Finite Group Databasebeta

← C22×C3×D3×Dic5 #13089243

Subgroups — C22×C3×D3×Dic5 #13089243

Order 1440.

998 conjugacy classes of subgroups; only the 456 normal subgroups are shown.

normal subgroups only

This group has 998 conjugacy classes of subgroups — too many to draw. Only its 456 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×C5×D3C211yesmaximal, characteristic
2C22×C3×Dic15C211yesmaximal, characteristic, complemented
3C22×C32×Dic5C211yesmaximal, characteristic, complemented
4C2×C3×D3×Dic5C2112yesmaximal, direct product factor
5C23×C3×Dic5—31nomaximal
6C22×D3×Dic5C311yesmaximal, characteristic, direct product factor
7C23×C32×C5C2211yescharacteristic
8C22×C3×C5×D3C418yescomplemented
9C22×C3×C5×D3C2216yes—
10C2×C3×Dic15C2216yescomplemented
11C2×C32×Dic5C2216yescomplemented
12C3×D3×Dic5C22116yesdirect product factor
13C22×C3×C4×D3—51nomaximal, Hall
14C24×C3×C5—31no—
15C23×C5×D3C2×C311yescharacteristic
16C22×Dic15C2×C311yescharacteristic, complemented
17C22×C3×Dic5—21no—
18C22×C3×Dic5—31no—
19C22×C3×Dic5—312no—
20C22×C3×Dic5D311yescharacteristic, direct product factor
21C22×C3×Dic5C2×C311yescharacteristic, complemented
22C2×D3×Dic5C2×C3112yesdirect product factor
23C22×C32×C5C2×C414yescomplemented
24C22×C32×C5C2313yes—
25C2×C3×C5×D3C2×C4124yescomplemented
26C2×C3×C5×D3C2314yes—
27C3×Dic15C2314yescomplemented
28C32×Dic5C2314yescomplemented
29C23×Dic5—31noHall
30C23×C3×D3D511yescharacteristic
31C22×C32×C4—51no—
32C22×C3×Dic3—51no—
33C2×C3×C4×D3—512no—
34C23×C3×C5—21no—
35C23×C3×C5—38no—
36C23×C3×C5—36no—
37C23×C3×C5C2×D311yescharacteristic
38C23×C3×C5C22×C311yescharacteristic
39C22×C5×D3C3×C418yescomplemented
40C22×C5×D3C22×C316yes—
41C2×Dic15C22×C316yescomplemented
42C2×C3×Dic5—26no—
43C2×C3×Dic5—36no—
44C2×C3×Dic5—316no—
45C2×C3×Dic5C2×D316yesdirect product factor
46C2×C3×Dic5C22×C316yescomplemented
47D3×Dic5C22×C3116yesdirect product factor
48C23×C3×C4—151no—
49C22×C4×D3—51no—
50C2×C32×C5C22×C416yescomplemented
51C2×C32×C5C2411yescharacteristic
52C3×C5×D3C22×C418yescomplemented
53C24×C5—31no—
54C22×Dic5—31no—
55C22×Dic5—312no—
56C22×Dic5C3×D311yescharacteristic, direct product factor
57C23×C32C2×D511yescharacteristic
58C22×C3×D3Dic518yesdirect product factor
59C22×C3×D3C2×D516yes—
60C2×C32×C4—56no—
61C2×C3×Dic3—56no—
62C3×C4×D3—516no—
63C22×C3×C5—24no—
64C22×C3×C5—23no—
65C22×C3×C5—324no—
66C22×C3×C5—34no—
67C22×C3×C5C4×D314yescomplemented
68C22×C3×C5C2×C3×C414yescomplemented
69C22×C3×C5C22×D313yes—
70C22×C3×C5C23×C313yes—
71C2×C5×D3C2×C3×C4124yescomplemented
72C2×C5×D3C23×C314yes—
73Dic15C23×C314yescomplemented
74C3×Dic5—24no—
75C3×Dic5—34no—
76C3×Dic5C22×D314yesdirect product factor
77C3×Dic5C23×C314yescomplemented
78C24×C3—31no—
79C23×D3C3×D511yescharacteristic
80C22×C3×C4—51no—
81C22×C3×C4—51no—
82C22×C3×C4—101no—
83C22×C3×C4—1512no—
84C22×C3×C4—151no—
85C22×Dic3—51no—
86C2×C4×D3—512no—
87C32×C5C23×C411yescharacteristic, complemented, Hall
88C23×C5—38no—
89C23×C5—36no—
90C23×C5C2×C3×D311yescharacteristic
91C2×Dic5—36no—
92C2×Dic5—316no—
93C2×Dic5C2×C3×D316yesdirect product factor
94C22×C32C2×Dic514yescomplemented
95C22×C32C22×D513yes—
96C2×C3×D3C2×Dic5124yesdirect product factor
97C2×C3×D3C22×D514yes—
98C32×C4—54no—
99C3×Dic3—54no—
100C23×C4—151noSylow
101C2×C3×C5—21no—
102C2×C3×C5—26no—
103C2×C3×C5—38no—
104C2×C3×C5C2×C4×D316yescomplemented
105C2×C3×C5C22×C3×C416yescomplemented
106C2×C3×C5C23×D311yescharacteristic
107C2×C3×C5C24×C311yescharacteristic
108C5×D3C22×C3×C418yescomplemented
109C23×C3—21no—
110C23×C3—38no—
111C23×C3—36no—
112C23×C3D3×D511yescharacteristic
113C23×C3C2×C3×D511yescharacteristic
114C22×D3C3×Dic518yesdirect product factor
115C22×D3C2×C3×D516yes—
116C2×C3×C4—56no—
117C2×C3×C4—56no—
118C2×C3×C4—106no—
119C2×C3×C4—1516no—
120C2×C3×C4—156no—
121C2×Dic3—56no—
122C4×D3—516no—
123C22×C5—34no—
124C22×C5—324no—
125C22×C5C3×C4×D314yescomplemented
126C22×C5C22×C3×D313yes—
127Dic5—34no—
128Dic5C22×C3×D314yesdirect product factor
129C2×C32C22×Dic516yescomplemented
130C2×C32C23×D511yescharacteristic
131C3×D3C22×Dic518yesdirect product factor
132C24—31no—
133C22×C4—51no—
134C22×C4—1512no—
135C22×C4—151no—
136C3×C5—21no—
137C3×C5C22×C4×D311yescharacteristic, complemented
138C3×C5C23×C3×C411yescharacteristic, complemented
139C22×C3—24no—
140C22×C3—23no—
141C22×C3—324no—
142C22×C3—34no—
143C22×C3D3×Dic514yesdirect product factor
144C22×C3C2×C3×Dic514yescomplemented
145C22×C3C2×D3×D513yes—
146C22×C3C22×C3×D513yes—
147C2×D3C2×C3×Dic5124yesdirect product factor
148C2×D3C22×C3×D514yes—
149C3×C4—54no—
150C3×C4—54no—
151C3×C4—104no—
152C3×C4—154no—
153Dic3—54no—
154C2×C5—38no—
155C2×C5C2×C3×C4×D316yescomplemented
156C2×C5C23×C3×D311yescharacteristic
157C32C23×Dic511yescharacteristic, complemented, Sylow
158C23—38no—
159C23—36no—
160C23C3×D3×D511yescharacteristic
161C2×C4—56no—
162C2×C4—1516no—
163C2×C4—156no—
164C2×C3—21no—
165C2×C3—26no—
166C2×C3—38no—
167C2×C3C2×D3×Dic516yesdirect product factor
168C2×C3C22×C3×Dic516yescomplemented
169C2×C3C22×D3×D511yescharacteristic
170C2×C3C23×C3×D511yescharacteristic
171D3C22×C3×Dic518yesdirect product factor
172C5C22×C3×C4×D311yescharacteristic, complemented, Sylow
173C22—34no—
174C22—324no—
175C22C3×D3×Dic514yesdirect product factor
176C22C2×C3×D3×D513yes—
177C4—54no—
178C4—154no—
179C3—21no—
180C3C22×D3×Dic511yescharacteristic, direct product factor
181C3C23×C3×Dic511yescharacteristic, complemented
182C2—38no—
183C2C2×C3×D3×Dic516yesdirect 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×C5×D3C2
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×Dic15C2semidirect
3. C22×C32×Dic5C2semidirect
4. C2×C3×D3×Dic5C2direct (N × H)
6. C22×D3×Dic5C3direct (N × H)
8. C22×C3×C5×D3C4semidirect
10. C2×C3×Dic15C22semidirect
11. C2×C32×Dic5C22semidirect
12. C3×D3×Dic5C22direct (N × H)
16. C22×Dic15C2×C3semidirect
20. C22×C3×Dic5D3direct (N × H)
21. C22×C3×Dic5C2×C3semidirect
22. C2×D3×Dic5C2×C3direct (N × H)
23. C22×C32×C5C2×C4semidirect
25. C2×C3×C5×D3C2×C4semidirect
27. C3×Dic15C23semidirect
28. C32×Dic5C23semidirect
39. C22×C5×D3C3×C4semidirect
41. C2×Dic15C22×C3semidirect
45. C2×C3×Dic5C2×D3direct (N × H)
46. C2×C3×Dic5C22×C3semidirect
47. D3×Dic5C22×C3direct (N × H)
50. C2×C32×C5C22×C4semidirect
52. C3×C5×D3C22×C4semidirect
56. C22×Dic5C3×D3direct (N × H)
58. C22×C3×D3Dic5direct (N × H)
67. C22×C3×C5C4×D3semidirect
68. C22×C3×C5C2×C3×C4semidirect
71. C2×C5×D3C2×C3×C4semidirect
73. Dic15C23×C3semidirect
76. C3×Dic5C22×D3direct (N × H)
77. C3×Dic5C23×C3semidirect
87. C32×C5C23×C4semidirect
93. C2×Dic5C2×C3×D3direct (N × H)
94. C22×C32C2×Dic5semidirect
96. C2×C3×D3C2×Dic5direct (N × H)
104. C2×C3×C5C2×C4×D3semidirect
105. C2×C3×C5C22×C3×C4semidirect
108. C5×D3C22×C3×C4semidirect
114. C22×D3C3×Dic5direct (N × H)
125. C22×C5C3×C4×D3semidirect
128. Dic5C22×C3×D3direct (N × H)
129. C2×C32C22×Dic5semidirect
131. C3×D3C22×Dic5direct (N × H)
137. C3×C5C22×C4×D3semidirect
138. C3×C5C23×C3×C4semidirect
143. C22×C3D3×Dic5direct (N × H)
144. C22×C3C2×C3×Dic5semidirect
147. C2×D3C2×C3×Dic5direct (N × H)
155. C2×C5C2×C3×C4×D3semidirect
157. C32C23×Dic5semidirect
167. C2×C3C2×D3×Dic5direct (N × H)
168. C2×C3C22×C3×Dic5semidirect
171. D3C22×C3×Dic5direct (N × H)
172. C5C22×C3×C4×D3semidirect
175. C22C3×D3×Dic5direct (N × H)
180. C3C22×D3×Dic5direct (N × H)
181. C3C23×C3×Dic5semidirect
183. C2C2×C3×D3×Dic5direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×C5×D32
C22×C3×Dic152
C22×C32×Dic52
C2×C3×D3×Dic52
C23×C3×Dic53
C22×D3×Dic53
C22×C3×C4×D35