Finite Group Databasebeta

← C2×C3×C5×#213 #13087947

Subgroups — C2×C3×C5×#213 #13087947

Order 1440.

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

normal subgroups only

This group has 358 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
1C2×C32×C5×D4C211yesmaximal, characteristic, complemented
2C2×C3×C5×D12C211yesmaximal, characteristic, complemented
3C3×C5×#213C214yesmaximal, direct product factor
4C2×C3×C5×C3⋊C8C211yesmaximal, characteristic, complemented
5C2×C3×C5×D8—31nomaximal
6C2×C5×#213C311yesmaximal, characteristic, direct product factor
7C23×C32×C5—21no—
8C22×C3×C5×D3—21no—
9C32×C5×D4—21no—
10C32×C5×D4C2212yescomplemented
11C2×C32×C4×C5C2211yescharacteristic
12C3×C5×D12—21no—
13C3×C5×D12C2212yescomplemented
14C3×C5×C3⋊C8C2212yescomplemented
15C2×C3×#213C511yesmaximal, characteristic, direct product factor, Hall
16C2×C3×C5×D4—21no—
17C2×C3×C5×D4—31no—
18C2×C3×C5×D4D311yescharacteristic, complemented
19C2×C3×C5×D4C2×C311yescharacteristic, complemented
20C2×C5×D12C2×C311yescharacteristic, complemented
21C3×C5×D8—34no—
22C2×C3×C5×C8—31no—
23C5×#213C2×C314yesdirect product factor
24C2×C5×C3⋊C8C2×C311yescharacteristic, complemented
25C22×C32×C5—22no—
26C22×C32×C5—42no—
27C22×C32×C5D411yescharacteristic
28C2×C3×C5×D3—22no—
29C2×C3×C5×D3—42no—
30C32×C4×C5D411yescharacteristic
31C32×C4×C5C2311yescharacteristic
32C2×C5×D8—31noHall
33C2×C32×D4C2×C511yescharacteristic, complemented
34C2×C3×D12C2×C511yescharacteristic, complemented
35C3×#213C2×C514yesdirect product factor
36C2×C3×C3⋊C8C2×C511yescharacteristic, complemented
37C23×C3×C5—22no—
38C23×C3×C5—21no—
39C23×C3×C5—21no—
40C23×C3×C5—61no—
41C22×C5×D3—21no—
42C3×C5×D4—22no—
43C3×C5×D4—22no—
44C3×C5×D4—21no—
45C3×C5×D4—21no—
46C3×C5×D4—32no—
47C3×C5×D4—61no—
48C3×C5×D4C2×D312yescomplemented
49C3×C5×D4C22×C312yescomplemented
50C2×C3×C4×C5—21no—
51C2×C3×C4×C5C2×D311yescharacteristic
52C2×C3×C4×C5C22×C311yescharacteristic
53C5×D12—21no—
54C5×D12C22×C312yescomplemented
55C3×C5×C8—32no—
56C5×C3⋊C8C22×C312yescomplemented
57C2×C3×D8—31no—
58C2×#213C3×C511yescharacteristic, direct product factor
59C2×C32×C5—42no—
60C2×C32×C5D812yescomplemented
61C2×C32×C5C2×D411yescharacteristic
62C3×C5×D3—42no—
63C2×C5×D4—31no—
64C2×C5×D4C3×D311yescharacteristic, complemented
65C5×D8—34no—
66C2×C5×C8—31no—
67C23×C32—21no—
68C22×C3×D3—21no—
69C32×D4—21no—
70C32×D4C22×C512yescomplemented
71C2×C32×C4C22×C511yescharacteristic
72C3×D12—21no—
73C3×D12C22×C512yescomplemented
74C3×C3⋊C8C22×C512yescomplemented
75C22×C3×C5—24no—
76C22×C3×C5—22no—
77C22×C3×C5—22no—
78C22×C3×C5—21no—
79C22×C3×C5—44no—
80C22×C3×C5—42no—
81C22×C3×C5—42no—
82C22×C3×C5—62no—
83C22×C3×C5—122no—
84C22×C3×C5#6711yescharacteristic
85C22×C3×C5C3×D411yescharacteristic
86C2×C5×D3—22no—
87C2×C5×D3—42no—
88C3×C4×C5—21no—
89C3×C4×C5—21no—
90C3×C4×C5#6711yescharacteristic
91C3×C4×C5C3×D411yescharacteristic
92C3×C4×C5C22×D311yescharacteristic
93C3×C4×C5C23×C311yescharacteristic
94C2×C3×D4—21no—
95C2×C3×D4—31no—
96C2×C3×D4C5×D311yescharacteristic, complemented
97C2×C3×D4C2×C3×C511yescharacteristic, complemented
98C2×D12C2×C3×C511yescharacteristic, complemented
99C3×D8—34no—
100C2×C3×C8—31no—
101#213C2×C3×C514yesdirect product factor
102C2×C3⋊C8C2×C3×C511yescharacteristic, complemented
103C32×C5C2×D811yescharacteristic, complemented, Hall
104C23×C5—21no—
105C23×C5—61no—
106C5×D4—21no—
107C5×D4—32no—
108C5×D4—61no—
109C5×D4C2×C3×D312yescomplemented
110C2×C4×C5C2×C3×D311yescharacteristic
111C5×C8—32no—
112C22×C32—22no—
113C22×C32—42no—
114C22×C32C5×D411yescharacteristic
115C2×C3×D3—22no—
116C2×C3×D3—42no—
117C32×C4C5×D411yescharacteristic
118C32×C4C23×C511yescharacteristic
119C2×D8—31noSylow
120C2×C3×C5—22no—
121C2×C3×C5—21no—
122C2×C3×C5—44no—
123C2×C3×C5—42no—
124C2×C3×C5—42no—
125C2×C3×C5—122no—
126C2×C3×C5#21312yesdirect product factor
127C2×C3×C5C3×D812yescomplemented
128C2×C3×C5C2×#6711yescharacteristic
129C2×C3×C5C2×C3×D411yescharacteristic
130C5×D3—42no—
131C23×C3—22no—
132C23×C3—21no—
133C23×C3—21no—
134C23×C3—61no—
135C22×D3—21no—
136C3×D4—22no—
137C3×D4—21no—
138C3×D4—21no—
139C3×D4—22no—
140C3×D4—32no—
141C3×D4—61no—
142C3×D4C2×C5×D312yescomplemented
143C3×D4C22×C3×C512yescomplemented
144C2×C3×C4—21no—
145C2×C3×C4C2×C5×D311yescharacteristic
146C2×C3×C4C22×C3×C511yescharacteristic
147D12—21no—
148D12C22×C3×C512yescomplemented
149C3×C8—32no—
150C3⋊C8C22×C3×C512yescomplemented
151C22×C5—22no—
152C22×C5—42no—
153C22×C5—62no—
154C22×C5—122no—
155C22×C5C3×#6711yescharacteristic
156C4×C5C3×#6711yescharacteristic
157C4×C5C22×C3×D311yescharacteristic
158C2×C32—42no—
159C2×C32C5×D812yescomplemented
160C2×C32C2×C5×D411yescharacteristic
161C3×D3—42no—
162C2×D4—31no—
163C2×D4C3×C5×D311yescharacteristic, complemented
164D8—34no—
165C2×C8—31no—
166C3×C5—21no—
167C3×C5C2×#21311yescharacteristic, direct product factor
168C3×C5C2×C3×D811yescharacteristic, complemented
169C22×C3—24no—
170C22×C3—22no—
171C22×C3—22no—
172C22×C3—21no—
173C22×C3—44no—
174C22×C3—42no—
175C22×C3—42no—
176C22×C3—62no—
177C22×C3—122no—
178C22×C3C5×#6711yescharacteristic
179C22×C3C3×C5×D411yescharacteristic
180C2×D3—22no—
181C2×D3—42no—
182C3×C4—21no—
183C3×C4—21no—
184C3×C4C5×#6711yescharacteristic
185C3×C4C3×C5×D411yescharacteristic
186C3×C4C22×C5×D311yescharacteristic
187C3×C4C23×C3×C511yescharacteristic
188C2×C5—42no—
189C2×C5—122no—
190C2×C5C3×#21312yesdirect product factor
191C2×C5C2×C3×#6711yescharacteristic
192C32C2×C5×D811yescharacteristic, complemented, Sylow
193C23—21no—
194C23—61no—
195D4—21no—
196D4—32no—
197D4—61no—
198D4C2×C3×C5×D312yescomplemented
199C2×C4C2×C3×C5×D311yescharacteristic
200C8—32no—
201C2×C3—21no—
202C2×C3—22no—
203C2×C3—44no—
204C2×C3—42no—
205C2×C3—42no—
206C2×C3—122no—
207C2×C3C5×#21312yesdirect product factor
208C2×C3C3×C5×D812yescomplemented
209C2×C3C2×C5×#6711yescharacteristic
210C2×C3C2×C3×C5×D411yescharacteristic
211D3—42no—
212C5C2×C3×#21311yescharacteristic, direct product factor, Sylow
213C22—22no—
214C22—42no—
215C22—62no—
216C22—122no—
217C22C3×C5×#6711yescharacteristic
218C4C3×C5×#6711yescharacteristic
219C4C22×C3×C5×D311yescharacteristic
220C3—21no—
221C3C2×C5×#21311yescharacteristic, direct product factor
222C3C2×C3×C5×D811yescharacteristic, complemented
223C2—42no—
224C2—122no—
225C2C3×C5×#21312yesdirect product factor
226C2C2×C3×C5×#6711yescharacteristic
subgroupG/H
centerC22×C3×C5#67
commutator, cosocleC3×C4C23×C3×C5
FrattiniC4C22×C3×C5×D3
FittingC2×C32×C5×D4C2
socleC22×C32×C5D4
hypercenterC2×C3×C5×D4D3
hypocenterC3C2×C3×C5×D8
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3×C5 ◁ C2×C3×C4×C5 ◁ C2×C3×C5×D4

Complements

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

NH ≅ G/Ntype
1. C2×C32×C5×D4C2semidirect
2. C2×C3×C5×D12C2semidirect
3. C3×C5×#213C2direct (N × H)
4. C2×C3×C5×C3⋊C8C2semidirect
6. C2×C5×#213C3direct (N × H)
10. C32×C5×D4C22semidirect
13. C3×C5×D12C22semidirect
14. C3×C5×C3⋊C8C22semidirect
15. C2×C3×#213C5direct (N × H)
18. C2×C3×C5×D4D3semidirect
19. C2×C3×C5×D4C2×C3semidirect
20. C2×C5×D12C2×C3semidirect
23. C5×#213C2×C3direct (N × H)
24. C2×C5×C3⋊C8C2×C3semidirect
33. C2×C32×D4C2×C5semidirect
34. C2×C3×D12C2×C5semidirect
35. C3×#213C2×C5direct (N × H)
36. C2×C3×C3⋊C8C2×C5semidirect
48. C3×C5×D4C2×D3semidirect
49. C3×C5×D4C22×C3semidirect
54. C5×D12C22×C3semidirect
56. C5×C3⋊C8C22×C3semidirect
58. C2×#213C3×C5direct (N × H)
60. C2×C32×C5D8semidirect
64. C2×C5×D4C3×D3semidirect
70. C32×D4C22×C5semidirect
73. C3×D12C22×C5semidirect
74. C3×C3⋊C8C22×C5semidirect
96. C2×C3×D4C5×D3semidirect
97. C2×C3×D4C2×C3×C5semidirect
98. C2×D12C2×C3×C5semidirect
101. #213C2×C3×C5direct (N × H)
102. C2×C3⋊C8C2×C3×C5semidirect
103. C32×C5C2×D8semidirect
109. C5×D4C2×C3×D3semidirect
126. C2×C3×C5#213direct (N × H)
127. C2×C3×C5C3×D8semidirect
142. C3×D4C2×C5×D3semidirect
143. C3×D4C22×C3×C5semidirect
148. D12C22×C3×C5semidirect
150. C3⋊C8C22×C3×C5semidirect
159. C2×C32C5×D8semidirect
163. C2×D4C3×C5×D3semidirect
167. C3×C5C2×#213direct (N × H)
168. C3×C5C2×C3×D8semidirect
190. C2×C5C3×#213direct (N × H)
192. C32C2×C5×D8semidirect
198. D4C2×C3×C5×D3semidirect
207. C2×C3C5×#213direct (N × H)
208. C2×C3C3×C5×D8semidirect
212. C5C2×C3×#213direct (N × H)
221. C3C2×C5×#213direct (N × H)
222. C3C2×C3×C5×D8semidirect
225. C2C3×C5×#213direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×C5×D42
C2×C3×C5×D122
C3×C5×#2132
C2×C3×C5×C3⋊C82
C2×C3×C5×D83
C2×C5×#2133
C2×C3×#2135