Finite Group Databasebeta

← C5×#63921 #13087193

Subgroups — C5×#63921 #13087193

Order 1440.

582 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 132 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
1C22×C5×D32C211yesmaximal, characteristic
2C2×C4×C5×C3⋊D3C211yesmaximal, characteristic, complemented
3C2×C5×#624C211yesmaximal, characteristic, complemented
4C3×C5×(C2×D3)⋊C4C212yesmaximal, complemented
5C5×(C2×D3)⋊Dic3C212yesmaximal, complemented
6C5×D3×C22⋊C4—32nomaximal
7C22×C5×C3⋊D3C2211yescharacteristic
8C22×C3×C5×D3C2212yes—
9C2×C5×D32—22no—
10C2×C5×D32—22no—
11C2×C5×D32C414yescomplemented
12C2×C32×C4×C5C2211yescharacteristic, complemented
13C2×C5×C3⋊Dic3C2211yescharacteristic, complemented
14C4×C5×C3⋊D3—22no—
15C2×C3×C5×Dic3C2212yescomplemented
16C5×#624—22no—
17#63921C511yesmaximal, characteristic, direct product factor, Hall
18C23×C5×D3—32no—
19C2×C4×C5×D3—32no—
20C2×C4×C5×D3—32no—
21C2×C4×C5×D3—61no—
22C3×C5×C22⋊C4—32no—
23C5×#217—32no—
24C5×(C2×D3)⋊C4—32no—
25C5×(C2×D3)⋊C4D312yescomplemented
26C22×C32×C5C2311yescharacteristic
27C2×C5×C3⋊D3C2×C412yes—
28C2×C5×C3⋊D3D412yes—
29C2×C5×C3⋊D3D412yes—
30C2×C3×C5×D3—22no—
31C2×C3×C5×D3—22no—
32C2×C3×C5×D3C2×C414yescomplemented
33C5×D32—28no—
34C32×C4×C5—21no—
35C5×C3⋊Dic3—21no—
36C3×C5×Dic3—22no—
37C2×C5×C22⋊C4—91noHall
38C22×D32C2×C511yescharacteristic
39C2×C4×C3⋊D3C2×C511yescharacteristic, complemented
40C2×#624C2×C511yescharacteristic, complemented
41C3×(C2×D3)⋊C4C2×C512yescomplemented
42(C2×D3)⋊Dic3C2×C512yescomplemented
43C23×C3×C5—32no—
44C22×C5×D3—32no—
45C22×C5×D3—38no—
46C22×C5×D3—64no—
47C22×C5×D3—64no—
48C22×C5×D3—61no—
49C22×C5×D3C2×D312yes—
50C2×C3×C4×C5—21no—
51C2×C3×C4×C5—32no—
52C2×C3×C4×C5C2×D312yescomplemented
53C2×C5×Dic3—32no—
54C2×C5×Dic3—61no—
55C2×C5×Dic3C2×D312yescomplemented
56C4×C5×D3—64no—
57C4×C5×D3—64no—
58C4×C5×D3—64no—
59D3×C22⋊C4—32no—
60C2×C32×C5C22×C411yescharacteristic
61C2×C32×C5C2×D411yescharacteristic
62C2×C32×C5C2×D411yescharacteristic
63C5×C3⋊D3C22⋊C414yescomplemented
64C3×C5×D3—24no—
65C24×C5—91no—
66C22×C4×C5—91no—
67C22×C4×C5—91no—
68C5×C22⋊C4—32no—
69C5×C22⋊C4—92no—
70C22×C3⋊D3C22×C511yescharacteristic
71C22×C3×D3C22×C512yes—
72C2×D32—22no—
73C2×D32—22no—
74C2×D32C4×C514yescomplemented
75C2×C32×C4C22×C511yescharacteristic, complemented
76C2×C3⋊Dic3C22×C511yescharacteristic, complemented
77C4×C3⋊D3—22no—
78C2×C3×Dic3C22×C512yescomplemented
79#624—22no—
80C22×C3×C5—21no—
81C22×C3×C5—34no—
82C22×C3×C5—62no—
83C22×C3×C5—62no—
84C22×C3×C5C22×D312yes—
85C2×C5×D3—22no—
86C2×C5×D3—22no—
87C2×C5×D3—34no—
88C2×C5×D3—34no—
89C2×C5×D3—34no—
90C2×C5×D3—62no—
91C2×C5×D3—62no—
92C2×C5×D3—62no—
93C2×C5×D3—616no—
94C2×C5×D3C4×D314yescomplemented
95C3×C4×C5—22no—
96C3×C4×C5—22no—
97C3×C4×C5—62no—
98C5×Dic3—22no—
99C5×Dic3—62no—
100C5×Dic3—62no—
101C23×D3—32no—
102C2×C4×D3—32no—
103C2×C4×D3—32no—
104C2×C4×D3—61no—
105C3×C22⋊C4—32no—
106#217—32no—
107(C2×D3)⋊C4—32no—
108(C2×D3)⋊C4C5×D312yescomplemented
109C32×C5C2×C22⋊C411yescharacteristic, complemented, Hall
110C23×C5—32no—
111C23×C5—94no—
112C23×C5—91no—
113C23×C5—182no—
114C23×C5—182no—
115C2×C4×C5—32no—
116C2×C4×C5—91no—
117C2×C4×C5—182no—
118C2×C4×C5—182no—
119C2×C4×C5D3211yescharacteristic, complemented
120C22×C32C23×C511yescharacteristic
121C2×C3⋊D3C2×C4×C512yes—
122C2×C3⋊D3C5×D412yes—
123C2×C3⋊D3C5×D412yes—
124C2×C3×D3—22no—
125C2×C3×D3—22no—
126C2×C3×D3C2×C4×C514yescomplemented
127D32—28no—
128C32×C4—21no—
129C3⋊Dic3—21no—
130C3×Dic3—22no—
131C2×C22⋊C4—91noSylow
132C2×C3×C5—21no—
133C2×C3×C5—21no—
134C2×C3×C5—21no—
135C2×C3×C5—64no—
136C2×C3×C5C2×C4×D312yes—
137C2×C3×C5D3×D412yes—
138C2×C3×C5D3×D412yes—
139C5×D3—24no—
140C5×D3—38no—
141C5×D3—64no—
142C23×C3—32no—
143C22×D3—32no—
144C22×D3—38no—
145C22×D3—61no—
146C22×D3—64no—
147C22×D3—64no—
148C22×D3C2×C5×D312yes—
149C2×C3×C4—21no—
150C2×C3×C4—32no—
151C2×C3×C4C2×C5×D312yescomplemented
152C2×Dic3—32no—
153C2×Dic3—61no—
154C2×Dic3C2×C5×D312yescomplemented
155C4×D3—64no—
156C4×D3—64no—
157C4×D3—64no—
158C22×C5—34no—
159C22×C5—62no—
160C22×C5—62no—
161C22×C5—92no—
162C22×C5—92no—
163C22×C5—92no—
164C22×C5—188no—
165C22×C5C2×D3211yescharacteristic
166C4×C5—21no—
167C4×C5—62no—
168C4×C5—181no—
169C2×C32C22×C4×C511yescharacteristic
170C2×C32C2×C5×D411yescharacteristic
171C2×C32C2×C5×D411yescharacteristic
172C3⋊D3C5×C22⋊C414yescomplemented
173C3×D3—24no—
174C24—91no—
175C22×C4—91no—
176C22×C4—91no—
177C22⋊C4—32no—
178C22⋊C4—92no—
179C3×C5—21no—
180C3×C5D3×C22⋊C412yescomplemented
181C22×C3—21no—
182C22×C3—34no—
183C22×C3—62no—
184C22×C3—62no—
185C22×C3C22×C5×D312yes—
186C2×D3—22no—
187C2×D3—22no—
188C2×D3—34no—
189C2×D3—34no—
190C2×D3—34no—
191C2×D3—62no—
192C2×D3—62no—
193C2×D3—62no—
194C2×D3—616no—
195C2×D3C4×C5×D314yescomplemented
196C3×C4—22no—
197C3×C4—22no—
198C3×C4—62no—
199Dic3—22no—
200Dic3—62no—
201Dic3—62no—
202C2×C5—64no—
203C2×C5—94no—
204C2×C5C4×D3211yescharacteristic
205C2×C5#380211yescharacteristic
206C2×C5#381111yescharacteristic
207C32C2×C5×C22⋊C411yescharacteristic, complemented, Sylow
208C23—32no—
209C23—94no—
210C23—91no—
211C23—182no—
212C23—182no—
213C2×C4—32no—
214C2×C4—91no—
215C2×C4—182no—
216C2×C4—182no—
217C2×C4C5×D3211yescharacteristic, complemented
218C2×C3—21no—
219C2×C3—21no—
220C2×C3—21no—
221C2×C3—64no—
222C2×C3C2×C4×C5×D312yes—
223C2×C3C5×D3×D412yes—
224C2×C3C5×D3×D412yes—
225D3—24no—
226D3—38no—
227D3—64no—
228C5#6392111yescharacteristic, direct product factor, Sylow
229C22—34no—
230C22—62no—
231C22—62no—
232C22—92no—
233C22—92no—
234C22—92no—
235C22—188no—
236C22C2×C5×D3211yescharacteristic
237C4—21no—
238C4—62no—
239C4—181no—
240C3—21no—
241C3C5×D3×C22⋊C412yescomplemented
242C2—64no—
243C2—94no—
244C2C4×C5×D3211yescharacteristic
245C2C5×#380211yescharacteristic
246C2C5×#381111yescharacteristic
subgroupG/H
centerC22×C5C2×D32
commutatorC2×C32C22×C4×C5
FrattiniC22C2×C5×D32
FittingC2×C32×C4×C5C22
socleC22×C32×C5C23
prime radicalC22×C5×D32C2
cosocleC22×C32C23×C5
hypercenterC2×C4×C5D32
hypocenterC32C2×C5×C22⋊C4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C5 ◁ C2×C4×C5

Complements

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

NH ≅ G/Ntype
2. C2×C4×C5×C3⋊D3C2semidirect
3. C2×C5×#624C2semidirect
4. C3×C5×(C2×D3)⋊C4C2semidirect
5. C5×(C2×D3)⋊Dic3C2semidirect
11. C2×C5×D32C4semidirect
12. C2×C32×C4×C5C22semidirect
13. C2×C5×C3⋊Dic3C22semidirect
15. C2×C3×C5×Dic3C22semidirect
17. #63921C5direct (N × H)
25. C5×(C2×D3)⋊C4D3semidirect
32. C2×C3×C5×D3C2×C4semidirect
39. C2×C4×C3⋊D3C2×C5semidirect
40. C2×#624C2×C5semidirect
41. C3×(C2×D3)⋊C4C2×C5semidirect
42. (C2×D3)⋊Dic3C2×C5semidirect
52. C2×C3×C4×C5C2×D3semidirect
55. C2×C5×Dic3C2×D3semidirect
63. C5×C3⋊D3C22⋊C4semidirect
74. C2×D32C4×C5semidirect
75. C2×C32×C4C22×C5semidirect
76. C2×C3⋊Dic3C22×C5semidirect
78. C2×C3×Dic3C22×C5semidirect
94. C2×C5×D3C4×D3semidirect
108. (C2×D3)⋊C4C5×D3semidirect
109. C32×C5C2×C22⋊C4semidirect
119. C2×C4×C5D32semidirect
126. C2×C3×D3C2×C4×C5semidirect
151. C2×C3×C4C2×C5×D3semidirect
154. C2×Dic3C2×C5×D3semidirect
172. C3⋊D3C5×C22⋊C4semidirect
180. C3×C5D3×C22⋊C4semidirect
195. C2×D3C4×C5×D3semidirect
207. C32C2×C5×C22⋊C4semidirect
217. C2×C4C5×D32semidirect
228. C5#63921direct (N × H)
241. C3C5×D3×C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×C5×D322
C2×C4×C5×C3⋊D32
C2×C5×#6242
C3×C5×(C2×D3)⋊C42
C5×(C2×D3)⋊Dic32
C5×D3×C22⋊C43
#639215