Finite Group Databasebeta

← C5×#8526 #11931246

Subgroups — C5×#8526 #11931246

Order 1280.

784 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 142 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
1C2×C5×#540C211yesmaximal, characteristic
2C5×#1555C212yesmaximal, complemented
3C5×#1534C212yesmaximal, complemented
4C5×#1464C212yesmaximal, complemented
5C5×#540—24no—
6C5×#540—24no—
7C2×C5×#127C2211yescharacteristic
8C2×C5×#125C2211yescharacteristic
9C2×C5×#123—22no—
10C2×C5×#121—22no—
11C2×C5×#121C412yes—
12C2×C4×C5×D4C412yes—
13C2×C5×#117C2211yescharacteristic
14C5×#432C2214yes—
15C5×#331—24no—
16C5×#330—24no—
17C5×#324—22no—
18C5×#323—22no—
19#8526C511yesmaximal, characteristic, direct product factor, Sylow
20C22×C5×D4—22no—
21C22×C5×D4—22no—
22C22×C5×D4C2×C412yes—
23C23×C4×C5—22no—
24C23×C4×C5C2×C411yescharacteristic
25C5×#127—22no—
26C5×#127D414yes—
27C5×#125—22no—
28C5×#125D414yes—
29C5×#123—216no—
30C5×#121—216no—
31C5×#121—24no—
32C5×#121—24no—
33C4×C5×D4—24no—
34C4×C5×D4—24no—
35C5×#117—22no—
36C5×#117—22no—
37C2×C5×C4⋊C4—22no—
38C2×C5×C4⋊C4C2×C412yes—
39C2×C5×C4⋊C4D412yes—
40C2×C5×C22⋊C4—22no—
41C2×C5×C22⋊C4—22no—
42C2×C5×C22⋊C4D412yes—
43C2×C42×C5C2×C411yescharacteristic
44C2×C42×C5C2311yescharacteristic
45C5×C4⋊C8—24no—
46C5×C22⋊C8—24no—
47C5×#97—24no—
48C2×#540C2×C511yescharacteristic
49#1555C2×C512yescomplemented
50#1534C2×C512yescomplemented
51#1464C2×C512yescomplemented
52C24×C5—22no—
53C24×C5C22⋊C411yescharacteristic
54C2×C5×D4—28no—
55C2×C5×D4—22no—
56C2×C5×D4—28no—
57C2×C5×D4—44no—
58C2×C5×D4—84no—
59C2×C5×D4—84no—
60C2×C5×D4C22⋊C414yes—
61C22×C4×C5—22no—
62C22×C4×C5—28no—
63C22×C4×C5—21no—
64C22×C4×C5—21no—
65C22×C4×C5—42no—
66C22×C4×C5—84no—
67C22×C4×C5C22⋊C411yescharacteristic
68C22×C4×C5C22⋊C412yes—
69C22×C4×C5C22×C411yescharacteristic
70C22×C4×C5C2×D412yes—
71C2×C5×C8—44no—
72C5×C4⋊C4—22no—
73C5×C4⋊C4—28no—
74C5×C4⋊C4—24no—
75C5×C4⋊C4C22⋊C414yes—
76C5×C22⋊C4—28no—
77C5×C22⋊C4—28no—
78C5×C22⋊C4—24no—
79C42×C5—21no—
80C42×C5—21no—
81C42×C5C2×D412yes—
82C42×C5C2×D412yes—
83#540—24no—
84#540—24no—
85C2×#127C22×C511yescharacteristic
86C2×#125C22×C511yescharacteristic
87C2×#123—22no—
88C2×#121—22no—
89C2×#121C4×C512yes—
90C2×C4×D4C4×C512yes—
91C2×#117C22×C511yescharacteristic
92#432C22×C514yes—
93#331—24no—
94#330—24no—
95#324—22no—
96#323—22no—
97C23×C5—21no—
98C23×C5—24no—
99C23×C5—22no—
100C23×C5—44no—
101C23×C5—44no—
102C23×C5—42no—
103C23×C5—44no—
104C23×C5C2×C22⋊C411yescharacteristic
105C5×D4—44no—
106C5×D4—44no—
107C5×D4—88no—
108C5×D4—88no—
109C2×C4×C5—22no—
110C2×C4×C5—22no—
111C2×C4×C5—21no—
112C2×C4×C5—24no—
113C2×C4×C5—22no—
114C2×C4×C5—42no—
115C2×C4×C5—42no—
116C2×C4×C5—42no—
117C2×C4×C5—44no—
118C2×C4×C5—42no—
119C2×C4×C5—88no—
120C2×C4×C5C2×C22⋊C411yescharacteristic
121C2×C4×C5C2×C22⋊C411yescharacteristic
122C2×C4×C5#12014yes—
123C5×C8—84no—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4C2×C4×C512yes—
127C23×C4—22no—
128C23×C4C2×C4×C511yescharacteristic
129#127—22no—
130#127C5×D414yes—
131#125—22no—
132#125C5×D414yes—
133#123—216no—
134#121—216no—
135#121—24no—
136#121—24no—
137C4×D4—24no—
138C4×D4—24no—
139#117—22no—
140#117—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4C2×C4×C512yes—
143C2×C4⋊C4C5×D412yes—
144C2×C22⋊C4—22no—
145C2×C22⋊C4—22no—
146C2×C22⋊C4C5×D412yes—
147C2×C42C2×C4×C511yescharacteristic
148C2×C42C23×C511yescharacteristic
149C4⋊C8—24no—
150C22⋊C8—24no—
151#97—24no—
152C22×C5—22no—
153C22×C5—44no—
154C22×C5—42no—
155C22×C5—44no—
156C22×C5—41no—
157C22×C5—44no—
158C22×C5—82no—
159C22×C5—88no—
160C22×C5#37911yescharacteristic
161C22×C5#41312yes—
162C4×C5—24no—
163C4×C5—21no—
164C4×C5—21no—
165C4×C5—44no—
166C4×C5—42no—
167C4×C5—41no—
168C24—22no—
169C24C5×C22⋊C411yescharacteristic
170C2×D4—28no—
171C2×D4—22no—
172C2×D4—28no—
173C2×D4—44no—
174C2×D4—84no—
175C2×D4—84no—
176C2×D4C5×C22⋊C414yes—
177C22×C4—22no—
178C22×C4—28no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—42no—
182C22×C4—84no—
183C22×C4C5×C22⋊C411yescharacteristic
184C22×C4C5×C22⋊C412yes—
185C22×C4C22×C4×C511yescharacteristic
186C22×C4C2×C5×D412yes—
187C2×C8—44no—
188C4⋊C4—22no—
189C4⋊C4—28no—
190C4⋊C4—24no—
191C4⋊C4C5×C22⋊C414yes—
192C22⋊C4—28no—
193C22⋊C4—28no—
194C22⋊C4—24no—
195C42—21no—
196C42—21no—
197C42C2×C5×D412yes—
198C42C2×C5×D412yes—
199C2×C5—22no—
200C2×C5—81no—
201C2×C5—84no—
202C2×C5#178511yescharacteristic
203C2×C5#179512yes—
204C23—21no—
205C23—24no—
206C23—22no—
207C23—44no—
208C23—44no—
209C23—42no—
210C23—44no—
211C23C2×C5×C22⋊C411yescharacteristic
212D4—44no—
213D4—44no—
214D4—88no—
215D4—88no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—24no—
220C2×C4—22no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—44no—
225C2×C4—42no—
226C2×C4—88no—
227C2×C4C2×C5×C22⋊C411yescharacteristic
228C2×C4C2×C5×C22⋊C411yescharacteristic
229C2×C4C5×#12014yes—
230C8—84no—
231C5#852611yescharacteristic, direct product factor, Sylow
232C22—22no—
233C22—41no—
234C22—44no—
235C22—44no—
236C22—44no—
237C22—42no—
238C22—82no—
239C22—88no—
240C22C5×#37911yescharacteristic
241C22C5×#41312yes—
242C4—21no—
243C4—21no—
244C4—24no—
245C4—41no—
246C4—44no—
247C4—42no—
248C2—22no—
249C2—84no—
250C2—81no—
251C2C5×#178511yescharacteristic
252C2C5×#179512yes—
subgroupG/H
center, socleC22×C5#379
commutatorC22×C4C22×C4×C5
Frattini, cosocleC2×C42C23×C5
prime radicalC2×C5×#540C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G

Complements

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

NH ≅ G/Ntype
2. C5×#1555C2semidirect
3. C5×#1534C2semidirect
4. C5×#1464C2semidirect
19. #8526C5direct (N × H)
49. #1555C2×C5semidirect
50. #1534C2×C5semidirect
51. #1464C2×C5semidirect
231. C5#8526direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#5402
C5×#15552
C5×#15342
C5×#14642
#85265