Finite Group Databasebeta

← C22×C3×C5×D32 #72219437

Subgroups — C22×C3×C5×D32 #72219437

Order 2160.

1284 conjugacy classes of subgroups; only the 416 normal subgroups are shown.

normal subgroups only

This group has 1284 conjugacy classes of subgroups — too many to draw. Only its 416 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×C3×C5×C3⋊D3C211yesmaximal, characteristic, complemented
2C22×C32×C5×D3C212yesmaximal, complemented
3C2×C3×C5×D32C2112yesmaximal, direct product factor
4C23×C3×C5×D3—32nomaximal
5C22×C5×D32C311yesmaximal, characteristic, direct product factor
6C22×C33×C5C2211yescharacteristic, complemented
7C2×C3×C5×C3⋊D3C2216yescomplemented
8C2×C32×C5×D3C22112yescomplemented
9C3×C5×D32C22116yesdirect product factor
10C22×C3×D32C511yesmaximal, characteristic, direct product factor, Hall
11C23×C32×C5—32no—
12C22×C5×C3⋊D3C2×C311yescharacteristic, complemented
13C22×C3×C5×D3—22no—
14C22×C3×C5×D3—32no—
15C22×C3×C5×D3—324no—
16C22×C3×C5×D3—61no—
17C22×C3×C5×D3D312yesdirect product factor
18C22×C3×C5×D3C2×C312yescomplemented
19C2×C5×D32C2×C3112yesdirect product factor
20C2×C33×C5C2313yescomplemented
21C3×C5×C3⋊D3C2314yescomplemented
22C32×C5×D3C2318yescomplemented
23C24×C3×C5—91no—
24C23×C5×D3—32no—
25C22×C3×C3⋊D3C2×C511yescharacteristic, complemented
26C22×C32×D3C2×C512yescomplemented
27C2×C3×D32C2×C5112yesdirect product factor
28C22×C32×C5—21no—
29C22×C32×C5—22no—
30C22×C32×C5—312no—
31C22×C32×C5—41no—
32C22×C32×C5C2×D312yescomplemented
33C22×C32×C5C22×C311yescharacteristic, complemented
34C2×C5×C3⋊D3C22×C316yescomplemented
35C2×C3×C5×D3—212no—
36C2×C3×C5×D3—312no—
37C2×C3×C5×D3—332no—
38C2×C3×C5×D3—66no—
39C2×C3×C5×D3C2×D3112yesdirect product factor
40C2×C3×C5×D3C22×C3112yescomplemented
41C5×D32C22×C3116yesdirect product factor
42C23×C3×D3—32no—
43C22×D32C3×C511yescharacteristic, direct product factor
44C33×C5C2411yescharacteristic, complemented, Hall
45C23×C3×C5—32no—
46C23×C3×C5—32no—
47C23×C3×C5—62no—
48C23×C3×C5—912no—
49C23×C3×C5—91no—
50C22×C5×D3—32no—
51C22×C5×D3—324no—
52C22×C5×D3—61no—
53C22×C5×D3C3×D312yesdirect product factor
54C22×C33C22×C511yescharacteristic, complemented
55C2×C3×C3⋊D3C22×C516yescomplemented
56C2×C32×D3C22×C5112yescomplemented
57C3×D32C22×C5116yesdirect product factor
58C2×C32×C5—23no—
59C2×C32×C5—26no—
60C2×C32×C5—38no—
61C2×C32×C5—43no—
62C2×C32×C5C22×D316yescomplemented
63C2×C32×C5C23×C313yescomplemented
64C5×C3⋊D3C23×C314yescomplemented
65C3×C5×D3—28no—
66C3×C5×D3—38no—
67C3×C5×D3—64no—
68C3×C5×D3C22×D318yesdirect product factor
69C3×C5×D3C23×C318yescomplemented
70C24×C5—91noHall
71C23×C32—32no—
72C22×C3⋊D3C2×C3×C511yescharacteristic, complemented
73C22×C3×D3—22no—
74C22×C3×D3—32no—
75C22×C3×D3—324no—
76C22×C3×D3—61no—
77C22×C3×D3C5×D312yesdirect product factor
78C22×C3×D3C2×C3×C512yescomplemented
79C2×D32C2×C3×C5112yesdirect product factor
80C22×C3×C5—22no—
81C22×C3×C5—21no—
82C22×C3×C5—312no—
83C22×C3×C5—312no—
84C22×C3×C5—41no—
85C22×C3×C5—612no—
86C22×C3×C5—916no—
87C22×C3×C5—96no—
88C22×C3×C5D3211yescharacteristic, direct product factor
89C22×C3×C5C2×C3×D312yescomplemented
90C2×C5×D3—312no—
91C2×C5×D3—332no—
92C2×C5×D3—66no—
93C2×C5×D3C2×C3×D3112yesdirect product factor
94C2×C33C23×C513yescomplemented
95C3×C3⋊D3C23×C514yescomplemented
96C32×D3C23×C518yescomplemented
97C24×C3—91no—
98C23×D3—32no—
99C32×C5—21no—
100C32×C5—22no—
101C32×C5—41no—
102C32×C5C23×D312yescomplemented
103C32×C5C24×C311yescharacteristic, complemented
104C23×C5—32no—
105C23×C5—912no—
106C23×C5—91no—
107C22×C32—21no—
108C22×C32—22no—
109C22×C32—312no—
110C22×C32—41no—
111C22×C32C2×C5×D312yescomplemented
112C22×C32C22×C3×C511yescharacteristic, complemented
113C2×C3⋊D3C22×C3×C516yescomplemented
114C2×C3×D3—212no—
115C2×C3×D3—312no—
116C2×C3×D3—332no—
117C2×C3×D3—66no—
118C2×C3×D3C2×C5×D3112yesdirect product factor
119C2×C3×D3C22×C3×C5112yescomplemented
120D32C22×C3×C5116yesdirect product factor
121C2×C3×C5—26no—
122C2×C3×C5—23no—
123C2×C3×C5—38no—
124C2×C3×C5—38no—
125C2×C3×C5—43no—
126C2×C3×C5—68no—
127C2×C3×C5—94no—
128C2×C3×C5C2×D3213yesdirect product factor
129C2×C3×C5C22×C3×D316yescomplemented
130C5×D3—38no—
131C5×D3—64no—
132C5×D3C22×C3×D318yesdirect product factor
133C33C24×C511yescharacteristic, complemented, Sylow
134C23×C3—32no—
135C23×C3—32no—
136C23×C3—62no—
137C23×C3—912no—
138C23×C3—91no—
139C22×D3—32no—
140C22×D3—324no—
141C22×D3—61no—
142C22×D3C3×C5×D312yesdirect product factor
143C22×C5—312no—
144C22×C5—916no—
145C22×C5—96no—
146C22×C5C3×D3211yescharacteristic, direct product factor
147C2×C32—23no—
148C2×C32—26no—
149C2×C32—38no—
150C2×C32—43no—
151C2×C32C22×C5×D316yescomplemented
152C2×C32C23×C3×C513yescomplemented
153C3⋊D3C23×C3×C514yescomplemented
154C3×D3—28no—
155C3×D3—38no—
156C3×D3—64no—
157C3×D3C22×C5×D318yesdirect product factor
158C3×D3C23×C3×C518yescomplemented
159C24—91noSylow
160C3×C5—21no—
161C3×C5—22no—
162C3×C5—41no—
163C3×C5C22×D3211yescharacteristic, direct product factor
164C3×C5C23×C3×D312yescomplemented
165C22×C3—21no—
166C22×C3—22no—
167C22×C3—312no—
168C22×C3—312no—
169C22×C3—41no—
170C22×C3—612no—
171C22×C3—916no—
172C22×C3—96no—
173C22×C3C5×D3211yescharacteristic, direct product factor
174C22×C3C2×C3×C5×D312yescomplemented
175C2×D3—312no—
176C2×D3—332no—
177C2×D3—66no—
178C2×D3C2×C3×C5×D3112yesdirect product factor
179C2×C5—38no—
180C2×C5—94no—
181C2×C5C2×C3×D3213yesdirect product factor
182C32—21no—
183C32—22no—
184C32—41no—
185C32C23×C5×D312yescomplemented
186C32C24×C3×C511yescharacteristic, complemented
187C23—32no—
188C23—912no—
189C23—91no—
190C2×C3—23no—
191C2×C3—26no—
192C2×C3—38no—
193C2×C3—38no—
194C2×C3—43no—
195C2×C3—68no—
196C2×C3—94no—
197C2×C3C2×C5×D3213yesdirect product factor
198C2×C3C22×C3×C5×D316yescomplemented
199D3—38no—
200D3—64no—
201D3C22×C3×C5×D318yesdirect product factor
202C5C22×C3×D3211yescharacteristic, direct product factor, Sylow
203C22—312no—
204C22—916no—
205C22—96no—
206C22C3×C5×D3211yescharacteristic, direct product factor
207C3—21no—
208C3—22no—
209C3—41no—
210C3C22×C5×D3211yescharacteristic, direct product factor
211C3C23×C3×C5×D312yescomplemented
212C2—38no—
213C2—94no—
214C2C2×C3×C5×D3213yesdirect product factor
subgroupG/H
center, hypercenterC22×C3×C5D32
commutator, cosocle, hypocenterC32C24×C3×C5
FrattiniC1G
Fitting, socleC22×C33×C5C22
LCSG ▷ C32
UCSC1 ◁ C22×C3×C5

Complements

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

NH ≅ G/Ntype
1. C22×C3×C5×C3⋊D3C2semidirect
2. C22×C32×C5×D3C2semidirect
3. C2×C3×C5×D32C2direct (N × H)
5. C22×C5×D32C3direct (N × H)
6. C22×C33×C5C22semidirect
7. C2×C3×C5×C3⋊D3C22semidirect
8. C2×C32×C5×D3C22semidirect
9. C3×C5×D32C22direct (N × H)
10. C22×C3×D32C5direct (N × H)
12. C22×C5×C3⋊D3C2×C3semidirect
17. C22×C3×C5×D3D3direct (N × H)
18. C22×C3×C5×D3C2×C3semidirect
19. C2×C5×D32C2×C3direct (N × H)
20. C2×C33×C5C23semidirect
21. C3×C5×C3⋊D3C23semidirect
22. C32×C5×D3C23semidirect
25. C22×C3×C3⋊D3C2×C5semidirect
26. C22×C32×D3C2×C5semidirect
27. C2×C3×D32C2×C5direct (N × H)
32. C22×C32×C5C2×D3semidirect
33. C22×C32×C5C22×C3semidirect
34. C2×C5×C3⋊D3C22×C3semidirect
39. C2×C3×C5×D3C2×D3direct (N × H)
40. C2×C3×C5×D3C22×C3semidirect
41. C5×D32C22×C3direct (N × H)
43. C22×D32C3×C5direct (N × H)
44. C33×C5C24semidirect
53. C22×C5×D3C3×D3direct (N × H)
54. C22×C33C22×C5semidirect
55. C2×C3×C3⋊D3C22×C5semidirect
56. C2×C32×D3C22×C5semidirect
57. C3×D32C22×C5direct (N × H)
62. C2×C32×C5C22×D3semidirect
63. C2×C32×C5C23×C3semidirect
64. C5×C3⋊D3C23×C3semidirect
68. C3×C5×D3C22×D3direct (N × H)
69. C3×C5×D3C23×C3semidirect
72. C22×C3⋊D3C2×C3×C5semidirect
77. C22×C3×D3C5×D3direct (N × H)
78. C22×C3×D3C2×C3×C5semidirect
79. C2×D32C2×C3×C5direct (N × H)
88. C22×C3×C5D32direct (N × H)
89. C22×C3×C5C2×C3×D3semidirect
93. C2×C5×D3C2×C3×D3direct (N × H)
94. C2×C33C23×C5semidirect
95. C3×C3⋊D3C23×C5semidirect
96. C32×D3C23×C5semidirect
102. C32×C5C23×D3semidirect
103. C32×C5C24×C3semidirect
111. C22×C32C2×C5×D3semidirect
112. C22×C32C22×C3×C5semidirect
113. C2×C3⋊D3C22×C3×C5semidirect
118. C2×C3×D3C2×C5×D3direct (N × H)
119. C2×C3×D3C22×C3×C5semidirect
120. D32C22×C3×C5direct (N × H)
128. C2×C3×C5C2×D32direct (N × H)
129. C2×C3×C5C22×C3×D3semidirect
132. C5×D3C22×C3×D3direct (N × H)
133. C33C24×C5semidirect
142. C22×D3C3×C5×D3direct (N × H)
146. C22×C5C3×D32direct (N × H)
151. C2×C32C22×C5×D3semidirect
152. C2×C32C23×C3×C5semidirect
153. C3⋊D3C23×C3×C5semidirect
157. C3×D3C22×C5×D3direct (N × H)
158. C3×D3C23×C3×C5semidirect
163. C3×C5C22×D32direct (N × H)
164. C3×C5C23×C3×D3semidirect
173. C22×C3C5×D32direct (N × H)
174. C22×C3C2×C3×C5×D3semidirect
178. C2×D3C2×C3×C5×D3direct (N × H)
181. C2×C5C2×C3×D32direct (N × H)
185. C32C23×C5×D3semidirect
186. C32C24×C3×C5semidirect
197. C2×C3C2×C5×D32direct (N × H)
198. C2×C3C22×C3×C5×D3semidirect
201. D3C22×C3×C5×D3direct (N × H)
202. C5C22×C3×D32direct (N × H)
206. C22C3×C5×D32direct (N × H)
210. C3C22×C5×D32direct (N × H)
211. C3C23×C3×C5×D3semidirect
214. C2C2×C3×C5×D32direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C5×C3⋊D32
C22×C32×C5×D32
C2×C3×C5×D322
C23×C3×C5×D33
C22×C5×D323
C22×C3×D325