—
256C42—101no—
257C42—101no—
258C42—101no—
259C42—101no—
260C42C22×D511yescharacteristic
261C2×C5—81no—
262C2×C5#54811yescharacteristic
263C2×C5C2×#13711yescharacteristic
264C2×C5#57711yescharacteristic
265C23—21no—
266Q8—21no—
267Q8—21no—
268Q8—52no—
269Q8—81no—
270Q8—102no—
271Q8—101no—
272Q8—101no—
273Q8—101no—
274Q8—401no—
275Q8—401no—
276D4—21no—
277D4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—51no—
282C2×C4—51no—
283C2×C4—51no—
284C2×C4—51no—
285C2×C4—101no—
286C2×C4—101no—
287C2×C4—101no—
288C2×C4—101no—
289C2×C4—202no—
290C2×C4D4×D511yescharacteristic
291C2×C4D4×D511yescharacteristic
292C2×C4C23×D511yescharacteristic
293C8—21no—
294C8—21no—
295C8—101no—
296C8—101no—
297C5#336911yescharacteristic, complemented, Sylow
298C22—41no—
299C22—41no—
300C22—41no—
301C22C2×D4×D511yescharacteristic
302C4—21no—
303C4—21no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—52no—
308C4—101no—
309C4—101no—
310C4—101no—
311C4—102no—
312C4—201no—
313C4—201no—
314C4—201no—
315C4D5×#4111yescharacteristic
316C4#415011yescharacteristic
317C2—81no—
318C2#6601111yescharacteristic
319C2#6610911yescharacteristic
320C2D5×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#489C2
socleC22×C5C22×D4
prime radicalC2×C5×D4C23
hypercenter#489D5
hypocenterC5#3369
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #66056C2semidirect
4. #65383C2semidirect
6. #65082C2semidirect
7. #65081C2semidirect
8. #65080C2semidirect
9. #65078C2semidirect
14. #65005C2semidirect
15. #64990C2semidirect
297. C5#3369semidirect

Maximal subgroups

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

subgroupindex
#660562
#660052
C5×#4892
#653832
#653432
#650822
#650812
#650802
#650782
#650562
#650542
#650502
#650472
#650052
#649902
#33695