Finite Group Databasebeta

← C22×#67184 #13089517

Subgroups — C22×#67184 #13089517

Order 1440.

1416 conjugacy classes of subgroups; only the 375 normal subgroups are shown.

normal subgroups only

This group has 1416 conjugacy classes of subgroups — too many to draw. Only its 375 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
1C24×C32×C5C211yesmaximal, characteristic, complemented
2C23×C3⋊D15C211yesmaximal, characteristic, complemented
3C2×#67184C2112yesmaximal, direct product factor
4C22×C3⋊Dic15C211yesmaximal, characteristic, complemented
5C22×#1220—34nomaximal
6C23×C32×C5—24no—
7C23×C32×C5C2211yescharacteristic
8C23×C32×C5C2216yescomplemented
9C22×C3⋊D15—24no—
10C22×C3⋊D15C2216yescomplemented
11#67184C22116yesdirect product factor
12C2×C3⋊Dic15C2216yescomplemented
13C22×#638—51nomaximal, Hall
14C24×C3×C5D314yescomplemented
15C23×D15—34no—
16C2×#1220—348no—
17C22×Dic15—34no—
18C22×C32×C5—212no—
19C22×C32×C5D414yescomplemented
20C22×C32×C5C2313yes—
21C22×C32×C5C2314yescomplemented
22C2×C3⋊D15—212no—
23C2×C3⋊D15C2314yescomplemented
24C3⋊Dic15C2314yescomplemented
25C22×C22⋊D5—91noHall
26C24×C32D511yescharacteristic, complemented
27C23×C3⋊D3—51no—
28C2×#638—512no—
29C22×C3⋊Dic3—51no—
30C23×C3×C5—216no—
31C23×C3×C5C2×D314yes—
32C23×C3×C5C2×D3124yescomplemented
33C22×D15—324no—
34C22×D15—616no—
35#1220—364no—
36C2×Dic15—324no—
37C22×#67—154no—
38C2×C32×C5—24no—
39C2×C32×C5C2×D416yescomplemented
40C2×C32×C5C2411yescharacteristic
41C3⋊D15—24no—
42C24×C5C3⋊D311yescharacteristic, complemented
43C23×D5—91no—
44C2×C22⋊D5—912no—
45C22×Dic5—91no—
46C23×C32—24no—
47C23×C32C2×D511yescharacteristic
48C23×C32C2×D516yescomplemented
49C22×C3⋊D3—56no—
50C22×C3⋊D3—104no—
51#638—516no—
52C2×C3⋊Dic3—56no—
53C22×C3×C5—248no—
54C22×C3×C5#67116yescomplemented
55C22×C3×C5C22×D3112yes—
56C22×C3×C5C22×D3116yescomplemented
57C2×D15—316no—
58C2×D15—648no—
59Dic15—316no—
60C24×C3D1514yescomplemented
61C23×D3—154no—
62C2×#67—1548no—
63C22×Dic3—154no—
64C32×C5C22×D411yescharacteristic, complemented, Hall
65C23×C5—24no—
66C23×C5C2×C3⋊D311yescharacteristic
67C23×C5C2×C3⋊D316yescomplemented
68C22×D5—96no—
69C22×D5—184no—
70C22⋊D5—916no—
71C2×Dic5—96no—
72C22×C32—212no—
73C22×C32C22⋊D514yescomplemented
74C22×C32C22×D513yes—
75C22×C32C22×D514yescomplemented
76C2×C3⋊D3—54no—
77C2×C3⋊D3—1012no—
78C3⋊Dic3—54no—
79C22×D4—451noSylow
80C2×C3×C5—216no—
81C2×C3×C5C2×#67124yescomplemented
82C2×C3×C5C23×D314yes—
83D15—616no—
84C23×C3—216no—
85C23×C3C2×D1514yes—
86C23×C3C2×D15124yescomplemented
87C22×D3—1524no—
88C22×D3—3016no—
89#67—1564no—
90C2×Dic3—1524no—
91C22×C5—212no—
92C22×C5#63814yescomplemented
93C22×C5C22×C3⋊D313yes—
94C22×C5C22×C3⋊D314yescomplemented
95C2×D5—94no—
96C2×D5—1812no—
97Dic5—94no—
98C2×C32—24no—
99C2×C32C2×C22⋊D516yescomplemented
100C2×C32C23×D511yescharacteristic
101C3⋊D3—104no—
102C24—451no—
103C24C3⋊D1511yescharacteristic, complemented
104C2×D4—4512no—
105C22×C4—451no—
106C3×C5C22×#6714yescomplemented
107C22×C3—248no—
108C22×C3#1220116yescomplemented
109C22×C3C22×D15112yes—
110C22×C3C22×D15116yescomplemented
111C2×D3—1516no—
112C2×D3—3048no—
113Dic3—1516no—
114C2×C5—24no—
115C2×C5C2×#63816yescomplemented
116C2×C5C23×C3⋊D311yescharacteristic
117D5—184no—
118C32C22×C22⋊D511yescharacteristic, complemented, Sylow
119C23—24no—
120C23—456no—
121C23—904no—
122C23C2×C3⋊D1511yescharacteristic
123C23C2×C3⋊D1516yescomplemented
124D4—4516no—
125C2×C4—456no—
126C2×C3—216no—
127C2×C3C2×#1220124yescomplemented
128C2×C3C23×D1514yes—
129D3—3016no—
130C5C22×#63811yescharacteristic, complemented, Sylow
131C22—212no—
132C22—454no—
133C22—9012no—
134C22#6718414yesdirect product factor
135C22C22×C3⋊D1513yes—
136C22C22×C3⋊D1514yescomplemented
137C4—454no—
138C3C22×#122014yescomplemented
139C2—24no—
140C2—904no—
141C2C2×#6718416yesdirect product factor
142C2C23×C3⋊D1511yescharacteristic
subgroupG/H
centerC23C2×C3⋊D15
commutator, cosocleC2×C32×C5C24
FrattiniC2C23×C3⋊D15
FittingC24×C32×C5C2
socleC23×C32×C5C22
hypercenterC24C3⋊D15
hypocenterC32×C5C22×D4
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C24×C32×C5C2semidirect
2. C23×C3⋊D15C2semidirect
3. C2×#67184C2direct (N × H)
4. C22×C3⋊Dic15C2semidirect
8. C23×C32×C5C22semidirect
10. C22×C3⋊D15C22semidirect
11. #67184C22direct (N × H)
12. C2×C3⋊Dic15C22semidirect
14. C24×C3×C5D3semidirect
19. C22×C32×C5D4semidirect
21. C22×C32×C5C23semidirect
23. C2×C3⋊D15C23semidirect
24. C3⋊Dic15C23semidirect
26. C24×C32D5semidirect
32. C23×C3×C5C2×D3semidirect
39. C2×C32×C5C2×D4semidirect
42. C24×C5C3⋊D3semidirect
48. C23×C32C2×D5semidirect
54. C22×C3×C5#67semidirect
56. C22×C3×C5C22×D3semidirect
60. C24×C3D15semidirect
64. C32×C5C22×D4semidirect
67. C23×C5C2×C3⋊D3semidirect
73. C22×C32C22⋊D5semidirect
75. C22×C32C22×D5semidirect
81. C2×C3×C5C2×#67semidirect
86. C23×C3C2×D15semidirect
92. C22×C5#638semidirect
94. C22×C5C22×C3⋊D3semidirect
99. C2×C32C2×C22⋊D5semidirect
103. C24C3⋊D15semidirect
106. C3×C5C22×#67semidirect
108. C22×C3#1220semidirect
110. C22×C3C22×D15semidirect
115. C2×C5C2×#638semidirect
118. C32C22×C22⋊D5semidirect
123. C23C2×C3⋊D15semidirect
127. C2×C3C2×#1220semidirect
130. C5C22×#638semidirect
134. C22#67184direct (N × H)
136. C22C22×C3⋊D15semidirect
138. C3C22×#1220semidirect
141. C2C2×#67184direct (N × H)

Maximal subgroups

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

subgroupindex
C24×C32×C52
C23×C3⋊D152
C2×#671842
C22×C3⋊Dic152
C22×#12203
C22×#6385