Finite Group Databasebeta

← C3×C5×#237 #10624255

Subgroups — C3×C5×#237 #10624255

Order 720.

176 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 92 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
1C32×C5×D4C211yesmaximal, characteristic, complemented
2C3×C5×#67C212yesmaximal, complemented
3C2×C3×C5×Dic3C212yesmaximal, complemented
4C3×C4×C5×D3C211yesmaximal, characteristic, complemented
5C3×C5×Dic6C211yesmaximal, characteristic, complemented
6C3×C5×#41—31nomaximal
7C5×#237C311yesmaximal, characteristic, direct product factor
8C22×C32×C5C2212yes—
9C2×C3×C5×D3C2211yescharacteristic
10C32×C4×C5C2211yescharacteristic
11C3×C5×Dic3C2212yes—
12C3×C5×Dic3C2211yescharacteristic
13C3×#237C511yesmaximal, characteristic, direct product factor, Hall
14C3×C5×Q8—31no—
15C3×C5×D4—21no—
16C3×C5×D4—32no—
17C3×C5×D4D311yescharacteristic, complemented
18C3×C5×D4C2×C311yescharacteristic, complemented
19C2×C3×C4×C5—32no—
20C2×C3×C4×C5—31no—
21C5×#67C2×C312yescomplemented
22C2×C5×Dic3C2×C312yescomplemented
23C4×C5×D3C2×C311yescharacteristic, complemented
24C5×Dic6C2×C311yescharacteristic, complemented
25C2×C32×C5—22no—
26C2×C32×C5C2311yescharacteristic
27C3×C5×D3—21no—
28C5×#41—31noHall
29C32×D4C2×C511yescharacteristic, complemented
30C3×#67C2×C512yescomplemented
31C2×C3×Dic3C2×C512yescomplemented
32C3×C4×D3C2×C511yescharacteristic, complemented
33C3×Dic6C2×C511yescharacteristic, complemented
34C22×C3×C5—22no—
35C22×C3×C5—31no—
36C22×C3×C5C2×D312yes—
37C22×C3×C5C22×C312yes—
38C2×C5×D3C22×C311yescharacteristic
39C3×C4×C5—21no—
40C3×C4×C5—31no—
41C3×C4×C5—32no—
42C3×C4×C5C2×D311yescharacteristic
43C3×C4×C5C22×C311yescharacteristic
44C5×Dic3C22×C312yes—
45C5×Dic3C22×C311yescharacteristic
46C3×#41—31no—
47#237C3×C511yescharacteristic, direct product factor
48C32×C5#4111yescharacteristic, complemented, Hall
49C5×Q8—31no—
50C5×D4—32no—
51C5×D4C3×D311yescharacteristic, complemented
52C2×C4×C5—32no—
53C2×C4×C5—31no—
54C22×C32C22×C512yes—
55C2×C3×D3C22×C511yescharacteristic
56C32×C4C22×C511yescharacteristic
57C3×Dic3C22×C512yes—
58C3×Dic3C22×C511yescharacteristic
59C2×C3×C5—22no—
60C2×C3×C5—22no—
61C2×C3×C5—21no—
62C2×C3×C5—42no—
63C2×C3×C5—61no—
64C2×C3×C5C22×D311yescharacteristic
65C2×C3×C5C23×C311yescharacteristic
66C5×D3—21no—
67C3×Q8—31no—
68C3×D4—21no—
69C3×D4—32no—
70C3×D4C5×D311yescharacteristic, complemented
71C3×D4C2×C3×C511yescharacteristic, complemented
72C2×C3×C4—32no—
73C2×C3×C4—31no—
74#67C2×C3×C512yescomplemented
75C2×Dic3C2×C3×C512yescomplemented
76C4×D3C2×C3×C511yescharacteristic, complemented
77Dic6C2×C3×C511yescharacteristic, complemented
78C22×C5—31no—
79C22×C5C2×C3×D312yes—
80C4×C5—31no—
81C4×C5—32no—
82C4×C5C2×C3×D311yescharacteristic
83C2×C32—22no—
84C2×C32C23×C511yescharacteristic
85C3×D3—21no—
86#41—31noSylow
87C3×C5—21no—
88C3×C5#23711yescharacteristic, direct product factor
89C3×C5C3×#4111yescharacteristic, complemented
90C22×C3—22no—
91C22×C3—31no—
92C22×C3C2×C5×D312yes—
93C22×C3C22×C3×C512yes—
94C2×D3C22×C3×C511yescharacteristic
95C3×C4—21no—
96C3×C4—31no—
97C3×C4—32no—
98C3×C4C2×C5×D311yescharacteristic
99C3×C4C22×C3×C511yescharacteristic
100Dic3C22×C3×C512yes—
101Dic3C22×C3×C511yescharacteristic
102C2×C5—22no—
103C2×C5—61no—
104C2×C5C22×C3×D311yescharacteristic
105C32C5×#4111yescharacteristic, complemented, Sylow
106Q8—31no—
107D4—32no—
108D4C3×C5×D311yescharacteristic, complemented
109C2×C4—32no—
110C2×C4—31no—
111C2×C3—22no—
112C2×C3—22no—
113C2×C3—21no—
114C2×C3—42no—
115C2×C3—61no—
116C2×C3C22×C5×D311yescharacteristic
117C2×C3C23×C3×C511yescharacteristic
118D3—21no—
119C5C3×#23711yescharacteristic, direct product factor, Sylow
120C22—31no—
121C22C2×C3×C5×D312yes—
122C4—31no—
123C4—32no—
124C4C2×C3×C5×D311yescharacteristic
125C3—21no—
126C3C5×#23711yescharacteristic, direct product factor
127C3C3×C5×#4111yescharacteristic, complemented
128C2—22no—
129C2—61no—
130C2C22×C3×C5×D311yescharacteristic
subgroupG/H
centerC2×C3×C5C22×D3
commutator, cosocleC2×C3C23×C3×C5
FrattiniC2C22×C3×C5×D3
FittingC32×C5×D4C2
socleC2×C32×C5C23
hypercenterC3×C5×D4D3
hypocenterC3C3×C5×#41
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3×C5 ◁ C3×C5×D4

Complements

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

NH ≅ G/Ntype
1. C32×C5×D4C2semidirect
2. C3×C5×#67C2semidirect
3. C2×C3×C5×Dic3C2semidirect
4. C3×C4×C5×D3C2semidirect
5. C3×C5×Dic6C2semidirect
7. C5×#237C3direct (N × H)
13. C3×#237C5direct (N × H)
17. C3×C5×D4D3semidirect
18. C3×C5×D4C2×C3semidirect
21. C5×#67C2×C3semidirect
22. C2×C5×Dic3C2×C3semidirect
23. C4×C5×D3C2×C3semidirect
24. C5×Dic6C2×C3semidirect
29. C32×D4C2×C5semidirect
30. C3×#67C2×C5semidirect
31. C2×C3×Dic3C2×C5semidirect
32. C3×C4×D3C2×C5semidirect
33. C3×Dic6C2×C5semidirect
47. #237C3×C5direct (N × H)
48. C32×C5#41semidirect
51. C5×D4C3×D3semidirect
70. C3×D4C5×D3semidirect
71. C3×D4C2×C3×C5semidirect
74. #67C2×C3×C5semidirect
75. C2×Dic3C2×C3×C5semidirect
76. C4×D3C2×C3×C5semidirect
77. Dic6C2×C3×C5semidirect
88. C3×C5#237direct (N × H)
89. C3×C5C3×#41semidirect
105. C32C5×#41semidirect
108. D4C3×C5×D3semidirect
119. C5C3×#237direct (N × H)
126. C3C5×#237direct (N × H)
127. C3C3×C5×#41semidirect

Maximal subgroups

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

subgroupindex
C32×C5×D42
C3×C5×#672
C2×C3×C5×Dic32
C3×C4×C5×D32
C3×C5×Dic62
C3×C5×#413
C5×#2373
C3×#2375