Finite Group Databasebeta

← C5×#87705⋊A4 #37561432

Subgroups — C5×#87705⋊A4 #37561432

Order 23940.

112 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 32 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×C5×#87705C311yesmaximal, characteristic, complemented
2C5×C7×C22⋊(C19⋊C3)C311yesmaximal, characteristic, complemented
3C5×C19×C22⋊ΓL(1,𝔽8)C311yesmaximal, characteristic, complemented
4C5×C22⋊#87704C311yesmaximal, characteristic, complemented
5C5×ΓL(1,𝔽8)×C19⋊C3—41nomaximal, Hall
6#87705⋊A4C511yesmaximal, characteristic, direct product factor, Hall
7C2×C5×#87705—31no—
8C5×A4×C19⋊C3—71nomaximal, Hall
9C22×C5×C7×C19C3211yescharacteristic, complemented, Hall
10C5×#87704—41no—
11C5×#87705A411yescharacteristic, complemented
12C5×C19×ΓL(1,𝔽8)—41no—
13C5×C7×C19⋊C3—41no—
14C22⋊#87704C3×C511yescharacteristic, complemented
15C19×C22⋊ΓL(1,𝔽8)C3×C511yescharacteristic, complemented
16C7×C22⋊(C19⋊C3)C3×C511yescharacteristic, complemented
17C22×#87705C3×C511yescharacteristic, complemented
18C2×C5×C7×C19—31no—
19C5×A4×ΓL(1,𝔽8)—191nomaximal, Hall
20ΓL(1,𝔽8)×C19⋊C3—41noHall
21C5×C22⋊(C19⋊C3)—71no—
22C5×C22⋊(C19⋊C3)ΓL(1,𝔽8)11yescharacteristic, complemented
23C5×A4×C19—71no—
24C22×C5×C19⋊C3—71no—
25C3×C5×C19⋊C3—281noHall
26C2×#87705—31no—
27A4×C19⋊C3—71noHall
28C5×C7×C19C3×A411yescharacteristic, complemented, Hall
29C2×C5×C19⋊C3—211no—
30C22×C7×C19C32×C511yescharacteristic, complemented, Hall
31C5×C22⋊ΓL(1,𝔽8)—191no—
32C5×C22⋊ΓL(1,𝔽8)C19⋊C311yescharacteristic, complemented
33C5×C7×A4—191no—
34C22×C5×ΓL(1,𝔽8)—191no—
35#87705C5×A411yescharacteristic, complemented
36#87704—41no—
37C7×C19⋊C3—41no—
38C19×ΓL(1,𝔽8)—41no—
39C22×C5×C19C3×ΓL(1,𝔽8)11yescharacteristic, complemented, Hall
40C3×C5×ΓL(1,𝔽8)—761noHall
41C3×C5×C19—281no—
42C5×C19⋊C3—41no—
43C5×C19⋊C3—71no—
44C5×C19⋊C3—281no—
45C2×C7×C19—31no—
46A4×ΓL(1,𝔽8)—191noHall
47C22⋊(C19⋊C3)—71no—
48C22⋊(C19⋊C3)C5×ΓL(1,𝔽8)11yescharacteristic, complemented
49A4×C19—71no—
50C22×C19⋊C3—71no—
51C2×C5×ΓL(1,𝔽8)—571no—
52C2×C5×C19—31no—
53C3×C5×A4—1331noHall
54C3×C19⋊C3—281noHall
55C22×C5×C7C3×C19⋊C311yescharacteristic, complemented, Hall
56C7×C19C3×C5×A411yescharacteristic, complemented, Hall
57C2×C19⋊C3—211no—
58C3×C5×C7—761no—
59C5×ΓL(1,𝔽8)—41no—
60C5×ΓL(1,𝔽8)—191no—
61C5×ΓL(1,𝔽8)—761no—
62C5×C19A4×ΓL(1,𝔽8)11yescharacteristic, complemented, Hall
63C22⋊ΓL(1,𝔽8)—191no—
64C22⋊ΓL(1,𝔽8)C5×C19⋊C311yescharacteristic, complemented
65C7×A4—191no—
66C22×ΓL(1,𝔽8)—191no—
67C22×C19C3×C5×ΓL(1,𝔽8)11yescharacteristic, complemented, Hall
68C2×C5×C7—31no—
69C3×ΓL(1,𝔽8)—761noHall
70C22×C3×C5—1331no—
71C5×A4—71no—
72C5×A4—191no—
73C5×A4—1331no—
74C3×C19—281no—
75C19⋊C3—41no—
76C19⋊C3—71no—
77C19⋊C3—281no—
78C32×C5—5321noHall
79C2×ΓL(1,𝔽8)—571no—
80C2×C19—31no—
81C3×A4—1331noHall
82C5×C7A4×C19⋊C311yescharacteristic, complemented, Hall
83C2×C3×C5—3991no—
84C22×C7C3×C5×C19⋊C311yescharacteristic, complemented, Hall
85C3×C7—761no—
86ΓL(1,𝔽8)—41no—
87ΓL(1,𝔽8)—191no—
88ΓL(1,𝔽8)—761no—
89C22×C5ΓL(1,𝔽8)×C19⋊C311yescharacteristic, complemented, Hall
90C19C5×A4×ΓL(1,𝔽8)11yescharacteristic, complemented, Sylow
91C3×C5—5321no—
92C3×C5—281no—
93C3×C5—761no—
94C3×C5—1331no—
95C2×C7—31no—
96C22×C3—1331no—
97A4—71no—
98A4—191no—
99A4—1331no—
100C2×C5—31no—
101C32—5321noSylow
102C7C5×A4×C19⋊C311yescharacteristic, complemented, Sylow
103C2×C3—3991no—
104C5#87705⋊A411yescharacteristic, direct product factor, Sylow
105C22C5×ΓL(1,𝔽8)×C19⋊C311yescharacteristic, complemented, Sylow
106C3—5321no—
107C3—281no—
108C3—761no—
109C3—1331no—
110C2—31no—
subgroupG/H
center, hypercenterC5#87705⋊A4
commutator, cosocle, hypocenterC22×C7×C19C32×C5
FrattiniC1G
Fitting, socleC22×C5×C7×C19C32
LCSG ▷ C22×C7×C19
UCSC1 ◁ C5

Complements

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

NH ≅ G/Ntype
1. C22×C5×#87705C3semidirect
2. C5×C7×C22⋊(C19⋊C3)C3semidirect
3. C5×C19×C22⋊ΓL(1,𝔽8)C3semidirect
4. C5×C22⋊#87704C3semidirect
6. #87705⋊A4C5direct (N × H)
9. C22×C5×C7×C19C32semidirect
11. C5×#87705A4semidirect
14. C22⋊#87704C3×C5semidirect
15. C19×C22⋊ΓL(1,𝔽8)C3×C5semidirect
16. C7×C22⋊(C19⋊C3)C3×C5semidirect
17. C22×#87705C3×C5semidirect
22. C5×C22⋊(C19⋊C3)ΓL(1,𝔽8)semidirect
28. C5×C7×C19C3×A4semidirect
30. C22×C7×C19C32×C5semidirect
32. C5×C22⋊ΓL(1,𝔽8)C19⋊C3semidirect
35. #87705C5×A4semidirect
39. C22×C5×C19C3×ΓL(1,𝔽8)semidirect
48. C22⋊(C19⋊C3)C5×ΓL(1,𝔽8)semidirect
55. C22×C5×C7C3×C19⋊C3semidirect
56. C7×C19C3×C5×A4semidirect
62. C5×C19A4×ΓL(1,𝔽8)semidirect
64. C22⋊ΓL(1,𝔽8)C5×C19⋊C3semidirect
67. C22×C19C3×C5×ΓL(1,𝔽8)semidirect
82. C5×C7A4×C19⋊C3semidirect
84. C22×C7C3×C5×C19⋊C3semidirect
89. C22×C5ΓL(1,𝔽8)×C19⋊C3semidirect
90. C19C5×A4×ΓL(1,𝔽8)semidirect
102. C7C5×A4×C19⋊C3semidirect
104. C5#87705⋊A4direct (N × H)
105. C22C5×ΓL(1,𝔽8)×C19⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×C5×#877053
C5×C7×C22⋊(C19⋊C3)3
C5×C19×C22⋊ΓL(1,𝔽8)3
C5×C22⋊#877043
C5×ΓL(1,𝔽8)×C19⋊C34
#87705⋊A45
C5×A4×C19⋊C37
C5×A4×ΓL(1,𝔽8)19