Finite Group Databasebeta

← #10612901

Subgroups — #10612901

Order 640.

376 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#66023C211yesmaximal, characteristic, complemented
2#65880C211yesmaximal, characteristic
3C5×#478C211yesmaximal, characteristic, complemented
4#65352C211yesmaximal, characteristic, complemented
5#65296C211yesmaximal, characteristic, complemented
6D5×#107C211yesmaximal, characteristic, complemented
7#65164C211yesmaximal, characteristic, complemented
8#65150C211yesmaximal, characteristic, complemented
9#65146C211yesmaximal, characteristic, complemented
10#65139C211yesmaximal, characteristic, complemented
11#65128C211yesmaximal, characteristic, complemented
12#65061C211yesmaximal, characteristic
13#65060C211yesmaximal, characteristic
14#65055C211yesmaximal, characteristic, complemented
15#65048C211yesmaximal, characteristic, complemented
16C2×#711C2211yescharacteristic
17C5×#125C2211yescharacteristic
18C4×C5×D4C2211yescharacteristic
19D4×Dic5C2211yescharacteristic
20C4×C22⋊D5—21no—
21C2×#683—21no—
22C2×#676C2211yescharacteristic
23C2×C8×D5C2211yescharacteristic
24#4043—21no—
25#4042—21no—
26#4038C2211yescharacteristic
27D5×C4⋊C4C2211yescharacteristic
28D5×C4⋊C4C2211yescharacteristic
29#4036C2211yescharacteristic
30#4035—21no—
31#4034—21no—
32#4034C2211yescharacteristic
33#4033C2211yescharacteristic
34#4027—21no—
35#4023—21no—
36#4018C2211yescharacteristic
37#4015C2211yescharacteristic
38C4×Dic10C2211yescharacteristic
39C5×#107C2211yescharacteristic
40C5×#106C2211yescharacteristic
41C5×C4⋊C8C2211yescharacteristic
42C5×#102C2211yescharacteristic
43C5×#102C2211yescharacteristic
44#3964C2211yescharacteristic
45#3964C2211yescharacteristic
46#3950C2211yescharacteristic
47#3950C2211yescharacteristic
48#3948C2211yescharacteristic
49#3948C2211yescharacteristic
50#3946C2211yescharacteristic
51#3946C2211yescharacteristic
52#3942C2211yescharacteristic
53#3942C2211yescharacteristic
54#3940C2211yescharacteristic
55#3939C2211yescharacteristic
56#3939C2211yescharacteristic
57#3939C2211yescharacteristic
58C4⋊#168C2211yescharacteristic
59#3083—51nomaximal, Sylow
60C2×C5×D4C2311yescharacteristic
61C22×C4×C5—21no—
62C2×C22⋊D5—21no—
63C22×Dic5—21no—
64C22×Dic5—21no—
65#711—22no—
66#711Q814yescomplemented
67C2×C4×D5—21no—
68C2×C4×D5—21no—
69C2×C4×D5C2311yescharacteristic
70C2×Dic10—21no—
71C2×Dic10—21no—
72C2×Dic10C2311yescharacteristic
73C2×C5×C8C2311yescharacteristic
74C2×C5×C8C2311yescharacteristic
75C5×C4⋊C4—21no—
76C5×C4⋊C4—21no—
77C5×C4⋊C4C2311yescharacteristic
78C5×C4⋊C4C2311yescharacteristic
79C5×C4⋊C4C2311yescharacteristic
80C5×C22⋊C4—21no—
81C42×C5C2311yescharacteristic
82C22⋊Dic5—21no—
83#685—21no—
84#685D411yescharacteristic
85#685D411yescharacteristic
86C4⋊Dic5—21no—
87C4⋊Dic5—21no—
88C4⋊Dic5C2311yescharacteristic
89C4⋊Dic5C2311yescharacteristic
90C4⋊Dic5C2311yescharacteristic
91#683—21no—
92#683—21no—
93#683—21no—
94#683—21no—
95#683—21no—
96#683—21no—
97#683D411yescharacteristic
98#683D411yescharacteristic
99C4×Dic5—21no—
100C4×Dic5—21no—
101C4×Dic5—21no—
102C4×Dic5C2311yescharacteristic
103C2×#168C2311yescharacteristic
104C2×#168C2311yescharacteristic
105#676—22no—
106C8×D5—22no—
107#544—51no—
108#520—51no—
109#479—51no—
110#479—51no—
111#478—51no—
112#478D511yescharacteristic, complemented
113#477—51no—
114#477—51no—
115#474—51no—
116#474—51no—
117#428—51no—
118C2×#107—51no—
119#424—51no—
120#417—51no—
121#416—51no—
122C23×C5—21no—
123C22×D5C2×D411yescharacteristic
124C5×D4—21no—
125C5×D4C2×Q812yes—
126C2×C4×C5—21no—
127C2×C4×C5—21no—
128C2×C4×C5—21no—
129C2×C4×C5—81no—
130C2×C4×C5C2×D411yescharacteristic
131C2×C4×C5C2×D411yescharacteristic
132C2×C4×C5C2411yescharacteristic
133C22⋊D5—24no—
134C2×Dic5—21no—
135C2×Dic5—24no—
136C2×Dic5—21no—
137C2×Dic5—21no—
138C2×Dic5—21no—
139C2×Dic5—81no—
140C2×Dic5C2×D411yescharacteristic
141C2×Dic5C2×D411yescharacteristic
142C2×Dic5C2×D411yescharacteristic
143C4×D5—81no—
144C4×D5—81no—
145C4×D5C2×Q812yes—
146C4×D5#4112yes—
147Dic10—21no—
148Dic10—81no—
149Dic10—81no—
150Dic10C2×Q812yes—
151C5×C8—21no—
152C5×C8—21no—
153#168—21no—
154#168—21no—
155C2×#41—51no—
156C2×M16—51no—
157C22×C8—51no—
158C4⋊Q8—51no—
159C4⋊Q8—51no—
160C4⋊Q8—101no—
161#125—51no—
162#125—101no—
163#125C2×D511yescharacteristic
164C22⋊Q8—101no—
165C22⋊Q8—101no—
166C4×Q8—51no—
167C4×Q8—51no—
168C4×D4—51no—
169C4×D4—101no—
170C4×D4—101no—
171C4×D4C2×D511yescharacteristic
172#117—51no—
173#117—51no—
174#117—101no—
175C2×C4⋊C4—51no—
176C2×C4⋊C4—51no—
177C2×C4⋊C4—101no—
178#107—51no—
179#107—51no—
180#107—51no—
181#107—51no—
182#107—51no—
183#107C2×D511yescharacteristic
184#106—51no—
185#106C2×D511yescharacteristic
186C4⋊C8—51no—
187C4⋊C8—51no—
188C4⋊C8—51no—
189C4⋊C8C2×D511yescharacteristic
190#103—51no—
191#103—51no—
192#103—51no—
193#103—51no—
194#102—51no—
195#102—51no—
196#102C2×D511yescharacteristic
197#102C2×D511yescharacteristic
198C22×C5—22no—
199C22×C5—41no—
200C22×C5—41no—
201C22×C5C22×D411yescharacteristic
202C2×D5—21no—
203C2×D5—21no—
204C2×D5C22⋊Q812yes—
205C4×C5—21no—
206C4×C5—21no—
207C4×C5—41no—
208C4×C5—41no—
209C4×C5—41no—
210C4×C5C22×Q811yescharacteristic
211C4×C5C2×#4111yescharacteristic
212Dic5—22no—
213Dic5—21no—
214Dic5—21no—
215Dic5—41no—
216Dic5—41no—
217Dic5—41no—
218Dic5C22⋊Q812yes—
219#41—54no—
220#41—102no—
221C2×Q8—51no—
222C2×Q8—101no—
223C2×Q8—101no—
224C2×D4—101no—
225C2×D4C22×D511yescharacteristic
226C22×C4—21no—
227C22×C4—51no—
228C22×C4—101no—
229C22×C4—101no—
230C22×C4—101no—
231C22×C4—101no—
232M16—102no—
233C2×C8—51no—
234C2×C8—51no—
235C2×C8—102no—
236C2×C8C22×D511yescharacteristic
237C2×C8C22×D511yescharacteristic
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—51no—
241C4⋊C4—51no—
242C4⋊C4—51no—
243C4⋊C4—51no—
244C4⋊C4—51no—
245C4⋊C4—101no—
246C4⋊C4—101no—
247C4⋊C4—101no—
248C4⋊C4—101no—
249C4⋊C4—101no—
250C4⋊C4—101no—
251C4⋊C4—101no—
252C4⋊C4—101no—
253C4⋊C4C22×D511yescharacteristic
254C4⋊C4C22×D511yescharacteristic
255C4⋊C4C22×D511yescharacteristic
256C22⋊C4—21no—
257C22⋊C4—51no—
258C22⋊C4—51no—
259C22⋊C4—101no—
260C22⋊C4—101no—
261C42—51no—
262C42—101no—
263C42—101no—
264C42—101no—
265C42C22×D511yescharacteristic
266C2×C5—42no—
267C2×C5C2×C22⋊Q811yescharacteristic
268C2×C5#57611yescharacteristic
269C2×C5#57811yescharacteristic
270D5—22no—
271C23—21no—
272C23—51no—
273Q8—52no—
274Q8—101no—
275Q8—401no—
276Q8—401no—
277D4—21no—
278D4—104no—
279D4Q8×D512yes—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—51no—
284C2×C4—52no—
285C2×C4—52no—
286C2×C4—51no—
287C2×C4—51no—
288C2×C4—81no—
289C2×C4—104no—
290C2×C4—101no—
291C2×C4—101no—
292C2×C4—101no—
293C2×C4—101no—
294C2×C4—401no—
295C2×C4—401no—
296C2×C4—401no—
297C2×C4D4×D511yescharacteristic
298C2×C4D4×D511yescharacteristic
299C2×C4C23×D511yescharacteristic
300C8—21no—
301C8—21no—
302C8—101no—
303C8—101no—
304C5#308311yescharacteristic, complemented, Sylow
305C22—22no—
306C22—41no—
307C22—41no—
308C22—52no—
309C22—101no—
310C22—101no—
311C22C2×D4×D511yescharacteristic
312C4—21no—
313C4—21no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—52no—
318C4—101no—
319C4—101no—
320C4—102no—
321C4—201no—
322C4—201no—
323C4—201no—
324C4C2×Q8×D511yescharacteristic
325C4D5×#4111yescharacteristic
326C2—42no—
327C2—102no—
328C2D5×C22⋊Q811yescharacteristic
329C2#6610511yescharacteristic
330C2#6610811yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#478C2
socleC22×C5C22×D4
prime radicalC2×#711C22
hypercenter#478D5
hypocenterC5#3083
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #66023C2semidirect
3. C5×#478C2semidirect
4. #65352C2semidirect
5. #65296C2semidirect
6. D5×#107C2semidirect
7. #65164C2semidirect
8. #65150C2semidirect
9. #65146C2semidirect
10. #65139C2semidirect
11. #65128C2semidirect
14. #65055C2semidirect
15. #65048C2semidirect
66. #711Q8semidirect
112. #478D5semidirect
304. C5#3083semidirect

Maximal subgroups

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

subgroupindex
#660232
#658802
C5×#4782
#653522
#652962
D5×#1072
#651642
#651502
#651462
#651392
#651282
#650612
#650602
#650552
#650482
#30835