Finite Group Databasebeta

← C22×C3×#1200 #13089245

Subgroups — C22×C3×#1200 #13089245

Order 1440.

998 conjugacy classes of subgroups; only the 376 normal subgroups are shown.

normal subgroups only

This group has 998 conjugacy classes of subgroups — too many to draw. Only its 376 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
2C22×C3×C5×Dic3C211yesmaximal, characteristic, complemented
3C22×C32×Dic5C211yesmaximal, characteristic, complemented
4C2×C3×#1200C2112yesmaximal, direct product factor
5C22×C3×C4×D5—31nomaximal
6C22×#1200C311yesmaximal, characteristic, direct product factor
7C23×C32×C5C2211yescharacteristic
8C22×C3×D15C418yescomplemented
9C22×C3×D15C2216yes—
10C2×C3×C5×Dic3C2216yescomplemented
11C2×C32×Dic5C2216yescomplemented
12C3×#1200C22116yesdirect product factor
13C22×C3×C4×D3—51nomaximal, Hall
14C23×D15C2×C311yescharacteristic
15C23×C3×D5—31no—
16C22×C3×C4×C5—31no—
17C22×C5×Dic3C2×C311yescharacteristic, complemented
18C22×C3×Dic5—21no—
19C22×C3×Dic5D311yescharacteristic, complemented
20C22×C3×Dic5C2×C311yescharacteristic, complemented
21C2×C3×C4×D5—312no—
22C2×#1200C2×C3112yesdirect product factor
23C22×C32×C5C2×C414yescomplemented
24C22×C32×C5C2313yes—
25C2×C3×D15C2×C4124yescomplemented
26C2×C3×D15C2314yes—
27C3×C5×Dic3C2314yescomplemented
28C32×Dic5C2314yescomplemented
29C22×C4×D5—31noHall
30C23×C3×D3—51no—
31C22×C32×C4—51no—
32C22×C3×Dic3D511yescharacteristic, complemented
33C2×C3×C4×D3—512no—
34C23×C3×C5—21no—
35C23×C3×C5C2×D311yescharacteristic
36C23×C3×C5C22×C311yescharacteristic
37C22×D15C3×C418yescomplemented
38C22×D15C22×C316yes—
39C22×C3×D5—38no—
40C22×C3×D5—36no—
41C2×C3×C4×C5—36no—
42C2×C5×Dic3C22×C316yescomplemented
43C2×C3×Dic5—26no—
44C2×C3×Dic5C2×D316yescomplemented
45C2×C3×Dic5C22×C316yescomplemented
46C3×C4×D5—316no—
47#1200C22×C3116yesdirect product factor
48C23×C3×C4—151no—
49C22×C4×D3—51no—
50C2×C32×C5C22×C416yescomplemented
51C2×C32×C5C2411yescharacteristic
52C3×D15C22×C418yescomplemented
53C23×D5—31no—
54C22×C4×C5—31no—
55C22×Dic5C3×D311yescharacteristic, complemented
56C2×C4×D5—312no—
57C23×C32C2×D511yescharacteristic
58C22×C3×D3—58no—
59C22×C3×D3—56no—
60C2×C32×C4—56no—
61C2×C3×Dic3C2×D516yescomplemented
62C3×C4×D3—516no—
63C22×C3×C5—24no—
64C22×C3×C5—23no—
65C22×C3×C5C4×D314yescomplemented
66C22×C3×C5C2×C3×C414yescomplemented
67C22×C3×C5C22×D313yes—
68C22×C3×C5C23×C313yes—
69C2×D15C2×C3×C4124yescomplemented
70C2×D15C23×C314yes—
71C2×C3×D5—324no—
72C2×C3×D5—34no—
73C3×C4×C5—34no—
74C3×Dic5—24no—
75C3×Dic5C22×D314yescomplemented
76C3×Dic5C23×C314yescomplemented
77C5×Dic3C23×C314yescomplemented
78C24×C3—151no—
79C23×D3—51no—
80C22×C3×C4—31no—
81C22×C3×C4—51no—
82C22×C3×C4—51no—
83C22×C3×C4—101no—
84C22×C3×C4—1512no—
85C22×Dic3C3×D511yescharacteristic, complemented
86C2×C4×D3—512no—
87C32×C5C23×C411yescharacteristic, complemented, Hall
88C23×C5C2×C3×D311yescharacteristic
89C22×D5—38no—
90C22×D5—36no—
91C2×C4×C5—36no—
92C2×Dic5C2×C3×D316yescomplemented
93C4×D5—316no—
94C22×C32C4×D514yescomplemented
95C22×C32C22×D513yes—
96C2×C3×D3—524no—
97C2×C3×D3—54no—
98C32×C4—54no—
99C3×Dic3C22×D514yescomplemented
100C23×C4—151noSylow
101C2×C3×C5—21no—
102C2×C3×C5—26no—
103C2×C3×C5C2×C4×D316yescomplemented
104C2×C3×C5C22×C3×C416yescomplemented
105C2×C3×C5C23×D311yescharacteristic
106C2×C3×C5C24×C311yescharacteristic
107D15C22×C3×C418yescomplemented
108C3×D5—38no—
109C23×C3—21no—
110C23×C3—158no—
111C23×C3—156no—
112C23×C3D3×D511yescharacteristic
113C23×C3C2×C3×D511yescharacteristic
114C22×D3—58no—
115C22×D3—56no—
116C2×C3×C4—36no—
117C2×C3×C4—56no—
118C2×C3×C4—56no—
119C2×C3×C4—106no—
120C2×C3×C4—1516no—
121C2×Dic3C2×C3×D516yescomplemented
122C4×D3—516no—
123C22×C5C3×C4×D314yescomplemented
124C22×C5C22×C3×D313yes—
125C2×D5—34no—
126C2×D5—324no—
127C4×C5—34no—
128Dic5C22×C3×D314yescomplemented
129C2×C32C2×C4×D516yescomplemented
130C2×C32C23×D511yescharacteristic
131C3×D3—58no—
132C24—151no—
133C22×C4—31no—
134C22×C4—51no—
135C22×C4—1512no—
136C3×C5—21no—
137C3×C5C22×C4×D311yescharacteristic, complemented
138C3×C5C23×C3×C411yescharacteristic, complemented
139C22×C3—24no—
140C22×C3—23no—
141C22×C3—1524no—
142C22×C3—154no—
143C22×C3#120014yesdirect product factor
144C22×C3C3×C4×D514yescomplemented
145C22×C3C2×D3×D513yes—
146C22×C3C22×C3×D513yes—
147C2×D3—524no—
148C2×D3—54no—
149C3×C4—34no—
150C3×C4—54no—
151C3×C4—54no—
152C3×C4—104no—
153Dic3C22×C3×D514yescomplemented
154C2×C5C2×C3×C4×D316yescomplemented
155C2×C5C23×C3×D311yescharacteristic
156D5—38no—
157C32C22×C4×D511yescharacteristic, complemented, Sylow
158C23—158no—
159C23—156no—
160C23C3×D3×D511yescharacteristic
161C2×C4—36no—
162C2×C4—56no—
163C2×C4—1516no—
164C2×C3—21no—
165C2×C3—26no—
166C2×C3—158no—
167C2×C3C2×#120016yesdirect product factor
168C2×C3C2×C3×C4×D516yescomplemented
169C2×C3C22×D3×D511yescharacteristic
170C2×C3C23×C3×D511yescharacteristic
171D3—58no—
172C5C22×C3×C4×D311yescharacteristic, complemented, Sylow
173C22—154no—
174C22—1524no—
175C22C3×#120014yesdirect product factor
176C22C2×C3×D3×D513yes—
177C4—34no—
178C4—54no—
179C3—21no—
180C3C22×#120011yescharacteristic, direct product factor
181C3C22×C3×C4×D511yescharacteristic, complemented
182C2—158no—
183C2C2×C3×#120016yesdirect product factor
184C2C22×C3×D3×D511yescharacteristic
subgroupG/H
center, hypercenterC23×C3D3×D5
commutator, hypocenterC3×C5C23×C3×C4
FrattiniC2C22×C3×D3×D5
Fitting, socleC23×C32×C5C22
prime radicalC23×C3×D15C2
cosocleC2×C3×C5C24×C3
LCSG ▷ C3×C5
UCSC1 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
2. C22×C3×C5×Dic3C2semidirect
3. C22×C32×Dic5C2semidirect
4. C2×C3×#1200C2direct (N × H)
6. C22×#1200C3direct (N × H)
8. C22×C3×D15C4semidirect
10. C2×C3×C5×Dic3C22semidirect
11. C2×C32×Dic5C22semidirect
12. C3×#1200C22direct (N × H)
17. C22×C5×Dic3C2×C3semidirect
19. C22×C3×Dic5D3semidirect
20. C22×C3×Dic5C2×C3semidirect
22. C2×#1200C2×C3direct (N × H)
23. C22×C32×C5C2×C4semidirect
25. C2×C3×D15C2×C4semidirect
27. C3×C5×Dic3C23semidirect
28. C32×Dic5C23semidirect
32. C22×C3×Dic3D5semidirect
37. C22×D15C3×C4semidirect
42. C2×C5×Dic3C22×C3semidirect
44. C2×C3×Dic5C2×D3semidirect
45. C2×C3×Dic5C22×C3semidirect
47. #1200C22×C3direct (N × H)
50. C2×C32×C5C22×C4semidirect
52. C3×D15C22×C4semidirect
55. C22×Dic5C3×D3semidirect
61. C2×C3×Dic3C2×D5semidirect
65. C22×C3×C5C4×D3semidirect
66. C22×C3×C5C2×C3×C4semidirect
69. C2×D15C2×C3×C4semidirect
75. C3×Dic5C22×D3semidirect
76. C3×Dic5C23×C3semidirect
77. C5×Dic3C23×C3semidirect
85. C22×Dic3C3×D5semidirect
87. C32×C5C23×C4semidirect
92. C2×Dic5C2×C3×D3semidirect
94. C22×C32C4×D5semidirect
99. C3×Dic3C22×D5semidirect
103. C2×C3×C5C2×C4×D3semidirect
104. C2×C3×C5C22×C3×C4semidirect
107. D15C22×C3×C4semidirect
121. C2×Dic3C2×C3×D5semidirect
123. C22×C5C3×C4×D3semidirect
128. Dic5C22×C3×D3semidirect
129. C2×C32C2×C4×D5semidirect
137. C3×C5C22×C4×D3semidirect
138. C3×C5C23×C3×C4semidirect
143. C22×C3#1200direct (N × H)
144. C22×C3C3×C4×D5semidirect
153. Dic3C22×C3×D5semidirect
154. C2×C5C2×C3×C4×D3semidirect
157. C32C22×C4×D5semidirect
167. C2×C3C2×#1200direct (N × H)
168. C2×C3C2×C3×C4×D5semidirect
172. C5C22×C3×C4×D3semidirect
175. C22C3×#1200direct (N × H)
180. C3C22×#1200direct (N × H)
181. C3C22×C3×C4×D5semidirect
183. C2C2×C3×#1200direct (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
C22×C3×C5×Dic32
C22×C32×Dic52
C2×C3×#12002
C22×C3×C4×D53
C22×#12003
C22×C3×C4×D35