Finite Group Databasebeta

← C5×#64307 #13089320

Subgroups — C5×#64307 #13089320

Order 1440.

704 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×#3828C211yesmaximal, characteristic, complemented
2C3×C5×#235C211yesmaximal, characteristic, complemented
3C2×C3×C5×D12C211yesmaximal, characteristic, complemented
4C5×D3×#67C214yesmaximal, complemented
5C5×#3802C212yesmaximal, complemented
6C5×D3×D12C212yesmaximal, complemented
7C5×#3796C212yesmaximal, complemented
8C5×#3795C212yesmaximal, complemented
9C5×#1009—31nomaximal
10C5×#1004—31nomaximal
11C22×C3×C5×D3C2212yescomplemented
12C2×C5×D32C2214yescomplemented
13C2×C32×C4×C5C2211yescharacteristic, complemented
14C5×#638C2212yescomplemented
15C5×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
16C4×C5×C3⋊D3C2212yescomplemented
17C5×#634C2211yescharacteristic, complemented
18C3×C5×#67C2212yescomplemented
19C3×C5×D12C2212yescomplemented
20C3×C5×D12C2212yescomplemented
21C3×C5×D12C2211yescharacteristic, complemented
22C3×C4×C5×D3C2212yescomplemented
23C3×C5×Dic6C2211yescharacteristic, complemented
24C5×#626C2214yescomplemented
25C5×#625C2214yescomplemented
26C5×D3×Dic3C2214yescomplemented
27#64307C511yesmaximal, characteristic, direct product factor, Hall
28C3×C5×#41—31no—
29C2×C3×C5×D4—31no—
30C2×C5×#67—34no—
31C5×#239—32no—
32C5×#237—32no—
33C5×#237—32no—
34C5×D3×D4—34no—
35C5×D3×D4—32no—
36C5×D3×D4—32no—
37C5×D3×D4—32no—
38C5×#235—31no—
39C5×#235—31no—
40C5×#235—32no—
41C5×#235—61no—
42C5×#235D311yescharacteristic, complemented
43C2×C5×D12—32no—
44C2×C5×D12D311yescharacteristic, complemented
45C22×C32×C5C2311yescharacteristic
46C2×C5×C3⋊D3C2312yes—
47C2×C3×C5×D3—42no—
48C2×C3×C5×D3C2314yes—
49C2×C3×C5×D3C2312yes—
50C5×D32—44no—
51C32×C4×C5C2311yescharacteristic
52C32×C4×C5C2311yescharacteristic
53C5×C3⋊Dic3C2312yes—
54C3×C5×Dic3C2312yes—
55C5×H(5,𝔽2)—91noHall
56#3828C2×C511yescharacteristic, complemented
57C3×#235C2×C511yescharacteristic, complemented
58C2×C3×D12C2×C511yescharacteristic, complemented
59D3×#67C2×C514yescomplemented
60#3802C2×C512yescomplemented
61D3×D12C2×C512yescomplemented
62#3796C2×C512yescomplemented
63#3795C2×C512yescomplemented
64C23×C3×C5—32no—
65C22×C5×D3—34no—
66C22×C5×D3—34no—
67C22×C5×D3C2×D312yescomplemented
68C3×C5×Q8—31no—
69C3×C5×D4—32no—
70C3×C5×D4—31no—
71C3×C5×D4—32no—
72C3×C5×D4—32no—
73C2×C3×C4×C5—21no—
74C2×C3×C4×C5—32no—
75C2×C3×C4×C5C2×D311yescharacteristic, complemented
76C2×C3×C4×C5C2×D311yescharacteristic, complemented
77C5×#67—32no—
78C5×#67—32no—
79C5×#67—34no—
80C5×#67—34no—
81C5×#67—34no—
82C5×#67—62no—
83C5×#67C2×D312yescomplemented
84C2×C5×Dic3—34no—
85C5×D12—34no—
86C5×D12—31no—
87C5×D12—31no—
88C5×D12—61no—
89C5×D12C2×D312yescomplemented
90C5×D12C2×D312yescomplemented
91C5×D12C2×D311yescharacteristic, complemented
92C4×C5×D3—34no—
93C4×C5×D3—32no—
94C4×C5×D3—32no—
95C4×C5×D3—62no—
96C4×C5×D3C2×D312yescomplemented
97C5×Dic6—31no—
98C5×Dic6—31no—
99C5×Dic6—61no—
100C5×Dic6C2×D311yescharacteristic, complemented
101#1009—31no—
102#1004—31no—
103C2×C32×C5—21no—
104C2×C32×C5C2411yescharacteristic
105C5×C3⋊D3—22no—
106C3×C5×D3—22no—
107C3×C5×D3—24no—
108C5×#41—31no—
109C5×#41—92no—
110C5×#41—92no—
111C5×#41—91no—
112C2×C5×D4—31no—
113C2×C5×D4—92no—
114C2×C5×D4—92no—
115C2×C5×D4—94no—
116C22×C3×D3C22×C512yescomplemented
117C2×D32C22×C514yescomplemented
118C2×C32×C4C22×C511yescharacteristic, complemented
119#638C22×C512yescomplemented
120C4⋊(C3⋊D3)C22×C511yescharacteristic, complemented
121C4×C3⋊D3C22×C512yescomplemented
122#634C22×C511yescharacteristic, complemented
123C3×#67C22×C512yescomplemented
124C3×D12C22×C512yescomplemented
125C3×D12C22×C512yescomplemented
126C3×D12C22×C511yescharacteristic, complemented
127C3×C4×D3C22×C512yescomplemented
128C3×Dic6C22×C511yescharacteristic, complemented
129#626C22×C514yescomplemented
130#625C22×C514yescomplemented
131D3×Dic3C22×C514yescomplemented
132C22×C3×C5—21no—
133C22×C3×C5—34no—
134C22×C3×C5—32no—
135C22×C3×C5—122no—
136C22×C3×C5C22×D311yescharacteristic
137C22×C3×C5C22×D311yescharacteristic
138C2×C5×D3—32no—
139C2×C5×D3—32no—
140C2×C5×D3—42no—
141C2×C5×D3—62no—
142C2×C5×D3—124no—
143C2×C5×D3—124no—
144C2×C5×D3C22×D314yes—
145C2×C5×D3C22×D312yes—
146C3×C4×C5—21no—
147C3×C4×C5—21no—
148C3×C4×C5—32no—
149C3×C4×C5C22×D311yescharacteristic
150C3×C4×C5C22×D311yescharacteristic
151C3×C4×C5C22×D311yescharacteristic
152C3×C4×C5C22×D311yescharacteristic
153C5×Dic3—32no—
154C5×Dic3—32no—
155C5×Dic3—62no—
156C5×Dic3C22×D312yes—
157C3×#41—31no—
158C2×C3×D4—31no—
159C2×#67—34no—
160#239—32no—
161#237—32no—
162#237—32no—
163D3×D4—34no—
164D3×D4—32no—
165D3×D4—32no—
166D3×D4—32no—
167#235—31no—
168#235—31no—
169#235—32no—
170#235—61no—
171#235C5×D311yescharacteristic, complemented
172C2×D12—32no—
173C2×D12C5×D311yescharacteristic, complemented
174C32×C5H(5,𝔽2)11yescharacteristic, complemented, Hall
175C23×C5—32no—
176C23×C5—94no—
177C5×Q8—31no—
178C5×Q8—91no—
179C5×D4—32no—
180C5×D4—32no—
181C5×D4—32no—
182C5×D4—31no—
183C5×D4—94no—
184C5×D4—94no—
185C5×D4—92no—
186C5×D4—91no—
187C2×C4×C5—32no—
188C2×C4×C5—94no—
189C2×C4×C5—92no—
190C2×C4×C5D3211yescharacteristic, complemented
191C22×C32C23×C511yescharacteristic
192C2×C3⋊D3C23×C512yes—
193C2×C3×D3—42no—
194C2×C3×D3C23×C514yes—
195C2×C3×D3C23×C512yes—
196D32—44no—
197C32×C4C23×C511yescharacteristic
198C32×C4C23×C511yescharacteristic
199C3⋊Dic3C23×C512yes—
200C3×Dic3C23×C512yes—
201H(5,𝔽2)—91noSylow
202C2×C3×C5—21no—
203C2×C3×C5—21no—
204C2×C3×C5—21no—
205C2×C3×C5—41no—
206C2×C3×C5—62no—
207C2×C3×C5—64no—
208C2×C3×C5C23×D311yescharacteristic
209C2×C3×C5C23×D311yescharacteristic
210C5×D3—24no—
211C5×D3—22no—
212C5×D3—62no—
213C5×D3—62no—
214C5×D3—122no—
215C23×C3—32no—
216C22×D3—34no—
217C22×D3—34no—
218C22×D3C2×C5×D312yescomplemented
219C3×Q8—31no—
220C3×D4—32no—
221C3×D4—31no—
222C3×D4—32no—
223C3×D4—32no—
224C2×C3×C4—21no—
225C2×C3×C4—32no—
226C2×C3×C4C2×C5×D311yescharacteristic, complemented
227C2×C3×C4C2×C5×D311yescharacteristic, complemented
228#67—32no—
229#67—32no—
230#67—34no—
231#67—34no—
232#67—34no—
233#67—62no—
234#67C2×C5×D312yescomplemented
235C2×Dic3—34no—
236D12—34no—
237D12—31no—
238D12—31no—
239D12—61no—
240D12C2×C5×D312yescomplemented
241D12C2×C5×D312yescomplemented
242D12C2×C5×D311yescharacteristic, complemented
243C4×D3—34no—
244C4×D3—32no—
245C4×D3—32no—
246C4×D3—62no—
247C4×D3C2×C5×D312yescomplemented
248Dic6—31no—
249Dic6—31no—
250Dic6—61no—
251Dic6C2×C5×D311yescharacteristic, complemented
252C22×C5—32no—
253C22×C5—34no—
254C22×C5—92no—
255C22×C5—122no—
256C22×C5—364no—
257C22×C5C2×D3211yescharacteristic
258C4×C5—32no—
259C4×C5—92no—
260C4×C5C2×D3211yescharacteristic
261C4×C5C2×D3211yescharacteristic
262C2×C32—21no—
263C2×C32C24×C511yescharacteristic
264C3⋊D3—22no—
265C3×D3—22no—
266C3×D3—24no—
267#41—31no—
268#41—92no—
269#41—92no—
270#41—91no—
271C2×D4—31no—
272C2×D4—92no—
273C2×D4—92no—
274C2×D4—94no—
275C3×C5—21no—
276C3×C5#100411yescharacteristic, complemented
277C3×C5#100911yescharacteristic, complemented
278C22×C3—21no—
279C22×C3—34no—
280C22×C3—32no—
281C22×C3—122no—
282C22×C3C22×C5×D311yescharacteristic
283C22×C3C22×C5×D311yescharacteristic
284C2×D3—32no—
285C2×D3—32no—
286C2×D3—42no—
287C2×D3—62no—
288C2×D3—124no—
289C2×D3—124no—
290C2×D3C22×C5×D314yes—
291C2×D3C22×C5×D312yes—
292C3×C4—21no—
293C3×C4—21no—
294C3×C4—32no—
295C3×C4C22×C5×D311yescharacteristic
296C3×C4C22×C5×D311yescharacteristic
297C3×C4C22×C5×D311yescharacteristic
298C3×C4C22×C5×D311yescharacteristic
299Dic3—32no—
300Dic3—32no—
301Dic3—62no—
302Dic3C22×C5×D312yes—
303C2×C5—21no—
304C2×C5—64no—
305C2×C5—62no—
306C2×C5—182no—
307C2×C5C22×D3211yescharacteristic
308C32C5×H(5,𝔽2)11yescharacteristic, complemented, Sylow
309C23—32no—
310C23—94no—
311Q8—31no—
312Q8—91no—
313D4—32no—
314D4—32no—
315D4—32no—
316D4—31no—
317D4—94no—
318D4—94no—
319D4—92no—
320D4—91no—
321C2×C4—32no—
322C2×C4—94no—
323C2×C4—92no—
324C2×C4C5×D3211yescharacteristic, complemented
325C2×C3—21no—
326C2×C3—21no—
327C2×C3—21no—
328C2×C3—41no—
329C2×C3—62no—
330C2×C3—64no—
331C2×C3C23×C5×D311yescharacteristic
332C2×C3C23×C5×D311yescharacteristic
333D3—24no—
334D3—22no—
335D3—62no—
336D3—62no—
337D3—122no—
338C5#6430711yescharacteristic, direct product factor, Sylow
339C22—32no—
340C22—34no—
341C22—92no—
342C22—122no—
343C22—364no—
344C22C2×C5×D3211yescharacteristic
345C4—32no—
346C4—92no—
347C4C2×C5×D3211yescharacteristic
348C4C2×C5×D3211yescharacteristic
349C3—21no—
350C3C5×#100411yescharacteristic, complemented
351C3C5×#100911yescharacteristic, complemented
352C2—21no—
353C2—64no—
354C2—62no—
355C2—182no—
356C2C22×C5×D3211yescharacteristic
subgroupG/H
centerC2×C5C22×D32
commutator, cosocleC2×C32C24×C5
FrattiniC2C22×C5×D32
FittingC2×C32×C4×C5C22
socleC2×C32×C5C24
hypercenterC2×C4×C5D32
hypocenterC32C5×H(5,𝔽2)
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C5 ◁ C2×C4×C5

Complements

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

NH ≅ G/Ntype
1. C5×#3828C2semidirect
2. C3×C5×#235C2semidirect
3. C2×C3×C5×D12C2semidirect
4. C5×D3×#67C2semidirect
5. C5×#3802C2semidirect
6. C5×D3×D12C2semidirect
7. C5×#3796C2semidirect
8. C5×#3795C2semidirect
11. C22×C3×C5×D3C22semidirect
12. C2×C5×D32C22semidirect
13. C2×C32×C4×C5C22semidirect
14. C5×#638C22semidirect
15. C5×C4⋊(C3⋊D3)C22semidirect
16. C4×C5×C3⋊D3C22semidirect
17. C5×#634C22semidirect
18. C3×C5×#67C22semidirect
19. C3×C5×D12C22semidirect
20. C3×C5×D12C22semidirect
21. C3×C5×D12C22semidirect
22. C3×C4×C5×D3C22semidirect
23. C3×C5×Dic6C22semidirect
24. C5×#626C22semidirect
25. C5×#625C22semidirect
26. C5×D3×Dic3C22semidirect
27. #64307C5direct (N × H)
42. C5×#235D3semidirect
44. C2×C5×D12D3semidirect
56. #3828C2×C5semidirect
57. C3×#235C2×C5semidirect
58. C2×C3×D12C2×C5semidirect
59. D3×#67C2×C5semidirect
60. #3802C2×C5semidirect
61. D3×D12C2×C5semidirect
62. #3796C2×C5semidirect
63. #3795C2×C5semidirect
67. C22×C5×D3C2×D3semidirect
75. C2×C3×C4×C5C2×D3semidirect
76. C2×C3×C4×C5C2×D3semidirect
83. C5×#67C2×D3semidirect
89. C5×D12C2×D3semidirect
90. C5×D12C2×D3semidirect
91. C5×D12C2×D3semidirect
96. C4×C5×D3C2×D3semidirect
100. C5×Dic6C2×D3semidirect
116. C22×C3×D3C22×C5semidirect
117. C2×D32C22×C5semidirect
118. C2×C32×C4C22×C5semidirect
119. #638C22×C5semidirect
120. C4⋊(C3⋊D3)C22×C5semidirect
121. C4×C3⋊D3C22×C5semidirect
122. #634C22×C5semidirect
123. C3×#67C22×C5semidirect
124. C3×D12C22×C5semidirect
125. C3×D12C22×C5semidirect
126. C3×D12C22×C5semidirect
127. C3×C4×D3C22×C5semidirect
128. C3×Dic6C22×C5semidirect
129. #626C22×C5semidirect
130. #625C22×C5semidirect
131. D3×Dic3C22×C5semidirect
171. #235C5×D3semidirect
173. C2×D12C5×D3semidirect
174. C32×C5H(5,𝔽2)semidirect
190. C2×C4×C5D32semidirect
218. C22×D3C2×C5×D3semidirect
226. C2×C3×C4C2×C5×D3semidirect
227. C2×C3×C4C2×C5×D3semidirect
234. #67C2×C5×D3semidirect
240. D12C2×C5×D3semidirect
241. D12C2×C5×D3semidirect
242. D12C2×C5×D3semidirect
247. C4×D3C2×C5×D3semidirect
251. Dic6C2×C5×D3semidirect
276. C3×C5#1004semidirect
277. C3×C5#1009semidirect
308. C32C5×H(5,𝔽2)semidirect
324. C2×C4C5×D32semidirect
338. C5#64307direct (N × H)
350. C3C5×#1004semidirect
351. C3C5×#1009semidirect

Maximal subgroups

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

subgroupindex
C5×#38282
C3×C5×#2352
C2×C3×C5×D122
C5×D3×#672
C5×#38022
C5×D3×D122
C5×#37962
C5×#37952
C5×#10093
C5×#10043
#643075