Finite Group Databasebeta

← C2×C5×#3142 #11983337

Subgroups — C2×C5×#3142 #11983337

Order 1280.

2182 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
1C2×C5×#532C211yesmaximal, characteristic
2C22×C5×#125C211yesmaximal, characteristic, complemented
3C5×#3142C2116yesmaximal, direct product factor
4C2×C5×#491C212yesmaximal, complemented
5C2×C5×#490C212yesmaximal, complemented
6C2×C5×#488C214yesmaximal, complemented
7C2×C5×#417C214yesmaximal, complemented
8C2×C4×C5×M16C211yesmaximal, characteristic, complemented
9C22×C5×#41C2212yes—
10C22×C5×M16C2212yes—
11C5×#532—24no—
12C5×#532C2218yes—
13C2×C5×C4⋊Q8C2211yescharacteristic
14C2×C5×#127C2211yescharacteristic
15C2×C5×#125—22no—
16C2×C5×#125—24no—
17C2×C5×#125C2212yes—
18C2×C5×#125C2212yes—
19C2×C5×#125C2218yescomplemented
20C2×C5×#124C2212yes—
21C2×C5×#121—22no—
22C2×C4×C5×D4—22no—
23C22×C5×C4⋊C4—22no—
24C22×C5×C4⋊C4C2212yes—
25C22×C42×C5C2211yescharacteristic
26C5×#491C22116yescomplemented
27C5×#490C22116yescomplemented
28C5×#488C22132yescomplemented
29C5×#417C22132yescomplemented
30C2×C5×#103C2218yes—
31C2×C5×#102C2218yes—
32C4×C5×M16C2218yescomplemented
33C2×C5×#97C2212yes—
34C2×C4×C5×C8C2212yes—
35C2×#3142C511yesmaximal, characteristic, direct product factor, Sylow
36C2×C5×#41—24no—
37C2×C5×#41—216no—
38C2×C5×#41C2318yes—
39C22×C5×Q8C2312yes—
40C22×C5×D4—22no—
41C22×C5×D4—22no—
42C22×C5×D4C2312yes—
43C23×C4×C5—22no—
44C23×C4×C5—22no—
45C23×C4×C5D412yes—
46C23×C4×C5C2311yescharacteristic
47C2×C5×M16—28no—
48C2×C5×M16C2318yes—
49C22×C5×C8C2314yes—
50C5×C4⋊Q8—22no—
51C5×C4⋊Q8C2314yes—
52C5×#127—22no—
53C5×#127C2314yes—
54C5×#125—24no—
55C5×#125—24no—
56C5×#125—216no—
57C5×#125C2318yes—
58C5×#125C2318yes—
59C5×#124—24no—
60C5×#124C2318yes—
61C5×#121—216no—
62C4×C5×D4—216no—
63C2×C5×C4⋊C4—24no—
64C2×C5×C4⋊C4—24no—
65C2×C5×C4⋊C4—216no—
66C2×C5×C4⋊C4—24no—
67C2×C5×C4⋊C4—22no—
68C2×C5×C4⋊C4—22no—
69C2×C5×C4⋊C4C2314yes—
70C2×C5×C4⋊C4C2318yes—
71C2×C5×C22⋊C4—24no—
72C2×C42×C5D412yes—
73C2×C42×C5D414yes—
74C2×C42×C5C2311yescharacteristic
75C2×C42×C5C2311yescharacteristic
76C2×C42×C5C2314yes—
77C5×#103C23132yes—
78C5×#102C23132yes—
79C5×#97C2318yes—
80C4×C5×C8C2318yes—
81C2×#532C2×C511yescharacteristic
82C22×#125C2×C511yescharacteristic, complemented
83#3142C2×C5116yesdirect product factor
84C2×#491C2×C512yescomplemented
85C2×#490C2×C512yescomplemented
86C2×#488C2×C514yescomplemented
87C2×#417C2×C514yescomplemented
88C2×C4×M16C2×C511yescharacteristic, complemented
89C24×C5—22no—
90C24×C5C2×D411yescharacteristic
91C5×#41—232no—
92C5×#41—416no—
93C2×C5×Q8—22no—
94C2×C5×Q8—28no—
95C2×C5×Q8C2414yes—
96C2×C5×D4—28no—
97C2×C5×D4—28no—
98C2×C5×D4—22no—
99C2×C5×D4—28no—
100C2×C5×D4—48no—
101C2×C5×D4—48no—
102C2×C5×D4C2414yes—
103C22×C4×C5—28no—
104C22×C4×C5—22no—
105C22×C4×C5—24no—
106C22×C4×C5—28no—
107C22×C4×C5—44no—
108C22×C4×C5—48no—
109C22×C4×C5—48no—
110C22×C4×C5C2×D412yes—
111C22×C4×C5C2×D412yes—
112C22×C4×C5C2×D418yes—
113C22×C4×C5C2×D412yes—
114C22×C4×C5C2×D411yescharacteristic
115C22×C4×C5#4118yes—
116C22×C4×C5C2412yes—
117C22×C4×C5C2411yescharacteristic
118C5×M16—216no—
119C2×C5×C8—28no—
120C2×C5×C8C2418yes—
121C5×C4⋊C4—216no—
122C5×C4⋊C4—216no—
123C5×C4⋊C4—28no—
124C5×C4⋊C4—24no—
125C5×C4⋊C4—28no—
126C5×C4⋊C4C2418yes—
127C5×C22⋊C4—216no—
128C42×C5C2×D418yes—
129C42×C5C2×D412yes—
130C42×C5C2×D412yes—
131C42×C5C2412yes—
132C42×C5C2412yes—
133C22×#41C22×C512yes—
134C22×M16C22×C512yes—
135#532—24no—
136#532C22×C518yes—
137C2×C4⋊Q8C22×C511yescharacteristic
138C2×#127C22×C511yescharacteristic
139C2×#125—22no—
140C2×#125—24no—
141C2×#125C22×C512yes—
142C2×#125C22×C512yes—
143C2×#125C22×C518yescomplemented
144C2×#124C22×C512yes—
145C2×#121—22no—
146C2×C4×D4—22no—
147C22×C4⋊C4—22no—
148C22×C4⋊C4C22×C512yes—
149C22×C42C22×C511yescharacteristic
150#491C22×C5116yescomplemented
151#490C22×C5116yescomplemented
152#488C22×C5132yescomplemented
153#417C22×C5132yescomplemented
154C2×#103C22×C518yes—
155C2×#102C22×C518yes—
156C4×M16C22×C518yescomplemented
157C2×#97C22×C512yes—
158C2×C4×C8C22×C512yes—
159C23×C5—24no—
160C23×C5—24no—
161C23×C5—44no—
162C23×C5—48no—
163C23×C5#12414yes—
164C23×C5C22×D412yes—
165C23×C5C22×D411yescharacteristic
166C5×Q8—28no—
167C5×Q8—44no—
168C5×D4—28no—
169C5×D4—416no—
170C5×D4—416no—
171C5×D4—44no—
172C2×C4×C5—24no—
173C2×C4×C5—24no—
174C2×C4×C5—24no—
175C2×C4×C5—28no—
176C2×C4×C5—48no—
177C2×C4×C5—416no—
178C2×C4×C5—44no—
179C2×C4×C5—48no—
180C2×C4×C5—416no—
181C2×C4×C5#12414yes—
182C2×C4×C5#12418yes—
183C2×C4×C5C22×D414yes—
184C2×C4×C5C22×D414yes—
185C2×C4×C5C22×D412yes—
186C2×C4×C5C22×D411yescharacteristic
187C2×C4×C5C2×#4114yes—
188C2×C4×C5C2×#4118yes—
189C2×C4×C5C2511yescharacteristic
190C5×C8—28no—
191C2×#41—24no—
192C2×#41—216no—
193C2×#41C23×C518yes—
194C22×Q8C23×C512yes—
195C22×D4—22no—
196C22×D4—22no—
197C22×D4C23×C512yes—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4C5×D412yes—
201C23×C4C23×C511yescharacteristic
202C2×M16—28no—
203C2×M16C23×C518yes—
204C22×C8C23×C514yes—
205C4⋊Q8—22no—
206C4⋊Q8C23×C514yes—
207#127—22no—
208#127C23×C514yes—
209#125—24no—
210#125—24no—
211#125—216no—
212#125C23×C518yes—
213#125C23×C518yes—
214#124—24no—
215#124C23×C518yes—
216#121—216no—
217C4×D4—216no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—216no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4C23×C514yes—
225C2×C4⋊C4C23×C518yes—
226C2×C22⋊C4—24no—
227C2×C42C5×D412yes—
228C2×C42C5×D414yes—
229C2×C42C23×C511yescharacteristic
230C2×C42C23×C511yescharacteristic
231C2×C42C23×C514yes—
232#103C23×C5132yes—
233#102C23×C5132yes—
234#97C23×C518yes—
235C4×C8C23×C518yes—
236C22×C5—28no—
237C22×C5—24no—
238C22×C5—44no—
239C22×C5—48no—
240C22×C5—88no—
241C22×C5C2×#12414yes—
242C22×C5C2×#12412yes—
243C22×C5#57514yes—
244C22×C5C23×D411yescharacteristic
245C4×C5—24no—
246C4×C5—24no—
247C4×C5—44no—
248C4×C5—48no—
249C4×C5C2×#12414yes—
250C4×C5C2×#12412yes—
251C4×C5C22×#4112yes—
252C24—22no—
253C24C2×C5×D411yescharacteristic
254#41—232no—
255#41—416no—
256C2×Q8—22no—
257C2×Q8—28no—
258C2×Q8C24×C514yes—
259C2×D4—28no—
260C2×D4—28no—
261C2×D4—22no—
262C2×D4—28no—
263C2×D4—48no—
264C2×D4—48no—
265C2×D4C24×C514yes—
266C22×C4—22no—
267C22×C4—28no—
268C22×C4—24no—
269C22×C4—28no—
270C22×C4—44no—
271C22×C4—48no—
272C22×C4—48no—
273C22×C4C2×C5×D412yes—
274C22×C4C2×C5×D412yes—
275C22×C4C2×C5×D418yes—
276C22×C4C2×C5×D411yescharacteristic
277C22×C4C2×C5×D412yes—
278C22×C4C5×#4118yes—
279C22×C4C24×C511yescharacteristic
280C22×C4C24×C512yes—
281M16—216no—
282C2×C8—28no—
283C2×C8C24×C518yes—
284C4⋊C4—216no—
285C4⋊C4—216no—
286C4⋊C4—28no—
287C4⋊C4—24no—
288C4⋊C4—28no—
289C4⋊C4C24×C518yes—
290C22⋊C4—216no—
291C42C2×C5×D418yes—
292C42C2×C5×D412yes—
293C42C2×C5×D412yes—
294C42C24×C512yes—
295C42C24×C512yes—
296C2×C5—24no—
297C2×C5—84no—
298C2×C5#314214yesdirect product factor
299C2×C5C22×#12411yescharacteristic
300C2×C5C2×#57512yes—
301C23—24no—
302C23—24no—
303C23—44no—
304C23—48no—
305C23C5×#12414yes—
306C23C22×C5×D412yes—
307C23C22×C5×D411yescharacteristic
308Q8—28no—
309Q8—44no—
310D4—28no—
311D4—416no—
312D4—416no—
313D4—44no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—28no—
318C2×C4—48no—
319C2×C4—416no—
320C2×C4—44no—
321C2×C4—48no—
322C2×C4—416no—
323C2×C4C5×#12414yes—
324C2×C4C5×#12418yes—
325C2×C4C22×C5×D414yes—
326C2×C4C22×C5×D414yes—
327C2×C4C22×C5×D412yes—
328C2×C4C22×C5×D411yescharacteristic
329C2×C4C2×C5×#4114yes—
330C2×C4C2×C5×#4118yes—
331C2×C4C25×C511yescharacteristic
332C8—28no—
333C5C2×#314211yescharacteristic, direct product factor, Sylow
334C22—28no—
335C22—24no—
336C22—44no—
337C22—48no—
338C22—88no—
339C22C2×C5×#12414yes—
340C22C2×C5×#12412yes—
341C22C5×#57514yes—
342C22C23×C5×D411yescharacteristic
343C4—24no—
344C4—24no—
345C4—44no—
346C4—48no—
347C4C2×C5×#12414yes—
348C4C2×C5×#12412yes—
349C4C22×C5×#4112yes—
350C2—24no—
351C2—84no—
352C2C5×#314214yesdirect product factor
353C2C22×C5×#12411yescharacteristic
354C2C2×C5×#57512yes—
subgroupG/H
center, socleC23×C5C22×D4
commutator, Frattini, cosocleC2×C4C25×C5
prime radicalC2×C5×#532C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C5 ◁ C22×C42×C5 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×C5×#125C2semidirect
3. C5×#3142C2direct (N × H)
4. C2×C5×#491C2semidirect
5. C2×C5×#490C2semidirect
6. C2×C5×#488C2semidirect
7. C2×C5×#417C2semidirect
8. C2×C4×C5×M16C2semidirect
19. C2×C5×#125C22semidirect
26. C5×#491C22semidirect
27. C5×#490C22semidirect
28. C5×#488C22semidirect
29. C5×#417C22semidirect
32. C4×C5×M16C22semidirect
35. C2×#3142C5direct (N × H)
82. C22×#125C2×C5semidirect
83. #3142C2×C5direct (N × H)
84. C2×#491C2×C5semidirect
85. C2×#490C2×C5semidirect
86. C2×#488C2×C5semidirect
87. C2×#417C2×C5semidirect
88. C2×C4×M16C2×C5semidirect
143. C2×#125C22×C5semidirect
150. #491C22×C5semidirect
151. #490C22×C5semidirect
152. #488C22×C5semidirect
153. #417C22×C5semidirect
156. C4×M16C22×C5semidirect
298. C2×C5#3142direct (N × H)
333. C5C2×#3142direct (N × H)
352. C2C5×#3142direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#5322
C22×C5×#1252
C5×#31422
C2×C5×#4912
C2×C5×#4902
C2×C5×#4882
C2×C5×#4172
C2×C4×C5×M162
C2×#31425