Finite Group Databasebeta

← #10617756

Subgroups — #10617756

Order 640.

1030 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C5×#41C211yesmaximal, characteristic, complemented
2#66168C211yesmaximal, characteristic, complemented
3#66165C211yesmaximal, characteristic, complemented
4#65910C211yesmaximal, characteristic, complemented
5#65908C211yesmaximal, characteristic, complemented
6#65906C211yesmaximal, characteristic, complemented
7#65903C211yesmaximal, characteristic, complemented
8#65896C213yesmaximal, complemented
9#65895C213yesmaximal, complemented
10#65893C213yesmaximal, complemented
11#65892C213yesmaximal, complemented
12#65866C213yesmaximal, complemented
13#65863C213yesmaximal, complemented
14#65813C213yesmaximal, complemented
15#65813C213yesmaximal, complemented
16C2×C5×#41C2211yescharacteristic
17C2×#713C2211yescharacteristic
18C2×Q8×D5C2211yescharacteristic
19C4×C5×Q8C2211yescharacteristic
20C4×C5×Q8C2211yescharacteristic
21C4×C5×D4C2213yes—
22C4×C5×D4C2213yes—
23C5×#117C2213yes—
24C2×C42×C5C2213yes—
25Q8×Dic5C2211yescharacteristic
26Q8×Dic5C2211yescharacteristic
27#4086C2213yes—
28#4084C2213yes—
29#4084C2213yes—
30#4082C2213yes—
31#4076C2213yes—
32#4076C2213yes—
33#4075C2213yes—
34#4075C2213yes—
35#4075C2216yes—
36C4×C22⋊D5C2213yes—
37C4×C22⋊D5C2213yes—
38C4×C22⋊D5C2216yes—
39#4070C2213yes—
40#4070C2213yes—
41#4043C2213yes—
42#4041C2213yes—
43#4039C2213yes—
44#4038C2213yes—
45D5×C4⋊C4C2213yes—
46#4036C2213yes—
47#4034C2213yes—
48#4033C2213yes—
49#4031C2216yes—
50#4030C2216yes—
51#4029C2216yes—
52#4025C2216yes—
53#4022C2211yescharacteristic
54#4022C2213yes—
55#4021C2213yes—
56#4021C2213yes—
57C4×D20C2213yes—
58C4×D20C2213yes—
59#4018C2216yes—
60#4018C2213yes—
61#4018C2213yes—
62#4016C2213yes—
63#4016C2213yes—
64C4×Dic10C2213yes—
65C4×Dic10C2213yes—
66#3524—51nomaximal, Sylow
67C5×#41—24no—
68C2×C5×Q8C2311yescharacteristic
69C2×C5×D4C2313yes—
70C22×C4×C5C2313yes—
71C22×C4×C5C2313yes—
72C22×C4×C5C2313yes—
73C2×C22⋊D5C2313yes—
74C2×C22⋊D5C2313yes—
75C2×C22⋊D5C2316yes—
76#713—24no—
77Q8×D5—24no—
78C2×D20C2313yes—
79C2×D20C2313yes—
80C2×C4×D5C2313yes—
81C2×C4×D5C2313yes—
82C2×C4×D5C2316yes—
83C2×Dic10C2313yes—
84C2×Dic10C2313yes—
85C5×C4⋊C4C2313yes—
86C5×C4⋊C4C2313yes—
87C5×C22⋊C4C2316yes—
88C42×C5C2311yescharacteristic
89C42×C5C2313yes—
90C42×C5C2313yes—
91C42×C5C2313yes—
92C22⋊Dic5C2313yes—
93C22⋊Dic5C2316yes—
94C22⋊Dic5C2313yes—
95#685C2311yescharacteristic
96#685C2313yes—
97#685C2313yes—
98#685C2316yes—
99#685C2313yes—
100#685C2313yes—
101#685C2311yescharacteristic
102#685C2313yes—
103#685C2311yescharacteristic
104C4⋊Dic5C2313yes—
105C4⋊Dic5C2313yes—
106C4⋊Dic5C2313yes—
107C4⋊Dic5C2313yes—
108#683C2311yescharacteristic
109#683C2313yes—
110#683C2313yes—
111#683C2311yescharacteristic
112#683C2313yes—
113#683C2313yes—
114#683C2316yes—
115#683C2313yes—
116#683C2311yescharacteristic
117C4×Dic5C2316yes—
118C4×Dic5C2313yes—
119C4×Dic5C2313yes—
120#559—53no—
121#557—51no—
122#555—53no—
123#553—53no—
124#552—53no—
125#550—51no—
126#548—51no—
127#548—51no—
128#542—53no—
129#540—53no—
130#529—53no—
131#529—53no—
132#520—51no—
133#519—51no—
134C4×#41D511yescharacteristic, complemented
135C23×C5C2413yes—
136C22×D5C2411yescharacteristic
137C22×D5C2413yes—
138C5×Q8#4114yes—
139C5×D4—26no—
140C2×C4×C5—26no—
141C2×C4×C5—43no—
142C2×C4×C5—43no—
143C2×C4×C5C2413yes—
144C2×C4×C5C2411yescharacteristic
145C2×C4×C5C2411yescharacteristic
146C2×C4×C5C2411yescharacteristic
147C2×C4×C5C2413yes—
148C2×C4×C5C2413yes—
149C22⋊D5—46no—
150C22⋊D5—43no—
151C22⋊D5—43no—
152C2×Dic5C2411yescharacteristic
153C2×Dic5C2413yes—
154C2×Dic5C2411yescharacteristic
155C2×Dic5C2413yes—
156C2×Dic5C2413yes—
157C2×Dic5C2411yescharacteristic
158D20—26no—
159D20—43no—
160C4×D5—26no—
161C4×D5—26no—
162C4×D5—46no—
163Dic10—26no—
164Dic10—43no—
165C2×#41—51no—
166C2×#41C2×D511yescharacteristic
167C22×Q8—51no—
168C4⋊Q8—53no—
169#127—53no—
170#126—56no—
171#126—51no—
172#126—53no—
173#125—53no—
174#125—53no—
175#125—53no—
176#124—53no—
177#124—56no—
178#124—53no—
179#124—53no—
180#123—56no—
181#123—53no—
182#123—53no—
183#123—56no—
184C22⋊Q8—53no—
185C22⋊Q8—53no—
186C22⋊Q8—53no—
187#121—53no—
188#121—53no—
189#121—53no—
190#121—53no—
191#121—53no—
192#121—56no—
193C4×Q8—51no—
194C4×Q8—51no—
195C4×Q8—53no—
196C4×Q8—53no—
197C4×Q8—53no—
198C4×Q8C2×D511yescharacteristic
199C4×Q8C2×D511yescharacteristic
200C4×D4—53no—
201C4×D4—53no—
202C4×D4—56no—
203C4×D4—53no—
204C4×D4—53no—
205C4×D4—53no—
206C4×D4C2×D513yes—
207C4×D4C2×D513yes—
208#117—53no—
209#117—53no—
210#117—53no—
211#117—56no—
212#117C2×D513yes—
213C2×C4⋊C4—53no—
214C2×C42C2×D513yes—
215C22×C5—23no—
216C22×C5—23no—
217C22×C5—23no—
218C22×C5C2511yescharacteristic
219C2×D5—21no—
220C2×D5—23no—
221C2×D5—23no—
222C2×D5—21no—
223C2×D5—23no—
224C2×D5—21no—
225C4×C5—21no—
226C4×C5—23no—
227C4×C5—23no—
228C4×C5—21no—
229C4×C5—21no—
230C4×C5C2×#4116yes—
231Dic5—23no—
232Dic5—21no—
233Dic5—21no—
234Dic5—23no—
235Dic5—23no—
236Dic5—21no—
237#41—24no—
238#41—104no—
239C2×Q8—53no—
240C2×Q8—53no—
241C2×Q8—104no—
242C2×Q8C22×D511yescharacteristic
243C2×D4—53no—
244C2×D4—56no—
245C2×D4—53no—
246C2×D4—53no—
247C2×D4—53no—
248C2×D4C22×D513yes—
249C22×C4—53no—
250C22×C4—53no—
251C22×C4—56no—
252C22×C4C22×D513yes—
253C22×C4C22×D513yes—
254C22×C4C22×D513yes—
255C4⋊C4—53no—
256C4⋊C4—53no—
257C4⋊C4—51no—
258C4⋊C4—53no—
259C4⋊C4—53no—
260C4⋊C4—53no—
261C4⋊C4—53no—
262C4⋊C4—56no—
263C4⋊C4—51no—
264C4⋊C4—53no—
265C4⋊C4—53no—
266C4⋊C4—51no—
267C4⋊C4—53no—
268C4⋊C4C22×D513yes—
269C4⋊C4C22×D513yes—
270C22⋊C4—53no—
271C22⋊C4—56no—
272C22⋊C4—53no—
273C22⋊C4—51no—
274C22⋊C4—53no—
275C22⋊C4—53no—
276C22⋊C4—53no—
277C22⋊C4—53no—
278C22⋊C4—56no—
279C22⋊C4—51no—
280C22⋊C4—53no—
281C22⋊C4—51no—
282C22⋊C4C22×D516yes—
283C42—56no—
284C42—53no—
285C42—53no—
286C42C22×D511yescharacteristic
287C42C22×D513yes—
288C42C22×D513yes—
289C42C22×D513yes—
290C2×C5—43no—
291C2×C5C22×#4111yescharacteristic
292C2×C5#58511yescharacteristic
293C2×C5#58511yescharacteristic
294D5—41no—
295D5—43no—
296C23—51no—
297C23—53no—
298C23C23×D513yes—
299Q8—106no—
300Q8—203no—
301Q8#70914yes—
302D4—26no—
303D4—106no—
304D4—203no—
305D4—203no—
306D4—203no—
307D4—206no—
308C2×C4—26no—
309C2×C4—43no—
310C2×C4—43no—
311C2×C4—51no—
312C2×C4—53no—
313C2×C4—51no—
314C2×C4—53no—
315C2×C4—53no—
316C2×C4—51no—
317C2×C4—106no—
318C2×C4—106no—
319C2×C4—206no—
320C2×C4C23×D513yes—
321C2×C4C23×D511yescharacteristic
322C2×C4C23×D511yescharacteristic
323C2×C4C23×D511yescharacteristic
324C2×C4C23×D513yes—
325C2×C4C23×D513yes—
326C5#352411yescharacteristic, complemented, Sylow
327C22—23no—
328C22—23no—
329C22—23no—
330C22—101no—
331C22—103no—
332C22—103no—
333C22—101no—
334C22—103no—
335C22—101no—
336C22C24×D511yescharacteristic
337C4—21no—
338C4—21no—
339C4—21no—
340C4—23no—
341C4—23no—
342C4—101no—
343C4—103no—
344C4—103no—
345C4—101no—
346C4—103no—
347C4—101no—
348C4C2×#70916yes—
349C2—43no—
350C2—201no—
351C2—203no—
352C2C22×#70911yescharacteristic
353C2#6628311yescharacteristic
354C2#6628311yescharacteristic
subgroupG/H
center, FrattiniC22C24×D5
commutator, socle, cosocleC22×C5C25
FittingC4×C5×#41C2
hypercenterC4×#41D5
hypocenterC5#3524
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C4×#41

Complements

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

NH ≅ G/Ntype
1. C4×C5×#41C2semidirect
2. #66168C2semidirect
3. #66165C2semidirect
4. #65910C2semidirect
5. #65908C2semidirect
6. #65906C2semidirect
7. #65903C2semidirect
8. #65896C2semidirect
9. #65895C2semidirect
10. #65893C2semidirect
11. #65892C2semidirect
12. #65866C2semidirect
13. #65863C2semidirect
14. #65813C2semidirect
15. #65813C2semidirect
134. C4×#41D5semidirect
326. C5#3524semidirect

Maximal subgroups

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

subgroupindex
C4×C5×#412
#661682
#661652
#659102
#659082
#659062
#659032
#658962
#658952
#658932
#658922
#658662
#658632
#658132
#35245