Finite Group Databasebeta

← C2×C5×D3×A4 #10624350

Subgroups — C2×C5×D3×A4 #10624350

Order 720.

164 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 42 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
1C2×C3×C5×A4C211yesmaximal, characteristic, complemented
2C5×D3×A4C212yesmaximal, direct product factor
3C23×C5×D3C311yesmaximal, characteristic, complemented
4C22×C5×A4—31nomaximal
5C2×C3×C5×D3—41nomaximal
6C3×C5×A4C2211yescharacteristic, complemented
7C2×D3×A4C511yesmaximal, characteristic, direct product factor, Hall
8C23×C3×C5C2×C311yescharacteristic, complemented
9C22×C5×D3—31no—
10C22×C5×D3—31no—
11C22×C5×D3—32no—
12C22×C5×D3C2×C312yescomplemented
13C2×C5×A4—21no—
14C2×C5×A4—32no—
15C2×C5×A4D311yescharacteristic, direct product factor
16C2×C32×C5—41no—
17C3×C5×D3—42no—
18C24×C5—31noHall
19C2×C3×A4C2×C511yescharacteristic, complemented
20D3×A4C2×C512yesdirect product factor
21C22×C3×C5—31no—
22C22×C3×C5—31no—
23C22×C3×C5—121no—
24C22×C3×C5C22×C311yescharacteristic, complemented
25C2×C5×D3—32no—
26C2×C5×D3—31no—
27C2×C5×D3—32no—
28C2×C5×D3—32no—
29C2×C5×D3—32no—
30C2×C5×D3A411yescharacteristic, direct product factor
31C5×A4—21no—
32C5×A4C2×D311yescharacteristic, direct product factor
33C23×D3C3×C511yescharacteristic, complemented
34C22×A4—31no—
35C32×C5—41noHall
36C23×C5—32no—
37C23×C5—91no—
38C23×C5—92no—
39C23×C5—91no—
40C23×C5C3×D311yescharacteristic, complemented
41C2×C3×D3—41no—
42C3×A4C22×C511yescharacteristic, complemented
43C2×C3×C5—31no—
44C2×C3×C5—31no—
45C2×C3×C5—41no—
46C2×C3×C5—81no—
47C2×C3×C5—122no—
48C2×C3×C5C2×A411yescharacteristic, complemented
49C5×D3—32no—
50C5×D3C2×A412yesdirect product factor
51C23×C3C2×C3×C511yescharacteristic, complemented
52C22×D3—31no—
53C22×D3—31no—
54C22×D3—32no—
55C22×D3C2×C3×C512yescomplemented
56C2×A4—21no—
57C2×A4—32no—
58C2×A4C5×D311yescharacteristic, direct product factor
59C22×C5—31no—
60C22×C5—31no—
61C22×C5—31no—
62C22×C5—92no—
63C22×C5—92no—
64C22×C5—92no—
65C22×C5—92no—
66C22×C5—91no—
67C22×C5C2×C3×D311yescharacteristic, complemented
68C2×C32—41no—
69C3×D3—42no—
70C24—31noSylow
71C3×C5—41no—
72C3×C5—81no—
73C3×C5C22×A411yescharacteristic, complemented
74C22×C3—31no—
75C22×C3—31no—
76C22×C3—121no—
77C22×C3C22×C3×C511yescharacteristic, complemented
78C2×D3—32no—
79C2×D3—31no—
80C2×D3—32no—
81C2×D3—32no—
82C2×D3—32no—
83C2×D3C5×A411yescharacteristic, direct product factor
84A4—21no—
85A4C2×C5×D311yescharacteristic, direct product factor
86C2×C5—31no—
87C2×C5—31no—
88C2×C5—32no—
89C2×C5—92no—
90C2×C5D3×A411yescharacteristic, direct product factor
91C32—41noSylow
92C23—32no—
93C23—91no—
94C23—92no—
95C23—91no—
96C23C3×C5×D311yescharacteristic, complemented
97C2×C3—31no—
98C2×C3—31no—
99C2×C3—41no—
100C2×C3—81no—
101C2×C3—122no—
102C2×C3C2×C5×A411yescharacteristic, complemented
103D3—32no—
104D3C2×C5×A412yesdirect product factor
105C5C2×D3×A411yescharacteristic, direct product factor, Sylow
106C22—31no—
107C22—31no—
108C22—31no—
109C22—92no—
110C22—92no—
111C22—92no—
112C22—92no—
113C22—91no—
114C22C2×C3×C5×D311yescharacteristic, complemented
115C3—41no—
116C3—81no—
117C3C22×C5×A411yescharacteristic, complemented
118C2—31no—
119C2—31no—
120C2—32no—
121C2—92no—
122C2C5×D3×A411yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C5D3×A4
commutator, cosocle, hypocenterC22×C3C22×C3×C5
FrattiniC1G
Fitting, socleC23×C3×C5C2×C3
LCSG ▷ C22×C3
UCSC1 ◁ C2×C5

Complements

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

NH ≅ G/Ntype
1. C2×C3×C5×A4C2semidirect
2. C5×D3×A4C2direct (N × H)
3. C23×C5×D3C3semidirect
6. C3×C5×A4C22semidirect
7. C2×D3×A4C5direct (N × H)
8. C23×C3×C5C2×C3semidirect
12. C22×C5×D3C2×C3semidirect
15. C2×C5×A4D3direct (N × H)
19. C2×C3×A4C2×C5semidirect
20. D3×A4C2×C5direct (N × H)
24. C22×C3×C5C22×C3semidirect
30. C2×C5×D3A4direct (N × H)
32. C5×A4C2×D3direct (N × H)
33. C23×D3C3×C5semidirect
40. C23×C5C3×D3semidirect
42. C3×A4C22×C5semidirect
48. C2×C3×C5C2×A4semidirect
50. C5×D3C2×A4direct (N × H)
51. C23×C3C2×C3×C5semidirect
55. C22×D3C2×C3×C5semidirect
58. C2×A4C5×D3direct (N × H)
67. C22×C5C2×C3×D3semidirect
73. C3×C5C22×A4semidirect
77. C22×C3C22×C3×C5semidirect
83. C2×D3C5×A4direct (N × H)
85. A4C2×C5×D3direct (N × H)
90. C2×C5D3×A4direct (N × H)
96. C23C3×C5×D3semidirect
102. C2×C3C2×C5×A4semidirect
104. D3C2×C5×A4direct (N × H)
105. C5C2×D3×A4direct (N × H)
114. C22C2×C3×C5×D3semidirect
117. C3C22×C5×A4semidirect
122. C2C5×D3×A4direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C5×A42
C5×D3×A42
C23×C5×D33
C22×C5×A43
C2×C3×C5×D34
C2×D3×A45