Finite Group Databasebeta

← C3×C5×#121 #91850

Subgroups — C3×C5×#121 #91850

Order 480.

188 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 96 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×D4C212yesmaximal, complemented
2C2×C3×C5×D4C211yesmaximal, characteristic, complemented
3C22×C3×C4×C5C211yesmaximal, characteristic, complemented
4C3×C5×C4⋊C4C211yesmaximal, characteristic, complemented
5C3×C5×C22⋊C4C212yesmaximal, complemented
6C5×#121C311yesmaximal, characteristic, direct product factor, Hall
7C23×C3×C5C2212yes—
8C23×C3×C5C2211yescharacteristic
9C3×C5×D4—24no—
10C3×C5×D4—22no—
11C2×C3×C4×C5—22no—
12C2×C3×C4×C5C2211yescharacteristic
13C2×C3×C4×C5C2212yes—
14C2×C3×C4×C5C2211yescharacteristic
15C3×#121C511yesmaximal, characteristic, direct product factor, Hall
16C2×C5×D4C2×C312yescomplemented
17C2×C5×D4C2×C311yescharacteristic, complemented
18C22×C4×C5C2×C311yescharacteristic, complemented
19C5×C4⋊C4C2×C311yescharacteristic, complemented
20C5×C22⋊C4C2×C312yescomplemented
21C22×C3×C5—21no—
22C22×C3×C5—21no—
23C22×C3×C5—22no—
24C22×C3×C5—22no—
25C22×C3×C5—22no—
26C22×C3×C5D412yescomplemented
27C22×C3×C5C2311yescharacteristic
28C3×C4×C5—21no—
29C3×C4×C5—22no—
30C3×C4×C5D412yescomplemented
31C2×C3×D4C2×C512yescomplemented
32C2×C3×D4C2×C511yescharacteristic, complemented
33C22×C3×C4C2×C511yescharacteristic, complemented
34C3×C4⋊C4C2×C511yescharacteristic, complemented
35C3×C22⋊C4C2×C512yescomplemented
36C23×C5C22×C312yes—
37C23×C5C22×C311yescharacteristic
38C5×D4—24no—
39C5×D4—22no—
40C2×C4×C5—22no—
41C2×C4×C5C22×C311yescharacteristic
42C2×C4×C5C22×C312yes—
43C2×C4×C5C22×C311yescharacteristic
44#121C3×C511yescharacteristic, direct product factor, Sylow
45C2×C3×C5—22no—
46C2×C3×C5—42no—
47C2×C3×C5C2×D411yescharacteristic
48C2×C3×C5C2×D411yescharacteristic
49C2×C3×C5#4111yescharacteristic
50C23×C3C22×C512yes—
51C23×C3C22×C511yescharacteristic
52C3×D4—24no—
53C3×D4—22no—
54C2×C3×C4—22no—
55C2×C3×C4C22×C511yescharacteristic
56C2×C3×C4C22×C512yes—
57C2×C3×C4C22×C511yescharacteristic
58C22×C5—21no—
59C22×C5—21no—
60C22×C5—22no—
61C22×C5—22no—
62C22×C5—22no—
63C22×C5C3×D412yescomplemented
64C22×C5C23×C311yescharacteristic
65C4×C5—21no—
66C4×C5—22no—
67C4×C5C3×D412yescomplemented
68C2×D4C2×C3×C512yescomplemented
69C2×D4C2×C3×C511yescharacteristic, complemented
70C22×C4C2×C3×C511yescharacteristic, complemented
71C4⋊C4C2×C3×C511yescharacteristic, complemented
72C22⋊C4C2×C3×C512yescomplemented
73C3×C5#12111yescharacteristic, direct product factor, Hall
74C22×C3—21no—
75C22×C3—21no—
76C22×C3—22no—
77C22×C3—22no—
78C22×C3—22no—
79C22×C3C5×D412yescomplemented
80C22×C3C23×C511yescharacteristic
81C3×C4—21no—
82C3×C4—22no—
83C3×C4C5×D412yescomplemented
84C2×C5—22no—
85C2×C5—42no—
86C2×C5C2×C3×D411yescharacteristic
87C2×C5C2×C3×D411yescharacteristic
88C2×C5C3×#4111yescharacteristic
89C23C22×C3×C512yes—
90C23C22×C3×C511yescharacteristic
91D4—24no—
92D4—22no—
93C2×C4—22no—
94C2×C4C22×C3×C511yescharacteristic
95C2×C4C22×C3×C512yes—
96C2×C4C22×C3×C511yescharacteristic
97C2×C3—22no—
98C2×C3—42no—
99C2×C3C2×C5×D411yescharacteristic
100C2×C3C2×C5×D411yescharacteristic
101C2×C3C5×#4111yescharacteristic
102C5C3×#12111yescharacteristic, direct product factor, Sylow
103C22—21no—
104C22—21no—
105C22—22no—
106C22—22no—
107C22—22no—
108C22C3×C5×D412yescomplemented
109C22C23×C3×C511yescharacteristic
110C4—21no—
111C4—22no—
112C4C3×C5×D412yescomplemented
113C3C5×#12111yescharacteristic, direct product factor, Sylow
114C2—22no—
115C2—42no—
116C2C2×C3×C5×D411yescharacteristic
117C2C2×C3×C5×D411yescharacteristic
118C2C3×C5×#4111yescharacteristic
subgroupG/H
center, socleC22×C3×C5C23
commutator, Frattini, cosocleC22C23×C3×C5
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C3×C5 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×C5×D4C2semidirect
2. C2×C3×C5×D4C2semidirect
3. C22×C3×C4×C5C2semidirect
4. C3×C5×C4⋊C4C2semidirect
5. C3×C5×C22⋊C4C2semidirect
6. C5×#121C3direct (N × H)
15. C3×#121C5direct (N × H)
16. C2×C5×D4C2×C3semidirect
17. C2×C5×D4C2×C3semidirect
18. C22×C4×C5C2×C3semidirect
19. C5×C4⋊C4C2×C3semidirect
20. C5×C22⋊C4C2×C3semidirect
26. C22×C3×C5D4semidirect
30. C3×C4×C5D4semidirect
31. C2×C3×D4C2×C5semidirect
32. C2×C3×D4C2×C5semidirect
33. C22×C3×C4C2×C5semidirect
34. C3×C4⋊C4C2×C5semidirect
35. C3×C22⋊C4C2×C5semidirect
44. #121C3×C5direct (N × H)
63. C22×C5C3×D4semidirect
67. C4×C5C3×D4semidirect
68. C2×D4C2×C3×C5semidirect
69. C2×D4C2×C3×C5semidirect
70. C22×C4C2×C3×C5semidirect
71. C4⋊C4C2×C3×C5semidirect
72. C22⋊C4C2×C3×C5semidirect
73. C3×C5#121direct (N × H)
79. C22×C3C5×D4semidirect
83. C3×C4C5×D4semidirect
102. C5C3×#121direct (N × H)
108. C22C3×C5×D4semidirect
112. C4C3×C5×D4semidirect
113. C3C5×#121direct (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×D42
C22×C3×C4×C52
C3×C5×C4⋊C42
C3×C5×C22⋊C42
C5×#1213
C3×#1215