Finite Group Databasebeta

← C2×C5×#5752 #14509963

Subgroups — C2×C5×#5752 #14509963

Order 1920.

3024 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
1C2×C5×#1009C212yesmaximal, complemented
2C2×C3×C5×#131C211yesmaximal, characteristic, complemented
3C5×#5752C2116yesmaximal, direct product factor
4C2×C5×#908C216yesmaximal, complemented
5C2×C5×#904C213yesmaximal, complemented
6C22×C5×D24C213yesmaximal, complemented
7C2×C5×#576—31nomaximal, Hall
8C2×C3×C5×#41C2211yescharacteristic
9C5×#1009—28no—
10C5×#1009C22116yescomplemented
11C2×C5×#239—22no—
12C2×C5×D3×D4—26no—
13C2×C5×#235C2216yescomplemented
14C22×C5×D12C2216yescomplemented
15C3×C5×#131C2218yescomplemented
16C2×C3×C5×M16C2213yescomplemented
17C22×C3×C5×C8C2213yescomplemented
18C5×#908C22148yescomplemented
19C2×C5×Dic12C2211yescharacteristic, complemented
20C5×#904C22124yescomplemented
21C2×C5×D24C2213yescomplemented
22C2×C5×D24C2216yescomplemented
23C2×C5×D24C22124yescomplemented
24C2×C5×#204C2216yescomplemented
25C2×#5752C511yesmaximal, characteristic, direct product factor, Hall
26C2×C5×H(5,𝔽2)—32no—
27C5×#576—316no—
28C2×C5×AGL(1,ℤ/8)—36no—
29C2×C5×#135—33no—
30C22×C5×D8—33no—
31C2×C5×#131D311yescharacteristic, complemented
32C23×C5×D3—26no—
33C3×C5×#41D414yes—
34C3×C5×#41C2314yes—
35C2×C3×C5×Q8D411yescharacteristic
36C2×C3×C5×D4D413yes—
37C22×C3×C4×C5C2313yes—
38C2×C5×#67—26no—
39C5×#239—216no—
40C5×D3×D4—248no—
41C5×#235—212no—
42C5×#235C23124yescomplemented
43C2×C5×D12—26no—
44C2×C5×D12—212no—
45C2×C5×D12C2316yes—
46C2×C5×D12C23124yescomplemented
47C2×C4×C5×D3—26no—
48C2×C5×Dic6C2312yes—
49C3×C5×M16C23112yescomplemented
50C2×C3×C5×C8C2313yes—
51C2×C3×C5×C8C2311yescharacteristic
52C2×C3×C5×C8C23112yescomplemented
53C5×Dic12C2314yescomplemented
54C5×D24C23112yescomplemented
55C5×D24C23124yescomplemented
56C5×#204C23124yescomplemented
57C2×#1009C2×C512yescomplemented
58C2×C3×#131C2×C511yescharacteristic, complemented
59#5752C2×C5116yesdirect product factor
60C2×#908C2×C516yescomplemented
61C2×#904C2×C513yescomplemented
62C22×D24C2×C513yescomplemented
63C5×H(5,𝔽2)—316no—
64C5×H(5,𝔽2)—68no—
65C2×C5×#41—36no—
66C2×C5×#41—62no—
67C2×C5×#41C2×D311yescharacteristic
68C22×C5×D4—36no—
69C22×C5×D4—66no—
70C5×AGL(1,ℤ/8)—348no—
71C5×#135—324no—
72C2×C5×Q16—31no—
73C2×C5×SD16—36no—
74C2×C5×D8—33no—
75C2×C5×D8—36no—
76C2×C5×D8—324no—
77C5×#131C2×D318yescomplemented
78C2×C5×M16C2×D313yescomplemented
79C22×C5×C8C2×D313yescomplemented
80C2×#576—31noSylow
81C23×C3×C5C2×D413yes—
82C22×C5×D3—26no—
83C22×C5×D3—224no—
84C22×C5×D3—812no—
85C3×C5×Q8C2×D414yes—
86C3×C5×D4C2×D4112yes—
87C2×C3×C4×C5C2×D413yes—
88C2×C3×C4×C5C2×D416yes—
89C2×C3×C4×C5C2411yescharacteristic
90C2×C3×C4×C5C2416yes—
91C5×#67—224no—
92C2×C5×Dic3—22no—
93C5×D12—26no—
94C5×D12—224no—
95C5×D12C24112yes—
96C4×C5×D3—224no—
97C5×Dic6—22no—
98C5×Dic6C2414yes—
99C3×C5×C8C2416yes—
100C3×C5×C8C2412yes—
101C2×C3×#41C22×C511yescharacteristic
102#1009—28no—
103#1009C22×C5116yescomplemented
104C2×#239—22no—
105C2×D3×D4—26no—
106C2×#235C22×C516yescomplemented
107C22×D12C22×C516yescomplemented
108C3×#131C22×C518yescomplemented
109C2×C3×M16C22×C513yescomplemented
110C22×C3×C8C22×C513yescomplemented
111#908C22×C5148yescomplemented
112C2×Dic12C22×C511yescharacteristic, complemented
113#904C22×C5124yescomplemented
114C2×D24C22×C513yescomplemented
115C2×D24C22×C516yescomplemented
116C2×D24C22×C5124yescomplemented
117C2×#204C22×C516yescomplemented
118C24×C5—66no—
119C5×#41—324no—
120C5×#41—616no—
121C5×#41—612no—
122C5×#41D1214yes—
123C5×#41C22×D314yes—
124C2×C5×Q8—32no—
125C2×C5×Q8D1211yescharacteristic
126C2×C5×D4—36no—
127C2×C5×D4—324no—
128C2×C5×D4—66no—
129C2×C5×D4—648no—
130C2×C5×D4—66no—
131C2×C5×D4—612no—
132C2×C5×D4D1213yes—
133C22×C4×C5—66no—
134C22×C4×C5C22×D313yes—
135C5×Q16—34no—
136C5×SD16—324no—
137C5×D8—312no—
138C5×D8—324no—
139C5×M16C22×D3112yescomplemented
140C2×C5×C8C22×D311yescharacteristic
141C2×C5×C8C22×D313yes—
142C2×C5×C8C22×D3112yescomplemented
143C2×H(5,𝔽2)—32no—
144#576—316no—
145C2×AGL(1,ℤ/8)—36no—
146C2×#135—33no—
147C22×D8—33no—
148C2×#131C5×D311yescharacteristic, complemented
149C22×C3×C5—26no—
150C22×C3×C5C22×D416yes—
151C22×C3×C5C22×D411yescharacteristic
152C2×C5×D3—212no—
153C2×C5×D3—412no—
154C2×C5×D3—824no—
155C3×C4×C5C22×D411yescharacteristic
156C3×C4×C5C22×D416yes—
157C3×C4×C5C2511yescharacteristic
158C5×Dic3—24no—
159C23×D3—26no—
160C3×#41C5×D414yes—
161C3×#41C23×C514yes—
162C2×C3×Q8C5×D411yescharacteristic
163C2×C3×D4C5×D413yes—
164C22×C3×C4C23×C513yes—
165C2×#67—26no—
166#239—216no—
167D3×D4—248no—
168#235—212no—
169#235C23×C5124yescomplemented
170C2×D12—26no—
171C2×D12—212no—
172C2×D12C23×C516yes—
173C2×D12C23×C5124yescomplemented
174C2×C4×D3—26no—
175C2×Dic6C23×C512yes—
176C3×M16C23×C5112yescomplemented
177C2×C3×C8C23×C513yes—
178C2×C3×C8C23×C511yescharacteristic
179C2×C3×C8C23×C5112yescomplemented
180Dic12C23×C514yescomplemented
181D24C23×C5112yescomplemented
182D24C23×C5124yescomplemented
183#204C23×C5124yescomplemented
184C23×C5—66no—
185C23×C5—624no—
186C23×C5—2412no—
187C23×C5C2×D1213yes—
188C5×Q8—34no—
189C5×Q8—62no—
190C5×Q8C2×D1214yes—
191C5×D4—312no—
192C5×D4—624no—
193C5×D4—624no—
194C5×D4—66no—
195C5×D4C2×D12112yes—
196C2×C4×C5—62no—
197C2×C4×C5—624no—
198C2×C4×C5C2×D1216yes—
199C2×C4×C5C2×D1213yes—
200C2×C4×C5C23×D316yes—
201C2×C4×C5C23×D311yescharacteristic
202C5×C8C23×D312yes—
203C5×C8C23×D316yes—
204H(5,𝔽2)—316no—
205H(5,𝔽2)—68no—
206C2×#41—36no—
207C2×#41—62no—
208C2×#41C2×C5×D311yescharacteristic
209C22×D4—36no—
210C22×D4—66no—
211AGL(1,ℤ/8)—348no—
212#135—324no—
213C2×Q16—31no—
214C2×SD16—36no—
215C2×D8—33no—
216C2×D8—36no—
217C2×D8—324no—
218#131C2×C5×D318yescomplemented
219C2×M16C2×C5×D313yescomplemented
220C22×C8C2×C5×D313yescomplemented
221C2×C3×C5—26no—
222C2×C3×C5#57612yescomplemented
223C2×C3×C5C23×D411yescharacteristic
224C5×D3—412no—
225C23×C3C2×C5×D413yes—
226C22×D3—26no—
227C22×D3—224no—
228C22×D3—812no—
229C3×Q8C2×C5×D414yes—
230C3×D4C2×C5×D4112yes—
231C2×C3×C4C2×C5×D413yes—
232C2×C3×C4C2×C5×D416yes—
233C2×C3×C4C24×C511yescharacteristic
234C2×C3×C4C24×C516yes—
235#67—224no—
236C2×Dic3—22no—
237D12—26no—
238D12—224no—
239D12C24×C5112yes—
240C4×D3—224no—
241Dic6—22no—
242Dic6C24×C514yes—
243C3×C8C24×C516yes—
244C3×C8C24×C512yes—
245C22×C5—26no—
246C22×C5—612no—
247C22×C5—1212no—
248C22×C5—2424no—
249C22×C5C22×D1216yes—
250C22×C5C22×D1211yescharacteristic
251C4×C5—64no—
252C4×C5C22×D1216yes—
253C4×C5C22×D1211yescharacteristic
254C4×C5C24×D311yescharacteristic
255C24—66no—
256#41—324no—
257#41—616no—
258#41—612no—
259#41C5×D1214yes—
260#41C22×C5×D314yes—
261C2×Q8—32no—
262C2×Q8C5×D1211yescharacteristic
263C2×D4—36no—
264C2×D4—324no—
265C2×D4—66no—
266C2×D4—648no—
267C2×D4—66no—
268C2×D4—612no—
269C2×D4C5×D1213yes—
270C22×C4—66no—
271C22×C4C22×C5×D313yes—
272Q16—34no—
273SD16—324no—
274D8—312no—
275D8—324no—
276M16C22×C5×D3112yescomplemented
277C2×C8C22×C5×D311yescharacteristic
278C2×C8C22×C5×D313yes—
279C2×C8C22×C5×D3112yescomplemented
280C3×C5C2×#57611yescharacteristic, complemented, Hall
281C22×C3—26no—
282C22×C3C22×C5×D416yes—
283C22×C3C22×C5×D411yescharacteristic
284C2×D3—212no—
285C2×D3—412no—
286C2×D3—824no—
287C3×C4C22×C5×D411yescharacteristic
288C3×C4C22×C5×D416yes—
289C3×C4C25×C511yescharacteristic
290Dic3—24no—
291C2×C5—26no—
292C2×C5—1212no—
293C2×C5#575212yesdirect product factor
294C2×C5C23×D1211yescharacteristic
295C23—66no—
296C23—624no—
297C23—2412no—
298C23C2×C5×D1213yes—
299Q8—34no—
300Q8—62no—
301Q8C2×C5×D1214yes—
302D4—312no—
303D4—624no—
304D4—624no—
305D4—66no—
306D4C2×C5×D12112yes—
307C2×C4—62no—
308C2×C4—624no—
309C2×C4C2×C5×D1213yes—
310C2×C4C2×C5×D1216yes—
311C2×C4C23×C5×D311yescharacteristic
312C2×C4C23×C5×D316yes—
313C8C23×C5×D312yes—
314C8C23×C5×D316yes—
315C2×C3—26no—
316C2×C3C5×#57612yescomplemented
317C2×C3C23×C5×D411yescharacteristic
318D3—412no—
319C5C2×#575211yescharacteristic, direct product factor, Sylow
320C22—26no—
321C22—612no—
322C22—1212no—
323C22—2424no—
324C22C22×C5×D1216yes—
325C22C22×C5×D1211yescharacteristic
326C4—64no—
327C4C22×C5×D1216yes—
328C4C22×C5×D1211yescharacteristic
329C4C24×C5×D311yescharacteristic
330C3C2×C5×#57611yescharacteristic, complemented, Sylow
331C2—26no—
332C2—1212no—
333C2C5×#575212yesdirect product factor
334C2C23×C5×D1211yescharacteristic
subgroupG/H
centerC22×C5C22×D12
commutator, cosocleC3×C4C25×C5
FrattiniC4C24×C5×D3
FittingC2×C3×C5×#131C2
socleC22×C3×C5C22×D4
hypercenterC2×C5×#131D3
hypocenterC3C2×C5×#576
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C5 ◁ C2×C5×#41 ◁ C2×C5×#131

Complements

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

NH ≅ G/Ntype
1. C2×C5×#1009C2semidirect
2. C2×C3×C5×#131C2semidirect
3. C5×#5752C2direct (N × H)
4. C2×C5×#908C2semidirect
5. C2×C5×#904C2semidirect
6. C22×C5×D24C2semidirect
10. C5×#1009C22semidirect
13. C2×C5×#235C22semidirect
14. C22×C5×D12C22semidirect
15. C3×C5×#131C22semidirect
16. C2×C3×C5×M16C22semidirect
17. C22×C3×C5×C8C22semidirect
18. C5×#908C22semidirect
19. C2×C5×Dic12C22semidirect
20. C5×#904C22semidirect
21. C2×C5×D24C22semidirect
22. C2×C5×D24C22semidirect
23. C2×C5×D24C22semidirect
24. C2×C5×#204C22semidirect
25. C2×#5752C5direct (N × H)
31. C2×C5×#131D3semidirect
42. C5×#235C23semidirect
46. C2×C5×D12C23semidirect
49. C3×C5×M16C23semidirect
52. C2×C3×C5×C8C23semidirect
53. C5×Dic12C23semidirect
54. C5×D24C23semidirect
55. C5×D24C23semidirect
56. C5×#204C23semidirect
57. C2×#1009C2×C5semidirect
58. C2×C3×#131C2×C5semidirect
59. #5752C2×C5direct (N × H)
60. C2×#908C2×C5semidirect
61. C2×#904C2×C5semidirect
62. C22×D24C2×C5semidirect
77. C5×#131C2×D3semidirect
78. C2×C5×M16C2×D3semidirect
79. C22×C5×C8C2×D3semidirect
103. #1009C22×C5semidirect
106. C2×#235C22×C5semidirect
107. C22×D12C22×C5semidirect
108. C3×#131C22×C5semidirect
109. C2×C3×M16C22×C5semidirect
110. C22×C3×C8C22×C5semidirect
111. #908C22×C5semidirect
112. C2×Dic12C22×C5semidirect
113. #904C22×C5semidirect
114. C2×D24C22×C5semidirect
115. C2×D24C22×C5semidirect
116. C2×D24C22×C5semidirect
117. C2×#204C22×C5semidirect
139. C5×M16C22×D3semidirect
142. C2×C5×C8C22×D3semidirect
148. C2×#131C5×D3semidirect
169. #235C23×C5semidirect
173. C2×D12C23×C5semidirect
176. C3×M16C23×C5semidirect
179. C2×C3×C8C23×C5semidirect
180. Dic12C23×C5semidirect
181. D24C23×C5semidirect
182. D24C23×C5semidirect
183. #204C23×C5semidirect
218. #131C2×C5×D3semidirect
219. C2×M16C2×C5×D3semidirect
220. C22×C8C2×C5×D3semidirect
222. C2×C3×C5#576semidirect
276. M16C22×C5×D3semidirect
279. C2×C8C22×C5×D3semidirect
280. C3×C5C2×#576semidirect
293. C2×C5#5752direct (N × H)
316. C2×C3C5×#576semidirect
319. C5C2×#5752direct (N × H)
330. C3C2×C5×#576semidirect
333. C2C5×#5752direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#10092
C2×C3×C5×#1312
C5×#57522
C2×C5×#9082
C2×C5×#9042
C22×C5×D242
C2×C5×#5763
C2×#57525