Finite Group Databasebeta

← C5×#12627 #11935347

Subgroups — C5×#12627 #11935347

Order 1280.

564 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 162 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#2326C211yesmaximal, characteristic
2C2×C5×#354C212yesmaximal
3C4×C5×#99C212yesmaximal
4C5×#1393C212yesmaximal
5C2×C5×#124C2211yescharacteristic
6C2×C4×C5×Q8C411yescharacteristic, complemented
7C2×C4×C5×D4C2211yescharacteristic
8C2×C5×#99C2214yes—
9C5×#391C411yescharacteristic, complemented
10C5×#390C2211yescharacteristic
11C5×#388—22no—
12C5×#385—22no—
13C5×#384C411yescharacteristic, complemented
14C4×C5×C22⋊C4—22no—
15C5×#376C411yescharacteristic, complemented
16C5×#354C418yescomplemented
17C5×#342—22no—
18#12627C511yesmaximal, characteristic, direct product factor, Sylow
19C22×C5×Q8C2×C411yescharacteristic
20C22×C5×D4C2311yescharacteristic
21C23×C4×C5D412yes—
22C5×#124—22no—
23C5×#124C2×C414yes—
24C4×C5×Q8—42no—
25C4×C5×D4—42no—
26C2×C5×C4⋊C4—21no—
27C2×C5×C4⋊C4—22no—
28C2×C5×C4⋊C4C2×C411yescharacteristic
29C2×C5×C4⋊C4C2×C411yescharacteristic
30C2×C5×C22⋊C4—22no—
31C2×C5×C22⋊C4—24no—
32C2×C5×C22⋊C4—22no—
33C2×C5×C22⋊C4D412yes—
34C2×C42×C5—21no—
35C2×C42×C5—42no—
36C2×C42×C5C2×C411yescharacteristic
37C2×C42×C5C2×C411yescharacteristic
38C2×C42×C5C2×C411yescharacteristic
39C5×#99—24no—
40C5×#99C2×C418yes—
41C5×#95—21no—
42C5×#95—22no—
43C5×#95—21no—
44C5×#95—42no—
45#2326C2×C511yescharacteristic
46C2×#354C2×C512yes—
47C4×#99C2×C512yes—
48#1393C2×C512yes—
49C24×C5C2×D412yes—
50C2×C5×Q8—21no—
51C2×C5×Q8—41no—
52C2×C5×Q8—41no—
53C2×C5×Q8C4212yescomplemented
54C2×C5×D4—41no—
55C2×C5×D4—41no—
56C2×C5×D4C22×C412yes—
57C2×C5×D4#4112yes—
58C22×C4×C5—21no—
59C22×C4×C5—21no—
60C22×C4×C5—22no—
61C22×C4×C5—22no—
62C22×C4×C5—21no—
63C22×C4×C5—21no—
64C22×C4×C5—44no—
65C22×C4×C5—82no—
66C22×C4×C5C22⋊C411yescharacteristic
67C22×C4×C5C22⋊C412yes—
68C22×C4×C5C22⋊C411yescharacteristic
69C22×C4×C5C22×C411yescharacteristic
70C5×C4⋊C4—41no—
71C5×C4⋊C4—41no—
72C5×C4⋊C4—42no—
73C5×C4⋊C4—82no—
74C5×C22⋊C4—28no—
75C5×C22⋊C4—28no—
76C5×C22⋊C4—42no—
77C5×C22⋊C4—44no—
78C5×C22⋊C4—82no—
79C42×C5—22no—
80C42×C5—21no—
81C42×C5—41no—
82C42×C5—42no—
83C42×C5—84no—
84C42×C5C4212yescomplemented
85C2×#124C22×C511yescharacteristic
86C2×C4×Q8C4×C511yescharacteristic, complemented
87C2×C4×D4C22×C511yescharacteristic
88C2×#99C22×C514yes—
89#391C4×C511yescharacteristic, complemented
90#390C22×C511yescharacteristic
91#388—22no—
92#385—22no—
93#384C4×C511yescharacteristic, complemented
94C4×C22⋊C4—22no—
95#376C4×C511yescharacteristic, complemented
96#354C4×C518yescomplemented
97#342—22no—
98C23×C5—22no—
99C23×C5—22no—
100C23×C5—42no—
101C23×C5—42no—
102C23×C5C2×C22⋊C411yescharacteristic
103C23×C5C4×D414yes—
104C5×Q8—42no—
105C5×Q8—42no—
106C5×D4—42no—
107C5×D4—42no—
108C2×C4×C5—21no—
109C2×C4×C5—21no—
110C2×C4×C5—22no—
111C2×C4×C5—22no—
112C2×C4×C5—21no—
113C2×C4×C5—22no—
114C2×C4×C5—41no—
115C2×C4×C5—44no—
116C2×C4×C5—41no—
117C2×C4×C5—41no—
118C2×C4×C5—41no—
119C2×C4×C5—41no—
120C2×C4×C5—42no—
121C2×C4×C5—41no—
122C2×C4×C5—48no—
123C2×C4×C5—41no—
124C2×C4×C5—42no—
125C2×C4×C5—41no—
126C2×C4×C5—41no—
127C2×C4×C5—41no—
128C2×C4×C5—82no—
129C2×C4×C5—84no—
130C2×C4×C5—84no—
131C2×C4×C5#9912yes—
132C2×C4×C5C2×C4211yescharacteristic
133C2×C4×C5#11711yescharacteristic
134C22×Q8C2×C4×C511yescharacteristic
135C22×D4C23×C511yescharacteristic
136C23×C4C5×D412yes—
137#124—22no—
138#124C2×C4×C514yes—
139C4×Q8—42no—
140C4×D4—42no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4C2×C4×C511yescharacteristic
144C2×C4⋊C4C2×C4×C511yescharacteristic
145C2×C22⋊C4—22no—
146C2×C22⋊C4—24no—
147C2×C22⋊C4—22no—
148C2×C22⋊C4C5×D412yes—
149C2×C42—21no—
150C2×C42—42no—
151C2×C42C2×C4×C511yescharacteristic
152C2×C42C2×C4×C511yescharacteristic
153C2×C42C2×C4×C511yescharacteristic
154#99—24no—
155#99C2×C4×C518yes—
156#95—21no—
157#95—21no—
158#95—22no—
159#95—42no—
160C22×C5—21no—
161C22×C5—21no—
162C22×C5—24no—
163C22×C5—44no—
164C22×C5—42no—
165C22×C5—42no—
166C22×C5—82no—
167C22×C5C4×C22⋊C411yescharacteristic
168C22×C5C2×#9911yescharacteristic
169C22×C5#41011yescharacteristic
170C4×C5—22no—
171C4×C5—21no—
172C4×C5—21no—
173C4×C5—42no—
174C4×C5—42no—
175C4×C5—41no—
176C4×C5—42no—
177C4×C5—81no—
178C4×C5—88no—
179C4×C5—81no—
180C24C2×C5×D412yes—
181C2×Q8—21no—
182C2×Q8—41no—
183C2×Q8—41no—
184C2×Q8C42×C512yescomplemented
185C2×D4—41no—
186C2×D4—41no—
187C2×D4C22×C4×C512yes—
188C2×D4C5×#4112yes—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—44no—
196C22×C4—82no—
197C22×C4C5×C22⋊C411yescharacteristic
198C22×C4C5×C22⋊C412yes—
199C22×C4C5×C22⋊C411yescharacteristic
200C22×C4C22×C4×C511yescharacteristic
201C4⋊C4—41no—
202C4⋊C4—41no—
203C4⋊C4—42no—
204C4⋊C4—82no—
205C22⋊C4—28no—
206C22⋊C4—28no—
207C22⋊C4—42no—
208C22⋊C4—44no—
209C22⋊C4—82no—
210C42—21no—
211C42—22no—
212C42—41no—
213C42—42no—
214C42—84no—
215C42C42×C512yescomplemented
216C2×C5—21no—
217C2×C5—21no—
218C2×C5—44no—
219C2×C5C4×#9911yescharacteristic
220C2×C5#212711yescharacteristic
221C2×C5#212911yescharacteristic
222C23—22no—
223C23—22no—
224C23—42no—
225C23—42no—
226C23C2×C5×C22⋊C411yescharacteristic
227C23C4×C5×D414yes—
228Q8—42no—
229Q8—42no—
230D4—42no—
231D4—42no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—41no—
239C2×C4—44no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—48no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—82no—
253C2×C4—84no—
254C2×C4—84no—
255C2×C4C5×#9912yes—
256C2×C4C2×C42×C511yescharacteristic
257C2×C4C5×#11711yescharacteristic
258C5#1262711yescharacteristic, direct product factor, Sylow
259C22—21no—
260C22—21no—
261C22—24no—
262C22—44no—
263C22—42no—
264C22—42no—
265C22—82no—
266C22C4×C5×C22⋊C411yescharacteristic
267C22C2×C5×#9911yescharacteristic
268C22C5×#41011yescharacteristic
269C4—22no—
270C4—21no—
271C4—21no—
272C4—42no—
273C4—42no—
274C4—41no—
275C4—42no—
276C4—81no—
277C4—88no—
278C4—81no—
279C2—21no—
280C2—21no—
281C2—44no—
282C2C4×C5×#9911yescharacteristic
283C2C5×#212711yescharacteristic
284C2C5×#212911yescharacteristic
subgroupG/H
center, socleC22×C5C2×#99
commutatorC2×C4C2×C42×C5
Frattini, cosocleC22×D4C23×C5
prime radicalC22×C5×D4C23
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C2×C4×C5×D4 ◁ G

Complements

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

NH ≅ G/Ntype
6. C2×C4×C5×Q8C4semidirect
9. C5×#391C4semidirect
13. C5×#384C4semidirect
15. C5×#376C4semidirect
16. C5×#354C4semidirect
18. #12627C5direct (N × H)
53. C2×C5×Q8C42semidirect
84. C42×C5C42semidirect
86. C2×C4×Q8C4×C5semidirect
89. #391C4×C5semidirect
93. #384C4×C5semidirect
95. #376C4×C5semidirect
96. #354C4×C5semidirect
184. C2×Q8C42×C5semidirect
215. C42C42×C5semidirect
258. C5#12627direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#23262
C2×C5×#3542
C4×C5×#992
C5×#13932
#126275