Finite Group Databasebeta

← C22×C3×C5×S4 #13089608

Subgroups — C22×C3×C5×S4 #13089608

Order 1440.

544 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 104 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×A4C211yesmaximal, characteristic, complemented
2C2×C3×C5×S4C216yesmaximal, direct product factor
3C22×C5×S4C311yesmaximal, characteristic, direct product factor
4C22×C3×C5×D4—31nomaximal
5C22×C3×C5×D3—41nomaximal
6C2×C3×C5×A4C2213yescomplemented
7C3×C5×S4C2214yesdirect product factor
8C22×C3×S4C511yesmaximal, characteristic, direct product factor, Hall
9C24×C3×C5—31no—
10C24×C3×C5D311yescharacteristic, complemented
11C22×C5×A4—21no—
12C22×C5×A4C2×C311yescharacteristic, complemented
13C2×C5×S4C2×C316yesdirect product factor
14C2×C3×C5×D4—36no—
15C2×C3×C5×D4—36no—
16C22×C3×C4×C5—31no—
17C22×C32×C5—41no—
18C2×C3×C5×D3—46no—
19C3×C5×A4C2311yescharacteristic, complemented
20C22×C5×D4—31noHall
21C22×C3×A4C2×C511yescharacteristic, complemented
22C2×C3×S4C2×C516yesdirect product factor
23C23×C3×C5—31no—
24C23×C3×C5—33no—
25C23×C3×C5—36no—
26C23×C3×C5—63no—
27C23×C3×C5—61no—
28C23×C3×C5C2×D313yescomplemented
29C22×C5×D3—41no—
30C2×C5×A4—23no—
31C2×C5×A4C22×C313yescomplemented
32C5×S4C22×C314yesdirect product factor
33C3×C5×D4—312no—
34C3×C5×D4—34no—
35C2×C3×C4×C5—36no—
36C22×S4C3×C511yescharacteristic, direct product factor
37C22×C3×D4—31no—
38C2×C32×C5—43no—
39C3×C5×D3—44no—
40C24×C5—31no—
41C24×C5C3×D311yescharacteristic, complemented
42C2×C5×D4—36no—
43C2×C5×D4—36no—
44C22×C4×C5—31no—
45C22×C3×D3—41no—
46C2×C3×A4C22×C513yescomplemented
47C3×S4C22×C514yesdirect product factor
48C22×C3×C5—33no—
49C22×C3×C5—33no—
50C22×C3×C5—33no—
51C22×C3×C5—34no—
52C22×C3×C5—41no—
53C22×C3×C5—61no—
54C22×C3×C5—66no—
55C22×C3×C5—66no—
56C22×C3×C5—81no—
57C22×C3×C5S411yescharacteristic, direct product factor
58C22×C3×C5C22×D311yescharacteristic, complemented
59C2×C5×D3—46no—
60C5×A4—21no—
61C5×A4C23×C311yescharacteristic, complemented
62C3×C4×C5—34no—
63C24×C3—31no—
64C24×C3C5×D311yescharacteristic, complemented
65C22×A4—21no—
66C22×A4C2×C3×C511yescharacteristic, complemented
67C2×S4C2×C3×C516yesdirect product factor
68C2×C3×D4—36no—
69C2×C3×D4—36no—
70C22×C3×C4—31no—
71C32×C5—41noHall
72C23×C5—31no—
73C23×C5—33no—
74C23×C5—36no—
75C23×C5—63no—
76C23×C5—61no—
77C23×C5C2×C3×D313yescomplemented
78C5×D4—312no—
79C5×D4—34no—
80C2×C4×C5—36no—
81C22×C32—41no—
82C2×C3×D3—46no—
83C3×A4C23×C511yescharacteristic, complemented
84C22×D4—31noSylow
85C2×C3×C5—31no—
86C2×C3×C5—33no—
87C2×C3×C5—43no—
88C2×C3×C5—64no—
89C2×C3×C5—83no—
90C2×C3×C5C2×S413yesdirect product factor
91C5×D3—44no—
92C23×C3—31no—
93C23×C3—33no—
94C23×C3—36no—
95C23×C3—63no—
96C23×C3—61no—
97C23×C3C2×C5×D313yescomplemented
98C22×D3—41no—
99C2×A4—23no—
100C2×A4C22×C3×C513yescomplemented
101S4C22×C3×C514yesdirect product factor
102C3×D4—312no—
103C3×D4—34no—
104C2×C3×C4—36no—
105C22×C5—33no—
106C22×C5—33no—
107C22×C5—33no—
108C22×C5—34no—
109C22×C5—61no—
110C22×C5—66no—
111C22×C5—66no—
112C22×C5C3×S411yescharacteristic, direct product factor
113C22×C5C22×C3×D311yescharacteristic, complemented
114C4×C5—34no—
115C2×C32—43no—
116C3×D3—44no—
117C24—31no—
118C24C3×C5×D311yescharacteristic, complemented
119C2×D4—36no—
120C2×D4—36no—
121C22×C4—31no—
122C3×C5—41no—
123C3×C5—81no—
124C3×C5C22×S411yescharacteristic, direct product factor
125C22×C3—33no—
126C22×C3—33no—
127C22×C3—33no—
128C22×C3—34no—
129C22×C3—41no—
130C22×C3—61no—
131C22×C3—66no—
132C22×C3—66no—
133C22×C3—81no—
134C22×C3C5×S411yescharacteristic, direct product factor
135C22×C3C22×C5×D311yescharacteristic, complemented
136C2×D3—46no—
137A4—21no—
138A4C23×C3×C511yescharacteristic, complemented
139C3×C4—34no—
140C2×C5—31no—
141C2×C5—33no—
142C2×C5—64no—
143C2×C5C2×C3×S413yesdirect product factor
144C32—41noSylow
145C23—31no—
146C23—33no—
147C23—36no—
148C23—63no—
149C23—61no—
150C23C2×C3×C5×D313yescomplemented
151D4—312no—
152D4—34no—
153C2×C4—36no—
154C2×C3—31no—
155C2×C3—33no—
156C2×C3—43no—
157C2×C3—64no—
158C2×C3—83no—
159C2×C3C2×C5×S413yesdirect product factor
160D3—44no—
161C5C22×C3×S411yescharacteristic, direct product factor, Sylow
162C22—33no—
163C22—33no—
164C22—33no—
165C22—34no—
166C22—61no—
167C22—66no—
168C22—66no—
169C22C3×C5×S411yescharacteristic, direct product factor
170C22C22×C3×C5×D311yescharacteristic, complemented
171C4—34no—
172C3—41no—
173C3—81no—
174C3C22×C5×S411yescharacteristic, direct product factor
175C2—31no—
176C2—33no—
177C2—64no—
178C2C2×C3×C5×S413yesdirect product factor
subgroupG/H
center, hypercenterC22×C3×C5S4
commutator, cosocle, hypocenterA4C23×C3×C5
FrattiniC1G
Fitting, socleC24×C3×C5D3
LCSG ▷ A4
UCSC1 ◁ C22×C3×C5

Complements

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

NH ≅ G/Ntype
1. C22×C3×C5×A4C2semidirect
2. C2×C3×C5×S4C2direct (N × H)
3. C22×C5×S4C3direct (N × H)
6. C2×C3×C5×A4C22semidirect
7. C3×C5×S4C22direct (N × H)
8. C22×C3×S4C5direct (N × H)
10. C24×C3×C5D3semidirect
12. C22×C5×A4C2×C3semidirect
13. C2×C5×S4C2×C3direct (N × H)
19. C3×C5×A4C23semidirect
21. C22×C3×A4C2×C5semidirect
22. C2×C3×S4C2×C5direct (N × H)
28. C23×C3×C5C2×D3semidirect
31. C2×C5×A4C22×C3semidirect
32. C5×S4C22×C3direct (N × H)
36. C22×S4C3×C5direct (N × H)
41. C24×C5C3×D3semidirect
46. C2×C3×A4C22×C5semidirect
47. C3×S4C22×C5direct (N × H)
57. C22×C3×C5S4direct (N × H)
58. C22×C3×C5C22×D3semidirect
61. C5×A4C23×C3semidirect
64. C24×C3C5×D3semidirect
66. C22×A4C2×C3×C5semidirect
67. C2×S4C2×C3×C5direct (N × H)
77. C23×C5C2×C3×D3semidirect
83. C3×A4C23×C5semidirect
90. C2×C3×C5C2×S4direct (N × H)
97. C23×C3C2×C5×D3semidirect
100. C2×A4C22×C3×C5semidirect
101. S4C22×C3×C5direct (N × H)
112. C22×C5C3×S4direct (N × H)
113. C22×C5C22×C3×D3semidirect
118. C24C3×C5×D3semidirect
124. C3×C5C22×S4direct (N × H)
134. C22×C3C5×S4direct (N × H)
135. C22×C3C22×C5×D3semidirect
138. A4C23×C3×C5semidirect
143. C2×C5C2×C3×S4direct (N × H)
150. C23C2×C3×C5×D3semidirect
159. C2×C3C2×C5×S4direct (N × H)
161. C5C22×C3×S4direct (N × H)
169. C22C3×C5×S4direct (N × H)
170. C22C22×C3×C5×D3semidirect
174. C3C22×C5×S4direct (N × H)
178. C2C2×C3×C5×S4direct (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×A42
C2×C3×C5×S42
C22×C5×S43
C22×C3×C5×D43
C22×C3×C5×D34
C22×C3×S45