Finite Group Databasebeta

← C22×C3×#1203 #13089248

Subgroups — C22×C3×#1203 #13089248

Order 1440.

998 conjugacy classes of subgroups; only the 296 normal subgroups are shown.

normal subgroups only

This group has 998 conjugacy classes of subgroups — too many to draw. Only its 296 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
1C23×C3×D15C211yesmaximal, characteristic, complemented
2C23×C3×C5×D3C211yesmaximal, characteristic, complemented
3C22×C32×Dic5C211yesmaximal, characteristic, complemented
4C2×C3×#1203C2112yesmaximal, direct product factor
5C22×C3×C22⋊D5—31nomaximal
6C22×#1203C311yesmaximal, characteristic, direct product factor
7C23×C32×C5C2211yescharacteristic
8C22×C3×D15—24no—
9C22×C3×D15C2216yescomplemented
10C22×C3×C5×D3—24no—
11C22×C3×C5×D3C2216yescomplemented
12C2×C32×Dic5C2216yescomplemented
13C3×#1203C22116yesdirect product factor
14C22×C3×D12—51nomaximal, Hall
15C24×C3×C5—31no—
16C23×D15C2×C311yescharacteristic, complemented
17C23×C5×D3C2×C311yescharacteristic, complemented
18C23×C3×D5—31no—
19C2×C3×C22⋊D5—312no—
20C22×C3×Dic5—21no—
21C22×C3×Dic5D311yescharacteristic, complemented
22C22×C3×Dic5C2×C311yescharacteristic, complemented
23C2×#1203C2×C3112yesdirect product factor
24C22×C32×C5D414yescomplemented
25C22×C32×C5C2313yes—
26C2×C3×D15—212no—
27C2×C3×D15C2314yescomplemented
28C2×C3×C5×D3—212no—
29C2×C3×C5×D3C2314yescomplemented
30C32×Dic5C2314yescomplemented
31C22×C22⋊D5—31noHall
32C23×C3×D3—51no—
33C23×C3×D3D511yescharacteristic, complemented
34C22×C32×C4—51no—
35C2×C3×D12—512no—
36C23×C3×C5—21no—
37C23×C3×C5—36no—
38C23×C3×C5—64no—
39C23×C3×C5C2×D311yescharacteristic
40C23×C3×C5C22×C311yescharacteristic
41C22×D15—24no—
42C22×D15C22×C316yescomplemented
43C22×C5×D3—24no—
44C22×C5×D3C22×C316yescomplemented
45C22×C3×D5—36no—
46C22×C3×D5—64no—
47C3×C22⋊D5—316no—
48C2×C3×Dic5—26no—
49C2×C3×Dic5C2×D316yescomplemented
50C2×C3×Dic5C22×C316yescomplemented
51#1203C22×C3116yesdirect product factor
52C22×C3×D4—151no—
53C22×D12—51no—
54C2×C32×C5C2×D416yescomplemented
55C2×C32×C5C2411yescharacteristic
56C3×D15—24no—
57C3×C5×D3—24no—
58C24×C5—31no—
59C23×D5—31no—
60C2×C22⋊D5—312no—
61C22×Dic5C3×D311yescharacteristic, complemented
62C23×C32C2×D511yescharacteristic
63C22×C3×D3—24no—
64C22×C3×D3—56no—
65C22×C3×D3—104no—
66C22×C3×D3C2×D516yescomplemented
67C2×C32×C4—56no—
68C3×D12—516no—
69C22×C3×C5—24no—
70C22×C3×C5—23no—
71C22×C3×C5—34no—
72C22×C3×C5—612no—
73C22×C3×C5D1214yescomplemented
74C22×C3×C5C3×D414yescomplemented
75C22×C3×C5C22×D313yes—
76C22×C3×C5C23×C313yes—
77C2×D15—212no—
78C2×D15C23×C314yescomplemented
79C2×C5×D3—212no—
80C2×C5×D3C23×C314yescomplemented
81C2×C3×D5—34no—
82C2×C3×D5—612no—
83C3×Dic5—24no—
84C3×Dic5C22×D314yescomplemented
85C3×Dic5C23×C314yescomplemented
86C24×C3—31no—
87C24×C3—151no—
88C23×D3—51no—
89C23×D3C3×D511yescharacteristic, complemented
90C2×C3×D4—1512no—
91C22×C3×C4—51no—
92C22×C3×C4—51no—
93C22×C3×C4—101no—
94C2×D12—512no—
95C32×C5C22×D411yescharacteristic, complemented, Hall
96C23×C5—36no—
97C23×C5—64no—
98C23×C5C2×C3×D311yescharacteristic
99C22×D5—36no—
100C22×D5—64no—
101C22⋊D5—316no—
102C2×Dic5C2×C3×D316yescomplemented
103C22×C32C22⋊D514yescomplemented
104C22×C32C22×D513yes—
105C2×C3×D3—212no—
106C2×C3×D3—54no—
107C2×C3×D3—1012no—
108C2×C3×D3C22×D514yescomplemented
109C32×C4—54no—
110C22×D4—151noSylow
111C2×C3×C5—21no—
112C2×C3×C5—26no—
113C2×C3×C5—64no—
114C2×C3×C5C2×D1216yescomplemented
115C2×C3×C5C2×C3×D416yescomplemented
116C2×C3×C5C23×D311yescharacteristic
117C2×C3×C5C24×C311yescharacteristic
118D15—24no—
119C3×D5—64no—
120C5×D3—24no—
121C23×C3—21no—
122C23×C3—36no—
123C23×C3—64no—
124C23×C3—156no—
125C23×C3—304no—
126C23×C3D3×D511yescharacteristic
127C23×C3C2×C3×D511yescharacteristic
128C22×D3—24no—
129C22×D3—56no—
130C22×D3—104no—
131C22×D3C2×C3×D516yescomplemented
132C3×D4—1516no—
133C2×C3×C4—56no—
134C2×C3×C4—56no—
135C2×C3×C4—106no—
136D12—516no—
137C22×C5—34no—
138C22×C5—612no—
139C22×C5C3×D1214yescomplemented
140C22×C5C22×C3×D313yes—
141C2×D5—34no—
142C2×D5—612no—
143Dic5C22×C3×D314yescomplemented
144C2×C32C2×C22⋊D516yescomplemented
145C2×C32C23×D511yescharacteristic
146C3×D3—24no—
147C3×D3—104no—
148C24—31no—
149C24—151no—
150C2×D4—1512no—
151C22×C4—51no—
152C3×C5—21no—
153C3×C5C22×D1211yescharacteristic, complemented
154C3×C5C22×C3×D411yescharacteristic, complemented
155C22×C3—24no—
156C22×C3—23no—
157C22×C3—34no—
158C22×C3—612no—
159C22×C3—154no—
160C22×C3—3012no—
161C22×C3#120314yesdirect product factor
162C22×C3C3×C22⋊D514yescomplemented
163C22×C3C2×D3×D513yes—
164C22×C3C22×C3×D513yes—
165C2×D3—212no—
166C2×D3—54no—
167C2×D3—1012no—
168C2×D3C22×C3×D514yescomplemented
169C3×C4—54no—
170C3×C4—54no—
171C3×C4—104no—
172C2×C5—64no—
173C2×C5C2×C3×D1216yescomplemented
174C2×C5C23×C3×D311yescharacteristic
175D5—64no—
176C32C22×C22⋊D511yescharacteristic, complemented, Sylow
177C23—36no—
178C23—64no—
179C23—156no—
180C23—304no—
181C23C3×D3×D511yescharacteristic
182D4—1516no—
183C2×C4—56no—
184C2×C3—21no—
185C2×C3—26no—
186C2×C3—64no—
187C2×C3—304no—
188C2×C3C2×#120316yesdirect product factor
189C2×C3C2×C3×C22⋊D516yescomplemented
190C2×C3C22×D3×D511yescharacteristic
191C2×C3C23×C3×D511yescharacteristic
192D3—24no—
193D3—104no—
194C5C22×C3×D1211yescharacteristic, complemented, Sylow
195C22—34no—
196C22—612no—
197C22—154no—
198C22—3012no—
199C22C3×#120314yesdirect product factor
200C22C2×C3×D3×D513yes—
201C4—54no—
202C3—21no—
203C3C22×#120311yescharacteristic, direct product factor
204C3C22×C3×C22⋊D511yescharacteristic, complemented
205C2—64no—
206C2—304no—
207C2C2×C3×#120316yesdirect product factor
208C2C22×C3×D3×D511yescharacteristic
subgroupG/H
center, hypercenterC23×C3D3×D5
commutator, cosocleC2×C3×C5C24×C3
FrattiniC2C22×C3×D3×D5
Fitting, socleC23×C32×C5C22
hypocenterC3×C5C22×C3×D4
LCSG ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
1. C23×C3×D15C2semidirect
2. C23×C3×C5×D3C2semidirect
3. C22×C32×Dic5C2semidirect
4. C2×C3×#1203C2direct (N × H)
6. C22×#1203C3direct (N × H)
9. C22×C3×D15C22semidirect
11. C22×C3×C5×D3C22semidirect
12. C2×C32×Dic5C22semidirect
13. C3×#1203C22direct (N × H)
16. C23×D15C2×C3semidirect
17. C23×C5×D3C2×C3semidirect
21. C22×C3×Dic5D3semidirect
22. C22×C3×Dic5C2×C3semidirect
23. C2×#1203C2×C3direct (N × H)
24. C22×C32×C5D4semidirect
27. C2×C3×D15C23semidirect
29. C2×C3×C5×D3C23semidirect
30. C32×Dic5C23semidirect
33. C23×C3×D3D5semidirect
42. C22×D15C22×C3semidirect
44. C22×C5×D3C22×C3semidirect
49. C2×C3×Dic5C2×D3semidirect
50. C2×C3×Dic5C22×C3semidirect
51. #1203C22×C3direct (N × H)
54. C2×C32×C5C2×D4semidirect
61. C22×Dic5C3×D3semidirect
66. C22×C3×D3C2×D5semidirect
73. C22×C3×C5D12semidirect
74. C22×C3×C5C3×D4semidirect
78. C2×D15C23×C3semidirect
80. C2×C5×D3C23×C3semidirect
84. C3×Dic5C22×D3semidirect
85. C3×Dic5C23×C3semidirect
89. C23×D3C3×D5semidirect
95. C32×C5C22×D4semidirect
102. C2×Dic5C2×C3×D3semidirect
103. C22×C32C22⋊D5semidirect
108. C2×C3×D3C22×D5semidirect
114. C2×C3×C5C2×D12semidirect
115. C2×C3×C5C2×C3×D4semidirect
131. C22×D3C2×C3×D5semidirect
139. C22×C5C3×D12semidirect
143. Dic5C22×C3×D3semidirect
144. C2×C32C2×C22⋊D5semidirect
153. C3×C5C22×D12semidirect
154. C3×C5C22×C3×D4semidirect
161. C22×C3#1203direct (N × H)
162. C22×C3C3×C22⋊D5semidirect
168. C2×D3C22×C3×D5semidirect
173. C2×C5C2×C3×D12semidirect
176. C32C22×C22⋊D5semidirect
188. C2×C3C2×#1203direct (N × H)
189. C2×C3C2×C3×C22⋊D5semidirect
194. C5C22×C3×D12semidirect
199. C22C3×#1203direct (N × H)
203. C3C22×#1203direct (N × H)
204. C3C22×C3×C22⋊D5semidirect
207. C2C2×C3×#1203direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×D152
C23×C3×C5×D32
C22×C32×Dic52
C2×C3×#12032
C22×C3×C22⋊D53
C22×#12033
C22×C3×D125