Finite Group Databasebeta

← C2×C3×D5×Dic5 #11925557

Subgroups — C2×C3×D5×Dic5 #11925557

Order 1200.

248 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 112 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×D5C211yesmaximal, characteristic
2C2×C3×#1039C211yesmaximal, characteristic, complemented
3C2×C3×C5×Dic5C211yesmaximal, characteristic, complemented
4C3×D5×Dic5C214yesmaximal, direct product factor
5C2×D5×Dic5C311yesmaximal, characteristic, direct product factor, Hall
6C22×C3×C52C2211yescharacteristic
7C2×C3×C5×D5C414yescomplemented
8C2×C3×C5×D5C2212yes—
9C3×#1039C2212yescomplemented
10C3×C5×Dic5C2212yescomplemented
11C22×C3×Dic5—51nomaximal
12C2×C3×C4×D5—51nomaximal
13C22×C5×D5C2×C311yescharacteristic
14C2×#1039C2×C311yescharacteristic, complemented
15C2×C5×Dic5C2×C311yescharacteristic, complemented
16D5×Dic5C2×C314yesdirect product factor
17C2×C3×C52C2×C412yescomplemented
18C2×C3×C52C2311yescharacteristic
19C3×C5×D5C2×C414yescomplemented
20C23×C3×C5—51no—
21C22×C3×D5D511yescharacteristic
22C2×C3×C4×C5—51no—
23C2×C3×Dic5—51no—
24C2×C3×Dic5—51no—
25C2×C3×Dic5—54no—
26C2×C3×Dic5—102no—
27C2×C3×Dic5D511yescharacteristic, direct product factor
28C3×C4×D5—54no—
29C22×C52C22×C311yescharacteristic
30C2×C5×D5C3×C414yescomplemented
31C2×C5×D5C22×C312yes—
32#1039C22×C312yescomplemented
33C5×Dic5C22×C312yescomplemented
34C22×Dic5—51no—
35C2×C4×D5—51no—
36C3×C52C22×C411yescharacteristic, complemented, Hall
37C22×C3×C5—22no—
38C22×C3×C5—54no—
39C22×C3×C5—52no—
40C22×C3×C5C2×D511yescharacteristic
41C22×C3×C5C2×D511yescharacteristic
42C2×C3×D5Dic514yesdirect product factor
43C2×C3×D5C2×D512yes—
44C3×C4×C5—52no—
45C3×Dic5—52no—
46C3×Dic5—52no—
47C3×Dic5—104no—
48C3×Dic5C2×D512yesdirect product factor
49C2×C52C2×C3×C412yescomplemented
50C2×C52C23×C311yescharacteristic
51C5×D5C2×C3×C414yescomplemented
52C22×C3×C4—251noHall
53C23×C5—51no—
54C22×D5C3×D511yescharacteristic
55C2×C4×C5—51no—
56C2×Dic5—51no—
57C2×Dic5—51no—
58C2×Dic5—54no—
59C2×Dic5—102no—
60C2×Dic5C3×D511yescharacteristic, direct product factor
61C4×D5—54no—
62C2×C3×C5—24no—
63C2×C3×C5—22no—
64C2×C3×C5—54no—
65C2×C3×C5C4×D512yescomplemented
66C2×C3×C5C2×Dic512yescomplemented
67C2×C3×C5C22×D511yescharacteristic
68C2×C3×C5C22×D511yescharacteristic
69C3×D5C2×Dic514yesdirect product factor
70C52C22×C3×C411yescharacteristic, complemented, Sylow
71C23×C3—51no—
72C2×C3×C4—51no—
73C2×C3×C4—254no—
74C2×C3×C4—251no—
75C22×C5—22no—
76C22×C5—54no—
77C22×C5—52no—
78C22×C5C2×C3×D511yescharacteristic
79C22×C5C2×C3×D511yescharacteristic
80C2×D5C3×Dic514yesdirect product factor
81C2×D5C2×C3×D512yes—
82C4×C5—52no—
83Dic5—52no—
84Dic5—52no—
85Dic5—104no—
86Dic5C2×C3×D512yesdirect product factor
87C22×C4—251noSylow
88C3×C5—22no—
89C3×C5C2×C4×D511yescharacteristic, complemented
90C3×C5C22×Dic511yescharacteristic, complemented
91C22×C3—54no—
92C22×C3—52no—
93C22×C3D5211yescharacteristic
94C3×C4—52no—
95C3×C4—252no—
96C2×C5—22no—
97C2×C5—24no—
98C2×C5—54no—
99C2×C5C3×C4×D512yescomplemented
100C2×C5C2×C3×Dic512yescomplemented
101C2×C5C22×C3×D511yescharacteristic
102C2×C5C22×C3×D511yescharacteristic
103D5C2×C3×Dic514yesdirect product factor
104C23—51no—
105C2×C4—51no—
106C2×C4—254no—
107C2×C4—251no—
108C2×C3—54no—
109C2×C3D5×Dic512yesdirect product factor
110C2×C3C2×D5211yescharacteristic
111C5—22no—
112C5C2×C3×C4×D511yescharacteristic, complemented
113C5C22×C3×Dic511yescharacteristic, complemented
114C22—52no—
115C22—54no—
116C22C3×D5211yescharacteristic
117C4—52no—
118C4—252no—
119C3C2×D5×Dic511yescharacteristic, direct product factor, Sylow
120C2—54no—
121C2C3×D5×Dic512yesdirect product factor
122C2C2×C3×D5211yescharacteristic
subgroupG/H
center, hypercenterC22×C3D52
commutator, hypocenterC52C22×C3×C4
FrattiniC2C2×C3×D52
Fitting, socleC22×C3×C52C22
prime radicalC22×C3×C5×D5C2
cosocleC2×C52C23×C3
LCSG ▷ C52
UCSC1 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
2. C2×C3×#1039C2semidirect
3. C2×C3×C5×Dic5C2semidirect
4. C3×D5×Dic5C2direct (N × H)
5. C2×D5×Dic5C3direct (N × H)
7. C2×C3×C5×D5C4semidirect
9. C3×#1039C22semidirect
10. C3×C5×Dic5C22semidirect
14. C2×#1039C2×C3semidirect
15. C2×C5×Dic5C2×C3semidirect
16. D5×Dic5C2×C3direct (N × H)
17. C2×C3×C52C2×C4semidirect
19. C3×C5×D5C2×C4semidirect
27. C2×C3×Dic5D5direct (N × H)
30. C2×C5×D5C3×C4semidirect
32. #1039C22×C3semidirect
33. C5×Dic5C22×C3semidirect
36. C3×C52C22×C4semidirect
42. C2×C3×D5Dic5direct (N × H)
48. C3×Dic5C2×D5direct (N × H)
49. C2×C52C2×C3×C4semidirect
51. C5×D5C2×C3×C4semidirect
60. C2×Dic5C3×D5direct (N × H)
65. C2×C3×C5C4×D5semidirect
66. C2×C3×C5C2×Dic5semidirect
69. C3×D5C2×Dic5direct (N × H)
70. C52C22×C3×C4semidirect
80. C2×D5C3×Dic5direct (N × H)
86. Dic5C2×C3×D5direct (N × H)
89. C3×C5C2×C4×D5semidirect
90. C3×C5C22×Dic5semidirect
99. C2×C5C3×C4×D5semidirect
100. C2×C5C2×C3×Dic5semidirect
103. D5C2×C3×Dic5direct (N × H)
109. C2×C3D5×Dic5direct (N × H)
112. C5C2×C3×C4×D5semidirect
113. C5C22×C3×Dic5semidirect
119. C3C2×D5×Dic5direct (N × H)
121. C2C3×D5×Dic5direct (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×D52
C2×C3×#10392
C2×C3×C5×Dic52
C3×D5×Dic52
C2×D5×Dic53
C22×C3×Dic55
C2×C3×C4×D55