Finite Group Databasebeta

← C3×C5×#63120⋊C3 #26043563

Subgroups — C3×C5×#63120⋊C3 #26043563

Order 11700.

156 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 60 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
1C3×C5×D5×C13⋊C3C211yesmaximal, characteristic
2C3×C5×#63120C311yesmaximal, characteristic, complemented
3C5×#63120⋊C3C313yesmaximal, direct product factor
4C3×C52×C13⋊C3C411yescharacteristic, complemented, Hall
5C3×#63120⋊C3C511yesmaximal, characteristic, direct product factor
6C3×C5×Dic13⋊C3—51nomaximal
7C3×C5×D5×C13C2×C311yescharacteristic
8C5×D5×C13⋊C3C2×C313yes—
9C5×#63120C3211yescharacteristic, complemented, Hall
10C2×C3×C5×C13⋊C3—51no—
11C3×D5×C13⋊C3C2×C511yescharacteristic
12C3×C52×C13C3×C411yescharacteristic, complemented
13C52×C13⋊C3C3×C413yescomplemented
14C32×C5×AGL(1,𝔽5)—131nomaximal, Hall
15C3×#63120C3×C511yescharacteristic, complemented
16#63120⋊C3C3×C513yesdirect product factor
17C3×C5×Dic13—51no—
18C5×Dic13⋊C3—53no—
19C5×D5×C13C2×C3211yescharacteristic
20C3×C5×C13⋊C3—41no—
21C3×C5×C13⋊C3C4×C511yescharacteristic, complemented
22C3×C5×C13⋊C3AGL(1,𝔽5)11yescharacteristic, complemented
23C3×Dic13⋊C3—51noHall
24C32×C5×D5—131no—
25C2×C3×C5×C13—51no—
26C3×D5×C13C2×C3×C511yescharacteristic
27C2×C5×C13⋊C3—53no—
28D5×C13⋊C3C2×C3×C513yes—
29C52×C13C32×C411yescharacteristic, complemented, Hall
30C3×C5×AGL(1,𝔽5)—133no—
31C3×C5×AGL(1,𝔽5)—131no—
32#63120C32×C511yescharacteristic, complemented
33C5×Dic13—51no—
34C2×C3×C13⋊C3—51no—
35C32×C52—131noHall
36C3×C5×C13—41no—
37C3×C5×C13C3×C4×C511yescharacteristic, complemented
38C3×C5×C13C3×AGL(1,𝔽5)11yescharacteristic, complemented
39C5×C13⋊C3—43no—
40C5×C13⋊C3C3×C4×C513yescomplemented
41C5×C13⋊C3C3×AGL(1,𝔽5)13yescomplemented
42C32×AGL(1,𝔽5)—131no—
43C32×C4×C5—651no—
44C3×Dic13—51no—
45Dic13⋊C3—53no—
46C3×C5×D5—133no—
47C3×C5×D5D13⋊C311yescharacteristic
48C2×C5×C13—51no—
49D5×C13C2×C32×C511yescharacteristic
50C3×C13⋊C3C5×AGL(1,𝔽5)11yescharacteristic, complemented, Hall
51C5×AGL(1,𝔽5)—131noHall
52C2×C32×C5—651no—
53C32×D5—131no—
54C2×C3×C13—51no—
55C2×C13⋊C3—53no—
56C3×C52—133no—
57C3×C52Dic13⋊C311yescharacteristic, complemented
58C5×C13—41no—
59C5×C13C32×C4×C511yescharacteristic, complemented
60C5×C13C32×AGL(1,𝔽5)11yescharacteristic, complemented
61C3×AGL(1,𝔽5)—133no—
62C3×AGL(1,𝔽5)—131no—
63C3×C4×C5—651no—
64C3×C4×C5—653no—
65Dic13—51noHall
66C5×D5C3×D13⋊C311yescharacteristic
67C32×C5—131no—
68C32×C5—131no—
69C32×C5—521no—
70C3×C13C3×C5×AGL(1,𝔽5)11yescharacteristic, complemented
71C13⋊C3C3×C5×AGL(1,𝔽5)13yescomplemented
72C32×C4—651noHall
73C2×C3×C5—51no—
74C2×C3×C5—653no—
75C3×D5—133no—
76C3×D5C5×D13⋊C311yescharacteristic
77C2×C13—51no—
78C52C3×Dic13⋊C311yescharacteristic, complemented, Sylow
79AGL(1,𝔽5)—131no—
80C4×C5—651no—
81C2×C32—651no—
82C3×C5—41no—
83C3×C5—133no—
84C3×C5—133no—
85C3×C5—523no—
86C3×C5C5×Dic13⋊C311yescharacteristic, complemented
87C3×C5#63120⋊C311yescharacteristic, direct product factor
88C13C32×C5×AGL(1,𝔽5)11yescharacteristic, complemented, Sylow
89C3×C4—651no—
90C3×C4—653no—
91C2×C5—51no—
92D5C3×C5×D13⋊C311yescharacteristic
93C32—131noSylow
94C2×C3—51no—
95C2×C3—653no—
96C5—41no—
97C5C3×C5×Dic13⋊C311yescharacteristic, complemented
98C5C3×#63120⋊C311yescharacteristic, direct product factor
99C4—651noSylow
100C3—133no—
101C3C5×#63120⋊C311yescharacteristic, direct product factor
102C2—51no—
subgroupG/H
center, hypercenterC3×C5#63120⋊C3
commutator, hypocenterC5×C13C32×C4×C5
FrattiniC1G
Fitting, socleC3×C52×C13C3×C4
prime radicalC3×C5×D5×C13⋊C3C2
cosocleD5×C13C2×C32×C5
LCSG ▷ C5×C13
UCSC1 ◁ C3×C5

Complements

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

NH ≅ G/Ntype
2. C3×C5×#63120C3semidirect
3. C5×#63120⋊C3C3direct (N × H)
4. C3×C52×C13⋊C3C4semidirect
5. C3×#63120⋊C3C5direct (N × H)
9. C5×#63120C32semidirect
12. C3×C52×C13C3×C4semidirect
13. C52×C13⋊C3C3×C4semidirect
15. C3×#63120C3×C5semidirect
16. #63120⋊C3C3×C5direct (N × H)
21. C3×C5×C13⋊C3C4×C5semidirect
22. C3×C5×C13⋊C3AGL(1,𝔽5)semidirect
29. C52×C13C32×C4semidirect
32. #63120C32×C5semidirect
37. C3×C5×C13C3×C4×C5semidirect
38. C3×C5×C13C3×AGL(1,𝔽5)semidirect
40. C5×C13⋊C3C3×C4×C5semidirect
41. C5×C13⋊C3C3×AGL(1,𝔽5)semidirect
50. C3×C13⋊C3C5×AGL(1,𝔽5)semidirect
57. C3×C52Dic13⋊C3semidirect
59. C5×C13C32×C4×C5semidirect
60. C5×C13C32×AGL(1,𝔽5)semidirect
70. C3×C13C3×C5×AGL(1,𝔽5)semidirect
71. C13⋊C3C3×C5×AGL(1,𝔽5)semidirect
78. C52C3×Dic13⋊C3semidirect
86. C3×C5C5×Dic13⋊C3semidirect
87. C3×C5#63120⋊C3direct (N × H)
88. C13C32×C5×AGL(1,𝔽5)semidirect
97. C5C3×C5×Dic13⋊C3semidirect
98. C5C3×#63120⋊C3direct (N × H)
101. C3C5×#63120⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×D5×C13⋊C32
C3×C5×#631203
C5×#63120⋊C33
C3×#63120⋊C35
C3×C5×Dic13⋊C35
C32×C5×AGL(1,𝔽5)13