Finite Group Databasebeta

← #10615331

Subgroups — #10615331

Order 640.

382 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#552C211yesmaximal, characteristic, complemented
2#65885C211yesmaximal, characteristic
3#65358C211yesmaximal, characteristic, complemented
4#65355C211yesmaximal, characteristic, complemented
5#65341C211yesmaximal, characteristic, complemented
6#65340C211yesmaximal, characteristic, complemented
7#65336C211yesmaximal, characteristic, complemented
8#65336C211yesmaximal, characteristic, complemented
9#65320C211yesmaximal, characteristic, complemented
10#65315C211yesmaximal, characteristic, complemented
11#65300C211yesmaximal, characteristic, complemented
12#65291C211yesmaximal, characteristic, complemented
13#65285C211yesmaximal, characteristic, complemented
14#65259C211yesmaximal, characteristic, complemented
15#65255C211yesmaximal, characteristic, complemented
16C2×#709C2211yescharacteristic
17C2×#709C2211yescharacteristic
18C5×#126—21no—
19C5×#125—21no—
20C5×#125C2211yescharacteristic
21C5×#123—21no—
22C5×C22⋊Q8C2211yescharacteristic
23C5×C22⋊Q8C2211yescharacteristic
24C4×C5×Q8C2211yescharacteristic
25C4×C5×D4C2211yescharacteristic
26C5×#117—21no—
27C5×#117C2211yescharacteristic
28C2×C5×C4⋊C4C2211yescharacteristic
29C2×#689C2211yescharacteristic
30C2×#689C2211yescharacteristic
31C2×#688C2211yescharacteristic
32#4084—21no—
33#4070C2211yescharacteristic
34#4070C2211yescharacteristic
35C2×#681C2211yescharacteristic
36C22×#168C2211yescharacteristic
37D5×C4⋊C4—21no—
38#4029—21no—
39#4029—21no—
40C4×D20C2211yescharacteristic
41#4015C2211yescharacteristic
42#3967C2211yescharacteristic
43C22⋊#168C2211yescharacteristic
44C22⋊#168C2211yescharacteristic
45#3942C2211yescharacteristic
46#3942C2211yescharacteristic
47#3942C2211yescharacteristic
48#3942C2211yescharacteristic
49#3942C2211yescharacteristic
50#3942C2211yescharacteristic
51#3941C2211yescharacteristic
52#3941C2211yescharacteristic
53#3941C2211yescharacteristic
54#3940C2211yescharacteristic
55#3939C2211yescharacteristic
56#3939C2211yescharacteristic
57C4⋊#168C2211yescharacteristic
58#3935C2211yescharacteristic
59#3315—51nomaximal, Sylow
60C2×C5×Q8C2311yescharacteristic
61C2×C5×D4D411yescharacteristic
62C22×C4×C5—21no—
63C22×C4×C5C2311yescharacteristic
64C22×C4×C5C2311yescharacteristic
65C2×C22⋊D5—21no—
66C2×C22⋊D5—21no—
67#709—22no—
68#709—22no—
69#709—41no—
70#709—41no—
71C2×D20C2311yescharacteristic
72C2×C4×D5—21no—
73C2×C4×D5—21no—
74C2×C4×D5—21no—
75C2×Dic10C2311yescharacteristic
76C2×Dic10C2311yescharacteristic
77C5×C4⋊C4—21no—
78C5×C4⋊C4—21no—
79C5×C4⋊C4—21no—
80C5×C4⋊C4—21no—
81C5×C4⋊C4—21no—
82C5×C4⋊C4—21no—
83C5×C4⋊C4D411yescharacteristic
84C5×C4⋊C4C2311yescharacteristic
85C5×C4⋊C4C2311yescharacteristic
86C5×C4⋊C4C2311yescharacteristic
87C5×C22⋊C4—21no—
88C5×C22⋊C4—21no—
89C5×C22⋊C4—21no—
90C5×C22⋊C4D411yescharacteristic
91C5×C22⋊C4D411yescharacteristic
92C42×C5—21no—
93C42×C5—21no—
94C42×C5C2311yescharacteristic
95C22⋊Dic5—21no—
96C22⋊Dic5—21no—
97#689—41no—
98#689—41no—
99#688—22no—
100#685—21no—
101C4⋊Dic5—21no—
102C4⋊Dic5—21no—
103C4⋊Dic5C2311yescharacteristic
104#683—21no—
105#683—21no—
106#681—41no—
107C2×#168—41no—
108C2×#168C2311yescharacteristic
109C2×#168C2311yescharacteristic
110C2×#168C2311yescharacteristic
111C2×#168C2311yescharacteristic
112#552D511yescharacteristic, complemented
113#547—51no—
114#491—51no—
115#485—51no—
116#484—51no—
117#478—51no—
118#470—51no—
119#467—51no—
120#462—51no—
121#452—51no—
122#452—51no—
123#440—51no—
124#435—51no—
125#417—51no—
126#416—51no—
127C23×C5C2×D411yescharacteristic
128C23×C5C2×D411yescharacteristic
129C22×D5—21no—
130C5×Q8—21no—
131C5×Q8—21no—
132C5×D4—41no—
133C2×C4×C5—21no—
134C2×C4×C5—21no—
135C2×C4×C5—21no—
136C2×C4×C5—21no—
137C2×C4×C5—21no—
138C2×C4×C5—21no—
139C2×C4×C5—21no—
140C2×C4×C5—21no—
141C2×C4×C5—42no—
142C2×C4×C5C2×D411yescharacteristic
143C2×C4×C5C2×D411yescharacteristic
144C2×C4×C5C2×D411yescharacteristic
145C2×C4×C5C2×D411yescharacteristic
146C2×C4×C5C2411yescharacteristic
147C22⋊D5—42no—
148C22⋊D5—42no—
149C2×Dic5—21no—
150C2×Dic5—21no—
151C2×Dic5—21no—
152D20—21no—
153D20#4112yes—
154C4×D5—42no—
155C4×D5—42no—
156C4×D5—42no—
157Dic10—21no—
158Dic10—21no—
159Dic10—21no—
160Dic10—21no—
161#168—21no—
162#168—21no—
163#168—21no—
164#168—21no—
165C2×#41—51no—
166C2×#41—51no—
167C2×Q16—51no—
168C2×Q16—51no—
169C2×SD16—51no—
170C2×M16—51no—
171C22×C8—51no—
172C4⋊Q8—51no—
173#126—21no—
174#125—21no—
175#125C2×D511yescharacteristic
176#123—21no—
177#123—101no—
178#123—101no—
179C22⋊Q8—51no—
180C22⋊Q8—51no—
181C22⋊Q8C2×D511yescharacteristic
182C22⋊Q8C2×D511yescharacteristic
183#121—101no—
184C4×Q8C2×D511yescharacteristic
185C4×D4—51no—
186C4×D4C2×D511yescharacteristic
187#117—21no—
188#117C2×D511yescharacteristic
189C2×C4⋊C4—101no—
190C2×C4⋊C4C2×D511yescharacteristic
191#107—51no—
192#107—51no—
193#106—51no—
194C4⋊C8—51no—
195#103—51no—
196#103—51no—
197#103—51no—
198#103—51no—
199#103—51no—
200#103—51no—
201#103—51no—
202#102—51no—
203#102—51no—
204#102—51no—
205C22⋊C8—51no—
206C22⋊C8—51no—
207#97—51no—
208C22×C5—21no—
209C22×C5—21no—
210C22×C5—21no—
211C22×C5—21no—
212C22×C5—21no—
213C22×C5—21no—
214C22×C5C22×D411yescharacteristic
215C2×D5—22no—
216C2×D5—41no—
217C2×D5—41no—
218C4×C5—21no—
219C4×C5—21no—
220C4×C5—21no—
221C4×C5—21no—
222C4×C5—41no—
223C4×C5—41no—
224C4×C5—41no—
225C4×C5—41no—
226C4×C5C2×#4111yescharacteristic
227C4×C5H(5,𝔽2)11yescharacteristic
228Dic5—41no—
229Dic5—41no—
230Dic5—41no—
231#41—102no—
232#41—102no—
233#41—201no—
234#41—201no—
235C2×Q8—51no—
236C2×Q8—51no—
237C2×Q8C22×D511yescharacteristic
238C2×D4—51no—
239C2×D4—101no—
240C2×D4—101no—
241C2×D4C22⋊D511yescharacteristic
242C22×C4—21no—
243C22×C4—101no—
244C22×C4—101no—
245C22×C4—101no—
246C22×C4C22×D511yescharacteristic
247C22×C4C22×D511yescharacteristic
248Q16—201no—
249Q16—201no—
250SD16—102no—
251M16—201no—
252C2×C8—51no—
253C2×C8—51no—
254C2×C8—51no—
255C2×C8—51no—
256C2×C8—201no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—51no—
264C4⋊C4—101no—
265C4⋊C4—101no—
266C4⋊C4—101no—
267C4⋊C4—101no—
268C4⋊C4C22⋊D511yescharacteristic
269C4⋊C4C22×D511yescharacteristic
270C4⋊C4C22×D511yescharacteristic
271C4⋊C4C22×D511yescharacteristic
272C22⋊C4—21no—
273C22⋊C4—21no—
274C22⋊C4—21no—
275C22⋊C4—101no—
276C22⋊C4—101no—
277C22⋊C4—101no—
278C22⋊C4C22⋊D511yescharacteristic
279C22⋊C4C22⋊D511yescharacteristic
280C42—21no—
281C42—21no—
282C42C22×D511yescharacteristic
283C2×C5—41no—
284C2×C5—41no—
285C2×C5#54611yescharacteristic
286C2×C5#57511yescharacteristic
287C2×C5#57811yescharacteristic
288D5—42no—
289C23—101no—
290C23C2×C22⋊D511yescharacteristic
291C23C2×C22⋊D511yescharacteristic
292Q8—21no—
293Q8—21no—
294Q8—101no—
295Q8—101no—
296Q8—101no—
297Q8—101no—
298D4—41no—
299D4—52no—
300D4—101no—
301D4—202no—
302D4—202no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—42no—
312C2×C4—101no—
313C2×C4—101no—
314C2×C4—101no—
315C2×C4—202no—
316C2×C4—202no—
317C2×C4—202no—
318C2×C4C2×C22⋊D511yescharacteristic
319C2×C4C2×C22⋊D511yescharacteristic
320C2×C4C2×C22⋊D511yescharacteristic
321C2×C4C2×C22⋊D511yescharacteristic
322C2×C4C23×D511yescharacteristic
323C8—101no—
324C8—101no—
325C8—101no—
326C8—101no—
327C5#331511yescharacteristic, complemented, Sylow
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—102no—
335C22—201no—
336C22—201no—
337C22C22×C22⋊D511yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—201no—
347C4—201no—
348C4—201no—
349C4D5×#4111yescharacteristic
350C4#414911yescharacteristic
351C2—41no—
352C2—41no—
353C2—202no—
354C2#6615611yescharacteristic
355C2#6616311yescharacteristic
356C2#6617211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#552C2
socleC22×C5C22×D4
prime radical#65885C2
hypercenter#552D5
hypocenterC5#3315
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
1. C5×#552C2semidirect
3. #65358C2semidirect
4. #65355C2semidirect
5. #65341C2semidirect
6. #65340C2semidirect
7. #65336C2semidirect
8. #65336C2semidirect
9. #65320C2semidirect
10. #65315C2semidirect
11. #65300C2semidirect
12. #65291C2semidirect
13. #65285C2semidirect
14. #65259C2semidirect
15. #65255C2semidirect
112. #552D5semidirect
327. C5#3315semidirect

Maximal subgroups

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

subgroupindex
C5×#5522
#658852
#653582
#653552
#653412
#653402
#653362
#653202
#653152
#653002
#652912
#652852
#652592
#652552
#33155