Finite Group Databasebeta

← C22×C3×#1204 #13089249

Subgroups — C22×C3×#1204 #13089249

Order 1440.

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

normal subgroups only

This group has 678 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
1C22×C3×Dic15C211yesmaximal, characteristic
2C22×C3×C5×Dic3C211yesmaximal, characteristic
3C22×C32×Dic5C211yesmaximal, characteristic
4C2×C3×#1204C2112yesmaximal, direct product factor
5C22×C3×Dic10—31nomaximal
6C22×#1204C311yesmaximal, characteristic, direct product factor
7C23×C32×C5C2211yescharacteristic
8C2×C3×Dic15C2216yes—
9C2×C3×C5×Dic3C2216yes—
10C2×C32×Dic5C2216yes—
11C3×#1204C22116yesdirect product factor
12C22×C3×Dic6—51nomaximal, Hall
13C22×C3×C4×C5—31no—
14C22×Dic15C2×C311yescharacteristic
15C22×C5×Dic3C2×C311yescharacteristic
16C22×C3×Dic5—21no—
17C22×C3×Dic5—31no—
18C22×C3×Dic5D311yescharacteristic
19C22×C3×Dic5C2×C311yescharacteristic
20C2×C3×Dic10—312no—
21C2×#1204C2×C3112yesdirect product factor
22C22×C32×C5Q814yescomplemented
23C22×C32×C5C2313yes—
24C3×Dic15C2314yes—
25C3×C5×Dic3C2314yes—
26C32×Dic5C2314yes—
27C22×Dic10—31noHall
28C22×C32×C4—51no—
29C22×C3×Dic3—51no—
30C22×C3×Dic3D511yescharacteristic
31C2×C3×Dic6—512no—
32C23×C3×C5—21no—
33C23×C3×C5C2×D311yescharacteristic
34C23×C3×C5C22×C311yescharacteristic
35C2×C3×C4×C5—36no—
36C2×Dic15C22×C316yes—
37C2×C5×Dic3C22×C316yes—
38C2×C3×Dic5—26no—
39C2×C3×Dic5—36no—
40C2×C3×Dic5C2×D316yes—
41C2×C3×Dic5C22×C316yes—
42C3×Dic10—316no—
43#1204C22×C3116yesdirect product factor
44C22×C3×Q8—151no—
45C22×Dic6—51no—
46C2×C32×C5C2×Q816yescomplemented
47C2×C32×C5C2411yescharacteristic
48C22×C4×C5—31no—
49C22×Dic5—31no—
50C22×Dic5C3×D311yescharacteristic
51C2×Dic10—312no—
52C23×C32C2×D511yescharacteristic
53C2×C32×C4—56no—
54C2×C3×Dic3—56no—
55C2×C3×Dic3C2×D516yes—
56C3×Dic6—516no—
57C22×C3×C5—24no—
58C22×C3×C5—23no—
59C22×C3×C5Dic614yescomplemented
60C22×C3×C5C3×Q814yescomplemented
61C22×C3×C5C22×D313yes—
62C22×C3×C5C23×C313yes—
63C3×C4×C5—34no—
64Dic15C23×C314yes—
65C3×Dic5—24no—
66C3×Dic5—34no—
67C3×Dic5C22×D314yes—
68C3×Dic5C23×C314yes—
69C5×Dic3C23×C314yes—
70C2×C3×Q8—1512no—
71C22×C3×C4—31no—
72C22×C3×C4—51no—
73C22×C3×C4—51no—
74C22×C3×C4—101no—
75C22×C3×C4—151no—
76C22×Dic3—51no—
77C22×Dic3C3×D511yescharacteristic
78C2×Dic6—512no—
79C32×C5C22×Q811yescharacteristic, complemented, Hall
80C23×C5C2×C3×D311yescharacteristic
81C2×C4×C5—36no—
82C2×Dic5—36no—
83C2×Dic5C2×C3×D316yes—
84Dic10—316no—
85C22×C32Dic1014yescomplemented
86C22×C32C22×D513yes—
87C32×C4—54no—
88C3×Dic3—54no—
89C3×Dic3C22×D514yes—
90C22×Q8—151noSylow
91C2×C3×C5—21no—
92C2×C3×C5—26no—
93C2×C3×C5C2×Dic616yescomplemented
94C2×C3×C5C2×C3×Q816yescomplemented
95C2×C3×C5C23×D311yescharacteristic
96C2×C3×C5C24×C311yescharacteristic
97C23×C3—21no—
98C23×C3D3×D511yescharacteristic
99C23×C3C2×C3×D511yescharacteristic
100C3×Q8—1516no—
101C2×C3×C4—36no—
102C2×C3×C4—56no—
103C2×C3×C4—56no—
104C2×C3×C4—106no—
105C2×C3×C4—156no—
106C2×Dic3—56no—
107C2×Dic3C2×C3×D516yes—
108Dic6—516no—
109C22×C5C3×Dic614yescomplemented
110C22×C5C22×C3×D313yes—
111C4×C5—34no—
112Dic5—34no—
113Dic5C22×C3×D314yes—
114C2×C32C2×Dic1016yescomplemented
115C2×C32C23×D511yescharacteristic
116C2×Q8—1512no—
117C22×C4—31no—
118C22×C4—51no—
119C22×C4—151no—
120C3×C5—21no—
121C3×C5C22×Dic611yescharacteristic, complemented
122C3×C5C22×C3×Q811yescharacteristic, complemented
123C22×C3—24no—
124C22×C3—23no—
125C22×C3#120414yesdirect product factor
126C22×C3C3×Dic1014yescomplemented
127C22×C3C2×D3×D513yes—
128C22×C3C22×C3×D513yes—
129C3×C4—34no—
130C3×C4—54no—
131C3×C4—54no—
132C3×C4—104no—
133C3×C4—154no—
134Dic3—54no—
135Dic3C22×C3×D514yes—
136C2×C5C2×C3×Dic616yescomplemented
137C2×C5C23×C3×D311yescharacteristic
138C32C22×Dic1011yescharacteristic, complemented, Sylow
139C23C3×D3×D511yescharacteristic
140Q8—1516no—
141C2×C4—36no—
142C2×C4—56no—
143C2×C4—156no—
144C2×C3—21no—
145C2×C3—26no—
146C2×C3C2×#120416yesdirect product factor
147C2×C3C2×C3×Dic1016yescomplemented
148C2×C3C22×D3×D511yescharacteristic
149C2×C3C23×C3×D511yescharacteristic
150C5C22×C3×Dic611yescharacteristic, complemented, Sylow
151C22C3×#120414yesdirect product factor
152C22C2×C3×D3×D513yes—
153C4—34no—
154C4—54no—
155C4—154no—
156C3—21no—
157C3C22×#120411yescharacteristic, direct product factor
158C3C22×C3×Dic1011yescharacteristic, complemented
159C2C2×C3×#120416yesdirect product factor
160C2C22×C3×D3×D511yescharacteristic
subgroupG/H
center, hypercenterC23×C3D3×D5
commutator, cosocleC2×C3×C5C24×C3
FrattiniC2C22×C3×D3×D5
Fitting, socle, prime radicalC23×C32×C5C22
hypocenterC3×C5C22×C3×Q8
LCSG ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
4. C2×C3×#1204C2direct (N × H)
6. C22×#1204C3direct (N × H)
11. C3×#1204C22direct (N × H)
21. C2×#1204C2×C3direct (N × H)
22. C22×C32×C5Q8semidirect
43. #1204C22×C3direct (N × H)
46. C2×C32×C5C2×Q8semidirect
59. C22×C3×C5Dic6semidirect
60. C22×C3×C5C3×Q8semidirect
79. C32×C5C22×Q8semidirect
85. C22×C32Dic10semidirect
93. C2×C3×C5C2×Dic6semidirect
94. C2×C3×C5C2×C3×Q8semidirect
109. C22×C5C3×Dic6semidirect
114. C2×C32C2×Dic10semidirect
121. C3×C5C22×Dic6semidirect
122. C3×C5C22×C3×Q8semidirect
125. C22×C3#1204direct (N × H)
126. C22×C3C3×Dic10semidirect
136. C2×C5C2×C3×Dic6semidirect
138. C32C22×Dic10semidirect
146. C2×C3C2×#1204direct (N × H)
147. C2×C3C2×C3×Dic10semidirect
150. C5C22×C3×Dic6semidirect
151. C22C3×#1204direct (N × H)
157. C3C22×#1204direct (N × H)
158. C3C22×C3×Dic10semidirect
159. C2C2×C3×#1204direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×Dic152
C22×C3×C5×Dic32
C22×C32×Dic52
C2×C3×#12042
C22×C3×Dic103
C22×#12043
C22×C3×Dic65