Finite Group Databasebeta

← C5×#84000 #14418119

Subgroups — C5×#84000 #14418119

Order 1920.

916 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#575C211yesmaximal, characteristic
2C5×#5819C212yesmaximal, complemented
3C2×C5×#837C212yesmaximal, complemented
4C5×#5197C214yesmaximal, complemented
5C5×#5196C214yesmaximal, complemented
6C5×#5125C211yesmaximal, characteristic, complemented
7C5×Dic3×M16C211yesmaximal, characteristic, complemented
8C5×#2956—31nomaximal, Hall
9C2×C3×C5×#41C2212yes—
10C3×C5×#137C414yescomplemented
11C3×C5×AGL(1,ℤ/8)C414yescomplemented
12C3×C5×#135C2214yes—
13C2×C3×C5×M16C2211yescharacteristic
14C5×#948—24no—
15C5×#948C2218yescomplemented
16C2×C4×C5×Dic3C2211yescharacteristic
17C2×C5×#208—24no—
18C2×C5×#208C2212yes—
19C2×C5×#208C2211yescharacteristic
20C5×#837C2214yescomplemented
21C5×#837C2214yescomplemented
22C5×#819C2214yescomplemented
23C5×#815C2212yescomplemented
24C5×C8×Dic3C2212yescomplemented
25#84000C511yesmaximal, characteristic, direct product factor, Hall
26C5×#575D311yescharacteristic
27C5×#568—32no—
28C5×#444—34no—
29C5×#443—34no—
30C5×#430—31no—
31C2×C5×#104—32no—
32C4×C5×M16—31no—
33C3×C5×#41—22no—
34C3×C5×#41—22no—
35C3×C5×#41C2×C414yes—
36C3×C5×#41C2314yes—
37C2×C3×C5×Q8C2×C412yes—
38C2×C3×C5×D4—22no—
39C2×C3×C5×D4C2×C412yes—
40C22×C3×C4×C5—22no—
41C22×C3×C4×C5C2311yescharacteristic
42C22×C5×Dic3—21no—
43C3×C5×Q16C2×C414yescomplemented
44C3×C5×SD16C2×C418yescomplemented
45C3×C5×D8C2×C414yescomplemented
46C3×C5×M16C2×C414yes—
47C2×C3×C5×C8C2312yes—
48C4×C5×Dic3—21no—
49C4×C5×Dic3C2311yescharacteristic
50C4×C5×Dic3C2311yescharacteristic
51C5×#208—22no—
52C5×#208—28no—
53C5×#208D414yes—
54C5×#208C2314yes—
55C2×C5×C3⋊C8—22no—
56C2×C5×C3⋊C8—28no—
57C2×C5×C3⋊C8C2312yes—
58C3×#575C2×C511yescharacteristic
59#5819C2×C512yescomplemented
60C2×#837C2×C512yescomplemented
61#5197C2×C514yescomplemented
62#5196C2×C514yescomplemented
63#5125C2×C511yescharacteristic, complemented
64Dic3×M16C2×C511yescharacteristic, complemented
65C2×C5×#41C2×D312yes—
66C5×#137Dic314yescomplemented
67C5×AGL(1,ℤ/8)Dic314yescomplemented
68C5×#135C2×D314yes—
69C5×#131—38no—
70C5×#131—64no—
71C2×C5×M16—31no—
72C2×C5×M16—32no—
73C2×C5×M16—64no—
74C2×C5×M16C2×D311yescharacteristic
75C2×C42×C5—31no—
76C5×#108—34no—
77C5×#104—34no—
78C5×#104—34no—
79C5×#97—32no—
80C4×C5×C8—32no—
81#2956—31noSylow
82C23×C3×C5—22no—
83C23×C3×C5#4111yescharacteristic
84C3×C5×Q8—22no—
85C3×C5×Q8C22×C414yes—
86C3×C5×D4—24no—
87C3×C5×D4—24no—
88C3×C5×D4—22no—
89C3×C5×D4C22×C414yes—
90C2×C3×C4×C5—22no—
91C2×C3×C4×C5—24no—
92C2×C3×C4×C5—24no—
93C2×C3×C4×C5C22×C412yes—
94C2×C3×C4×C5C2×D412yes—
95C2×C3×C4×C5#4111yescharacteristic
96C2×C3×C4×C5C2411yescharacteristic
97C2×C5×Dic3—21no—
98C2×C5×Dic3—21no—
99C2×C5×Dic3—42no—
100C3×C5×C8C22×C414yes—
101C5×C3⋊C8—24no—
102C5×C3⋊C8C2×D414yes—
103C2×C3×#41C22×C512yes—
104C3×#137C4×C514yescomplemented
105C3×AGL(1,ℤ/8)C4×C514yescomplemented
106C3×#135C22×C514yes—
107C2×C3×M16C22×C511yescharacteristic
108#948—24no—
109#948C22×C518yescomplemented
110C2×C4×Dic3C22×C511yescharacteristic
111C2×#208—24no—
112C2×#208C22×C512yes—
113C2×#208C22×C511yescharacteristic
114#837C22×C514yescomplemented
115#837C22×C514yescomplemented
116#819C22×C514yescomplemented
117#815C22×C512yescomplemented
118C8×Dic3C22×C512yescomplemented
119C5×#41—22no—
120C5×#41—22no—
121C5×#41C2×Dic314yes—
122C5×#41C22×D314yes—
123C2×C5×Q8C2×Dic312yes—
124C2×C5×D4—22no—
125C2×C5×D4C2×Dic312yes—
126C22×C4×C5—22no—
127C22×C4×C5—61no—
128C22×C4×C5C22×D311yescharacteristic
129C5×Q16C2×Dic314yescomplemented
130C5×SD16C2×Dic318yescomplemented
131C5×D8C2×Dic314yescomplemented
132C5×M16—34no—
133C5×M16—34no—
134C5×M16—68no—
135C5×M16—62no—
136C5×M16C2×Dic314yes—
137C2×C5×C8—32no—
138C2×C5×C8—68no—
139C2×C5×C8—62no—
140C2×C5×C8C22×D312yes—
141C42×C5—31no—
142C42×C5—31no—
143C42×C5—61no—
144#575C5×D311yescharacteristic
145#568—32no—
146#444—34no—
147#443—34no—
148#430—31no—
149C2×#104—32no—
150C4×M16—31no—
151C22×C3×C5—24no—
152C22×C3×C5—41no—
153C22×C3×C5—82no—
154C22×C3×C5C4×D412yes—
155C22×C3×C5C2×#4111yescharacteristic
156C3×C4×C5—24no—
157C3×C4×C5C4×D412yes—
158C3×C4×C5C23×C411yescharacteristic
159C3×C4×C5C22×D411yescharacteristic
160C5×Dic3—22no—
161C5×Dic3—41no—
162C3×#41—22no—
163C3×#41—22no—
164C3×#41C2×C4×C514yes—
165C3×#41C23×C514yes—
166C2×C3×Q8C2×C4×C512yes—
167C2×C3×D4—22no—
168C2×C3×D4C2×C4×C512yes—
169C22×C3×C4—22no—
170C22×C3×C4C23×C511yescharacteristic
171C22×Dic3—21no—
172C3×Q16C2×C4×C514yescomplemented
173C3×SD16C2×C4×C518yescomplemented
174C3×D8C2×C4×C514yescomplemented
175C3×M16C2×C4×C514yes—
176C2×C3×C8C23×C512yes—
177C4×Dic3—21no—
178C4×Dic3C23×C511yescharacteristic
179C4×Dic3C23×C511yescharacteristic
180#208—22no—
181#208—28no—
182#208C5×D414yes—
183#208C23×C514yes—
184C2×C3⋊C8—22no—
185C2×C3⋊C8—28no—
186C2×C3⋊C8C23×C512yes—
187C23×C5—22no—
188C23×C5#23711yescharacteristic
189C5×Q8—22no—
190C5×Q8C22×Dic314yes—
191C5×D4—24no—
192C5×D4—24no—
193C5×D4—22no—
194C5×D4C22×Dic314yes—
195C2×C4×C5—22no—
196C2×C4×C5—24no—
197C2×C4×C5—24no—
198C2×C4×C5—61no—
199C2×C4×C5—61no—
200C2×C4×C5—122no—
201C2×C4×C5D3×D412yes—
202C2×C4×C5#23711yescharacteristic
203C2×C4×C5C22×Dic312yes—
204C2×C4×C5C23×D311yescharacteristic
205C5×C8—34no—
206C5×C8—64no—
207C5×C8C22×Dic314yes—
208C2×#41C2×C5×D312yes—
209#137C5×Dic314yescomplemented
210AGL(1,ℤ/8)C5×Dic314yescomplemented
211#135C2×C5×D314yes—
212#131—38no—
213#131—64no—
214C2×M16—31no—
215C2×M16—32no—
216C2×M16—64no—
217C2×M16C2×C5×D311yescharacteristic
218C2×C42—31no—
219#108—34no—
220#104—34no—
221#104—34no—
222#97—32no—
223C4×C8—32no—
224C2×C3×C5—21no—
225C2×C3×C5—22no—
226C2×C3×C5—44no—
227C2×C3×C5C2×C4×D411yescharacteristic
228C23×C3—22no—
229C23×C3C5×#4111yescharacteristic
230C3×Q8—22no—
231C3×Q8C22×C4×C514yes—
232C3×D4—24no—
233C3×D4—24no—
234C3×D4—22no—
235C3×D4C22×C4×C514yes—
236C2×C3×C4—22no—
237C2×C3×C4—24no—
238C2×C3×C4—24no—
239C2×C3×C4C22×C4×C512yes—
240C2×C3×C4C2×C5×D412yes—
241C2×C3×C4C5×#4111yescharacteristic
242C2×C3×C4C24×C511yescharacteristic
243C2×Dic3—21no—
244C2×Dic3—21no—
245C2×Dic3—42no—
246C3×C8C22×C4×C514yes—
247C3⋊C8—24no—
248C3⋊C8C2×C5×D414yes—
249C22×C5—24no—
250C22×C5—41no—
251C22×C5—82no—
252C22×C5D4×Dic312yes—
253C22×C5C2×#23711yescharacteristic
254C4×C5—24no—
255C4×C5—62no—
256C4×C5—121no—
257C4×C5D4×Dic312yes—
258C4×C5C2×D3×D411yescharacteristic
259C4×C5C23×Dic311yescharacteristic
260#41—22no—
261#41—22no—
262#41C2×C5×Dic314yes—
263#41C22×C5×D314yes—
264C2×Q8C2×C5×Dic312yes—
265C2×D4—22no—
266C2×D4C2×C5×Dic312yes—
267C22×C4—22no—
268C22×C4—61no—
269C22×C4C22×C5×D311yescharacteristic
270Q16C2×C5×Dic314yescomplemented
271SD16C2×C5×Dic318yescomplemented
272D8C2×C5×Dic314yescomplemented
273M16—34no—
274M16—34no—
275M16—68no—
276M16—62no—
277M16C2×C5×Dic314yes—
278C2×C8—32no—
279C2×C8—68no—
280C2×C8—62no—
281C2×C8C22×C5×D312yes—
282C42—31no—
283C42—31no—
284C42—61no—
285C3×C5#295611yescharacteristic, complemented, Hall
286C22×C3—24no—
287C22×C3—41no—
288C22×C3—82no—
289C22×C3C4×C5×D412yes—
290C22×C3C2×C5×#4111yescharacteristic
291C3×C4—24no—
292C3×C4C4×C5×D412yes—
293C3×C4C23×C4×C511yescharacteristic
294C3×C4C22×C5×D411yescharacteristic
295Dic3—22no—
296Dic3—41no—
297C2×C5—21no—
298C2×C5—22no—
299C2×C5—44no—
300C2×C5C2×D4×Dic311yescharacteristic
301C23—22no—
302C23C5×#23711yescharacteristic
303Q8—22no—
304Q8C22×C5×Dic314yes—
305D4—24no—
306D4—24no—
307D4—22no—
308D4C22×C5×Dic314yes—
309C2×C4—22no—
310C2×C4—24no—
311C2×C4—24no—
312C2×C4—61no—
313C2×C4—61no—
314C2×C4—122no—
315C2×C4C5×D3×D412yes—
316C2×C4C5×#23711yescharacteristic
317C2×C4C22×C5×Dic312yes—
318C2×C4C23×C5×D311yescharacteristic
319C8—34no—
320C8—64no—
321C8C22×C5×Dic314yes—
322C2×C3—21no—
323C2×C3—22no—
324C2×C3—44no—
325C2×C3C2×C4×C5×D411yescharacteristic
326C5#8400011yescharacteristic, direct product factor, Sylow
327C22—24no—
328C22—41no—
329C22—82no—
330C22C5×D4×Dic312yes—
331C22C2×C5×#23711yescharacteristic
332C4—24no—
333C4—62no—
334C4—121no—
335C4C5×D4×Dic312yes—
336C4C2×C5×D3×D411yescharacteristic
337C4C23×C5×Dic311yescharacteristic
338C3C5×#295611yescharacteristic, complemented, Sylow
339C2—21no—
340C2—22no—
341C2—44no—
342C2C2×C5×D4×Dic311yescharacteristic
subgroupG/H
centerC4×C5C2×D3×D4
commutatorC3×C4C23×C4×C5
FrattiniC2×C4C23×C5×D3
Fitting, prime radicalC3×C5×#575C2
socleC2×C3×C5C2×C4×D4
cosocleC2×C3×C4C24×C5
hypercenterC5×#575D3
hypocenterC3C5×#2956
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C5×#575

Complements

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

NH ≅ G/Ntype
2. C5×#5819C2semidirect
3. C2×C5×#837C2semidirect
4. C5×#5197C2semidirect
5. C5×#5196C2semidirect
6. C5×#5125C2semidirect
7. C5×Dic3×M16C2semidirect
10. C3×C5×#137C4semidirect
11. C3×C5×AGL(1,ℤ/8)C4semidirect
15. C5×#948C22semidirect
20. C5×#837C22semidirect
21. C5×#837C22semidirect
22. C5×#819C22semidirect
23. C5×#815C22semidirect
24. C5×C8×Dic3C22semidirect
25. #84000C5direct (N × H)
43. C3×C5×Q16C2×C4semidirect
44. C3×C5×SD16C2×C4semidirect
45. C3×C5×D8C2×C4semidirect
59. #5819C2×C5semidirect
60. C2×#837C2×C5semidirect
61. #5197C2×C5semidirect
62. #5196C2×C5semidirect
63. #5125C2×C5semidirect
64. Dic3×M16C2×C5semidirect
66. C5×#137Dic3semidirect
67. C5×AGL(1,ℤ/8)Dic3semidirect
104. C3×#137C4×C5semidirect
105. C3×AGL(1,ℤ/8)C4×C5semidirect
109. #948C22×C5semidirect
114. #837C22×C5semidirect
115. #837C22×C5semidirect
116. #819C22×C5semidirect
117. #815C22×C5semidirect
118. C8×Dic3C22×C5semidirect
129. C5×Q16C2×Dic3semidirect
130. C5×SD16C2×Dic3semidirect
131. C5×D8C2×Dic3semidirect
172. C3×Q16C2×C4×C5semidirect
173. C3×SD16C2×C4×C5semidirect
174. C3×D8C2×C4×C5semidirect
209. #137C5×Dic3semidirect
210. AGL(1,ℤ/8)C5×Dic3semidirect
270. Q16C2×C5×Dic3semidirect
271. SD16C2×C5×Dic3semidirect
272. D8C2×C5×Dic3semidirect
285. C3×C5#2956semidirect
326. C5#84000direct (N × H)
338. C3C5×#2956semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#5752
C5×#58192
C2×C5×#8372
C5×#51972
C5×#51962
C5×#51252
C5×Dic3×M162
C5×#29563
#840005