Finite Group Databasebeta

← #10612249

Subgroups — #10612249

Order 640.

372 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#66005C211yesmaximal, characteristic, complemented
2D4×Dic10C211yesmaximal, characteristic, complemented
3C5×#463C211yesmaximal, characteristic
4#65453C211yesmaximal, characteristic, complemented
5#65451C211yesmaximal, characteristic, complemented
6#65442C211yesmaximal, characteristic
7D8×Dic5C211yesmaximal, characteristic
8#65344C211yesmaximal, characteristic, complemented
9#65307C211yesmaximal, characteristic, complemented
10#65139C211yesmaximal, characteristic, complemented
11#65117C211yesmaximal, characteristic, complemented
12#65083C211yesmaximal, characteristic, complemented
13#65081C211yesmaximal, characteristic, complemented
14#65054C211yesmaximal, characteristic, complemented
15#65048C211yesmaximal, characteristic, complemented
16C22×Dic10—21no—
17C2×C5×SD16C2211yescharacteristic
18C2×C5×D8C2211yescharacteristic
19C5×#124C2211yescharacteristic
20C4×C5×D4C2211yescharacteristic
21Q8×Dic5C2211yescharacteristic
22#4090C2211yescharacteristic
23#4082C2211yescharacteristic
24D4×Dic5C2211yescharacteristic
25D4×Dic5C2211yescharacteristic
26D4×Dic5C2211yescharacteristic
27C2×#687C2211yescharacteristic
28#4070—21no—
29#4034C2211yescharacteristic
30#4034C2211yescharacteristic
31#4025—21no—
32#4025—21no—
33#4024—21no—
34#4024—21no—
35#4024—21no—
36#4023—21no—
37#4015C2211yescharacteristic
38C4×Dic10C2211yescharacteristic
39C4×Dic10C2211yescharacteristic
40C5×C4⋊C8C2211yescharacteristic
41C5×#103C2211yescharacteristic
42C5×#102C2211yescharacteristic
43#3967C2211yescharacteristic
44#3967C2211yescharacteristic
45#3964C2211yescharacteristic
46#3964C2211yescharacteristic
47#3964C2211yescharacteristic
48#3950C2211yescharacteristic
49#3950C2211yescharacteristic
50#3948C2211yescharacteristic
51#3948C2211yescharacteristic
52#3947C2211yescharacteristic
53#3946C2211yescharacteristic
54#3946C2211yescharacteristic
55C8×Dic5C2211yescharacteristic
56#3942C2211yescharacteristic
57#3939C2211yescharacteristic
58#3935C2211yescharacteristic
59#3367—51nomaximal, Sylow
60C2×C5×Q8C2311yescharacteristic
61C2×C5×D4C2311yescharacteristic
62C2×C5×D4C2311yescharacteristic
63C22×C4×C5—21no—
64C22×Dic5—21no—
65C22×Dic5—21no—
66C22×Dic5—21no—
67C2×Dic10—21no—
68C2×Dic10—21no—
69C2×Dic10—21no—
70C2×Dic10—44no—
71C2×Dic10D411yescharacteristic
72C2×Dic10C2311yescharacteristic
73C5×SD16—22no—
74C5×D8—22no—
75C2×C5×C8C2311yescharacteristic
76C2×C5×C8C2311yescharacteristic
77C5×C4⋊C4C2311yescharacteristic
78C5×C22⋊C4—21no—
79C5×C22⋊C4—21no—
80C5×C22⋊C4—21no—
81C42×C5C2311yescharacteristic
82C22⋊Dic5—21no—
83C22⋊Dic5—21no—
84C22⋊Dic5—21no—
85C22⋊Dic5—21no—
86C22⋊Dic5—21no—
87#687—22no—
88C4⋊Dic5—21no—
89C4⋊Dic5—21no—
90C4⋊Dic5D411yescharacteristic
91C4⋊Dic5C2311yescharacteristic
92C4⋊Dic5C2311yescharacteristic
93C4⋊Dic5C2311yescharacteristic
94#683—21no—
95#683—21no—
96#683—21no—
97#683—21no—
98#683—21no—
99#683D411yescharacteristic
100#683D411yescharacteristic
101C4×Dic5—21no—
102C4×Dic5—21no—
103C4×Dic5C2311yescharacteristic
104C4×Dic5C2311yescharacteristic
105C2×#168C2311yescharacteristic
106C2×#168C2311yescharacteristic
107#556—51no—
108D4×Q8—51no—
109#489—51no—
110#489—51no—
111#487—51no—
112#477—51no—
113#477—51no—
114#474—51no—
115#463—51no—
116#463D511yescharacteristic
117#461—51no—
118#446—51no—
119#440—51no—
120#440—51no—
121C4×D8—51no—
122C23×C5—21no—
123C23×C5—21no—
124C5×Q8—21no—
125C5×Q8—21no—
126C5×D4—21no—
127C5×D4—21no—
128C5×D4—21no—
129C5×D4#4112yes—
130C2×C4×C5—21no—
131C2×C4×C5—21no—
132C2×C4×C5—42no—
133C2×C4×C5C2×D411yescharacteristic
134C2×C4×C5C2×D411yescharacteristic
135C2×C4×C5C2411yescharacteristic
136C2×Dic5—21no—
137C2×Dic5—21no—
138C2×Dic5—21no—
139C2×Dic5—21no—
140C2×Dic5—42no—
141C2×Dic5—42no—
142C2×Dic5—42no—
143C2×Dic5C2×D411yescharacteristic
144C2×Dic5C2×D411yescharacteristic
145C2×Dic5C2×D411yescharacteristic
146C2×Dic5C2×D411yescharacteristic
147Dic10—22no—
148Dic10—21no—
149Dic10—21no—
150Dic10—42no—
151Dic10—42no—
152Dic10—42no—
153C5×C8—21no—
154C5×C8—21no—
155#168—21no—
156#168—21no—
157C22×Q8—101no—
158C2×SD16—51no—
159C2×SD16C2×D511yescharacteristic
160C2×D8C2×D511yescharacteristic
161C4⋊Q8—51no—
162C4⋊Q8—51no—
163C4⋊Q8—51no—
164C4⋊Q8—51no—
165#126—101no—
166#126—101no—
167#124—51no—
168#124C2×D511yescharacteristic
169C22⋊Q8—101no—
170C22⋊Q8—101no—
171C22⋊Q8—101no—
172C22⋊Q8—101no—
173C4×Q8—51no—
174C4×Q8—51no—
175C4×Q8—51no—
176C4×D4—51no—
177C4×D4—51no—
178C4×D4—51no—
179C4×D4C2×D511yescharacteristic
180#117—101no—
181#107—51no—
182#107—51no—
183#107—51no—
184C4⋊C8—51no—
185C4⋊C8—51no—
186C4⋊C8C2×D511yescharacteristic
187#103—51no—
188#103—51no—
189#103—51no—
190#103—51no—
191#103—51no—
192#103C2×D511yescharacteristic
193#102—51no—
194#102—51no—
195#102—51no—
196#102C2×D511yescharacteristic
197#97—51no—
198#97—51no—
199C4×C8—51no—
200C22×C5—22no—
201C22×C5—41no—
202C22×C5—41no—
203C22×C5—41no—
204C22×C5—41no—
205C22×C5—41no—
206C22×C5C22×D411yescharacteristic
207C4×C5—21no—
208C4×C5—21no—
209C4×C5—41no—
210C4×C5—41no—
211C4×C5C2×#4111yescharacteristic
212C4×C52−1+411yescharacteristic
213Dic5—21no—
214Dic5—21no—
215Dic5—21no—
216Dic5—41no—
217Dic5—41no—
218Dic5—41no—
219Dic5—41no—
220Dic5AGL(1,ℤ/8)12yes—
221C2×Q8—51no—
222C2×Q8—51no—
223C2×Q8—101no—
224C2×Q8—101no—
225C2×Q8—101no—
226C2×Q8—204no—
227C2×Q8C22×D511yescharacteristic
228C2×D4C22×D511yescharacteristic
229C2×D4C22×D511yescharacteristic
230C22×C4—21no—
231C22×C4—101no—
232C22×C4—101no—
233C22×C4—101no—
234SD16—22no—
235SD16—102no—
236D8—22no—
237C2×C8—51no—
238C2×C8—51no—
239C2×C8C22×D511yescharacteristic
240C2×C8C22×D511yescharacteristic
241C4⋊C4—51no—
242C4⋊C4—51no—
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
255C22⋊C4—21no—
256C22⋊C4—21no—
257C22⋊C4—21no—
258C22⋊C4—101no—
259C22⋊C4—101no—
260C22⋊C4—101no—
261C22⋊C4—101no—
262C22⋊C4—101no—
263C42—51no—
264C42—51no—
265C42—101no—
266C42—101no—
267C42C22×D511yescharacteristic
268C2×C5—42no—
269C2×C5—81no—
270C2×C5#54811yescharacteristic
271C2×C5C2×AGL(1,ℤ/8)11yescharacteristic
272C2×C5#57811yescharacteristic
273C23—21no—
274C23—21no—
275Q8—21no—
276Q8—21no—
277Q8—102no—
278Q8—101no—
279Q8—101no—
280Q8—202no—
281Q8—202no—
282Q8—202no—
283D4—21no—
284D4—21no—
285D4—21no—
286D4#71112yes—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—42no—
290C2×C4—51no—
291C2×C4—51no—
292C2×C4—51no—
293C2×C4—51no—
294C2×C4—101no—
295C2×C4—101no—
296C2×C4—101no—
297C2×C4—101no—
298C2×C4—202no—
299C2×C4—202no—
300C2×C4—202no—
301C2×C4D4×D511yescharacteristic
302C2×C4D4×D511yescharacteristic
303C2×C4C23×D511yescharacteristic
304C8—21no—
305C8—21no—
306C8—101no—
307C8—101no—
308C5#336711yescharacteristic, complemented, Sylow
309C22—22no—
310C22—41no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22C2×D4×D511yescharacteristic
316C4—21no—
317C4—21no—
318C4—41no—
319C4—41no—
320C4—52no—
321C4—101no—
322C4—101no—
323C4—101no—
324C4—201no—
325C4—201no—
326C4—201no—
327C4—201no—
328C4C2×#71111yescharacteristic
329C4#415011yescharacteristic
330C2—42no—
331C2—81no—
332C2#6593211yescharacteristic
333C2#6610511yescharacteristic
334C2D5×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#463C2
socleC22×C5C22×D4
prime radicalC2×C5×D8C22
hypercenter#463D5
hypocenterC5#3367
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. #66005C2semidirect
2. D4×Dic10C2semidirect
4. #65453C2semidirect
5. #65451C2semidirect
8. #65344C2semidirect
9. #65307C2semidirect
10. #65139C2semidirect
11. #65117C2semidirect
12. #65083C2semidirect
13. #65081C2semidirect
14. #65054C2semidirect
15. #65048C2semidirect
308. C5#3367semidirect

Maximal subgroups

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

subgroupindex
#660052
D4×Dic102
C5×#4632
#654532
#654512
#654422
D8×Dic52
#653442
#653072
#651392
#651172
#650832
#650812
#650542
#650482
#33675