Finite Group Databasebeta

← C3×C5×D5×#3861 #36347973

Subgroups — C3×C5×D5×#3861 #36347973

Order 22050.

600 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 144 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×C52×#3861C211yesmaximal, characteristic, complemented, Hall
2C3×C5×C72×D5C311yesmaximal, characteristic, complemented
3C5×D5×#3861C313yesmaximal, direct product factor
4C2×C3×C5×#3861—51nomaximal
5C3×D5×#3861C511yesmaximal, characteristic, direct product factor
6C52×#3861C2×C313yescomplemented
7C3×C52×C72C2×C311yescharacteristic, complemented
8C3×C5×D5×ΓL(1,𝔽8)—78nomaximal
9C5×C72×D5C3211yescharacteristic, complemented, Hall
10C3×C5×#3861—22no—
11C3×C5×#3861D511yescharacteristic, direct product factor
12C3×C5×#3861C2×C511yescharacteristic, complemented
13C3×C52×ΓL(1,𝔽8)—78no—
14C2×C3×C5×C72—51no—
15C3×C72×D5C3×C511yescharacteristic, complemented
16C2×C5×#3861—53no—
17D5×#3861C3×C513yesdirect product factor
18C52×C72C2×C3211yescharacteristic, complemented, Hall
19C3×C5×C7×D5ΓL(1,𝔽8)18yescomplemented
20C5×D5×ΓL(1,𝔽8)—724no—
21C2×C3×#3861—51noHall
22C3×C5×C72—22no—
23C3×C5×C72C3×D511yescharacteristic, complemented
24C3×C5×C72C2×C3×C511yescharacteristic, complemented
25C5×#3861—26no—
26C5×#3861C3×D513yesdirect product factor
27C5×#3861C2×C3×C513yescomplemented
28C2×C3×C5×ΓL(1,𝔽8)—358no—
29C3×D5×ΓL(1,𝔽8)—78no—
30C3×C52×C7C2×ΓL(1,𝔽8)18yescomplemented
31C52×ΓL(1,𝔽8)—724no—
32C2×C5×C72—51no—
33C72×D5C32×C511yescharacteristic, complemented
34C32×C5×D5—491noHall
35C3×#3861C5×D511yescharacteristic, direct product factor, Hall
36C5×C7×D5C3×ΓL(1,𝔽8)18yescomplemented
37C3×C5×ΓL(1,𝔽8)—78no—
38C3×C5×ΓL(1,𝔽8)—78no—
39C3×C5×ΓL(1,𝔽8)—1416no—
40C2×C3×C72—51no—
41C2×#3861—53no—
42C5×C72—22no—
43C5×C72C32×D511yescharacteristic, complemented
44C5×C72C2×C32×C511yescharacteristic, complemented
45C32×C52—491noHall
46C2×C3×C5×C7—58no—
47C3×C7×D5C5×ΓL(1,𝔽8)18yescomplemented
48C2×C5×ΓL(1,𝔽8)—3524no—
49D5×ΓL(1,𝔽8)—724no—
50C52×C7C2×C3×ΓL(1,𝔽8)18yescomplemented
51C3×C5×D5—493no—
52C3×C5×D5#386111yescharacteristic, direct product factor
53C3×C72C3×C5×D511yescharacteristic, complemented
54#3861C3×C5×D513yesdirect product factor
55C2×C3×ΓL(1,𝔽8)—358no—
56C3×C5×C7—216no—
57C3×C5×C7D5×ΓL(1,𝔽8)18yescomplemented
58C3×C5×C7C2×C5×ΓL(1,𝔽8)18yescomplemented
59C5×ΓL(1,𝔽8)—724no—
60C5×ΓL(1,𝔽8)—724no—
61C5×ΓL(1,𝔽8)—1448no—
62C2×C72—51noHall
63C2×C32×C5—2451no—
64C32×D5—491no—
65C3×C52—493no—
66C3×C52C2×#386111yescharacteristic, complemented
67C2×C5×C7—58no—
68C7×D5C3×C5×ΓL(1,𝔽8)18yescomplemented
69C3×ΓL(1,𝔽8)—78no—
70C5×D5C3×#386111yescharacteristic, direct product factor, Hall
71C72C32×C5×D511yescharacteristic, complemented, Sylow
72C32×C5—491no—
73C32×C5—491no—
74C32×C5—982no—
75C2×C3×C7—58no—
76C2×ΓL(1,𝔽8)—3524no—
77C5×C7—216no—
78C5×C7C3×D5×ΓL(1,𝔽8)18yescomplemented
79C5×C7C2×C3×C5×ΓL(1,𝔽8)18yescomplemented
80C2×C3×C5—51no—
81C2×C3×C5—2453no—
82C3×D5—493no—
83C3×D5C5×#386111yescharacteristic, direct product factor
84C52C2×C3×#386111yescharacteristic, complemented, Sylow
85C3×C7C5×D5×ΓL(1,𝔽8)18yescomplemented
86ΓL(1,𝔽8)—724no—
87C2×C32—2451noHall
88C3×C5—22no—
89C3×C5—493no—
90C3×C5—493no—
91C3×C5—986no—
92C3×C5D5×#386111yescharacteristic, direct product factor
93C3×C5C2×C5×#386111yescharacteristic, complemented
94C2×C7—58no—
95C2×C5—51no—
96D5C3×C5×#386111yescharacteristic, direct product factor
97C32—491noSylow
98C7C3×C5×D5×ΓL(1,𝔽8)18yescomplemented
99C2×C3—51no—
100C2×C3—2453no—
101C5—22no—
102C5C3×D5×#386111yescharacteristic, direct product factor
103C5C2×C3×C5×#386111yescharacteristic, complemented
104C3—493no—
105C3C5×D5×#386111yescharacteristic, direct product factor
106C2—51noSylow
subgroupG/H
center, hypercenterC3×C5D5×#3861
commutator, cosocle, hypocenterC5×C72C2×C32×C5
FrattiniC1G
Fitting, socleC3×C52×C72C2×C3
LCSG ▷ C5×C72
UCSC1 ◁ C3×C5

Complements

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

NH ≅ G/Ntype
1. C3×C52×#3861C2semidirect
2. C3×C5×C72×D5C3semidirect
3. C5×D5×#3861C3direct (N × H)
5. C3×D5×#3861C5direct (N × H)
6. C52×#3861C2×C3semidirect
7. C3×C52×C72C2×C3semidirect
9. C5×C72×D5C32semidirect
11. C3×C5×#3861D5direct (N × H)
12. C3×C5×#3861C2×C5semidirect
15. C3×C72×D5C3×C5semidirect
17. D5×#3861C3×C5direct (N × H)
18. C52×C72C2×C32semidirect
19. C3×C5×C7×D5ΓL(1,𝔽8)semidirect
23. C3×C5×C72C3×D5semidirect
24. C3×C5×C72C2×C3×C5semidirect
26. C5×#3861C3×D5direct (N × H)
27. C5×#3861C2×C3×C5semidirect
30. C3×C52×C7C2×ΓL(1,𝔽8)semidirect
33. C72×D5C32×C5semidirect
35. C3×#3861C5×D5direct (N × H)
36. C5×C7×D5C3×ΓL(1,𝔽8)semidirect
43. C5×C72C32×D5semidirect
44. C5×C72C2×C32×C5semidirect
47. C3×C7×D5C5×ΓL(1,𝔽8)semidirect
50. C52×C7C2×C3×ΓL(1,𝔽8)semidirect
52. C3×C5×D5#3861direct (N × H)
53. C3×C72C3×C5×D5semidirect
54. #3861C3×C5×D5direct (N × H)
57. C3×C5×C7D5×ΓL(1,𝔽8)semidirect
58. C3×C5×C7C2×C5×ΓL(1,𝔽8)semidirect
66. C3×C52C2×#3861semidirect
68. C7×D5C3×C5×ΓL(1,𝔽8)semidirect
70. C5×D5C3×#3861direct (N × H)
71. C72C32×C5×D5semidirect
78. C5×C7C3×D5×ΓL(1,𝔽8)semidirect
79. C5×C7C2×C3×C5×ΓL(1,𝔽8)semidirect
83. C3×D5C5×#3861direct (N × H)
84. C52C2×C3×#3861semidirect
85. C3×C7C5×D5×ΓL(1,𝔽8)semidirect
92. C3×C5D5×#3861direct (N × H)
93. C3×C5C2×C5×#3861semidirect
96. D5C3×C5×#3861direct (N × H)
98. C7C3×C5×D5×ΓL(1,𝔽8)semidirect
102. C5C3×D5×#3861direct (N × H)
103. C5C2×C3×C5×#3861semidirect
105. C3C5×D5×#3861direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C52×#38612
C3×C5×C72×D53
C5×D5×#38613
C2×C3×C5×#38615
C3×D5×#38615
C3×C5×D5×ΓL(1,𝔽8)7