Finite Group Databasebeta

← C2×C3×C5×#11716322 #37561445

Subgroups — C2×C3×C5×#11716322 #37561445

Order 23940.

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

normal subgroups only

This group has 240 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×#87705C211yesmaximal, characteristic, complemented
2C3×C5×#11716322C212yesmaximal, direct product factor
3C2×C3×C5×C7×D19C311yesmaximal, characteristic, complemented
4C2×C5×#11716322C313yesmaximal, direct product factor
5C3×C5×#87705C2211yescharacteristic, complemented, Hall
6C2×C3×#11716322C511yesmaximal, characteristic, direct product factor, Hall
7C5×#11716322C2×C316yesdirect product factor
8C3×C5×C7×D19C2×C312yescomplemented
9C2×C5×#87705C2×C313yescomplemented
10C2×C3×C5×C7×C19C2×C311yescharacteristic, complemented
11C2×C3×C5×D19⋊C3—71nomaximal, Hall
12C2×C5×C7×D19C3211yescharacteristic, complemented, Hall
13C2×C3×#87705C2×C511yescharacteristic, complemented
14C3×#11716322C2×C512yesdirect product factor
15C3×C5×C7×C19C22×C311yescharacteristic, complemented
16C5×#87705C22×C313yescomplemented
17C2×C3×C5×C19⋊C3—71no—
18C3×C5×D19⋊C3—72no—
19C2×C3×C7×D19C3×C511yescharacteristic, complemented
20C2×#11716322C3×C513yesdirect product factor
21C2×C5×C7×C19C2×C3211yescharacteristic, complemented
22C5×C7×D19C2×C3212yescomplemented
23C22×C3×C5×ΓL(1,𝔽8)—191nomaximal, Hall
24C3×#87705C22×C511yescharacteristic, complemented, Hall
25C2×C3×C5×D19ΓL(1,𝔽8)11yescharacteristic, complemented
26C2×C5×D19⋊C3—73no—
27C3×C5×C19⋊C3—71noHall
28C2×C3×C7×C19C2×C3×C511yescharacteristic, complemented
29C3×C7×D19C2×C3×C512yescomplemented
30C2×#87705C2×C3×C513yescomplemented
31#11716322C2×C3×C516yesdirect product factor
32C2×C3×D19⋊C3—71noHall
33C5×C7×C19C22×C3211yescharacteristic, complemented, Hall
34C2×C3×C5×ΓL(1,𝔽8)—192no—
35C2×C3×C5×ΓL(1,𝔽8)—191no—
36C2×C3×C5×C19C2×ΓL(1,𝔽8)11yescharacteristic, complemented
37C3×C5×D19C2×ΓL(1,𝔽8)12yescomplemented
38C2×C5×C19⋊C3—73no—
39C5×D19⋊C3—76no—
40C2×C7×D19C32×C511yescharacteristic, complemented, Hall
41C22×C3×C5×C7—571no—
42C22×C5×ΓL(1,𝔽8)—193no—
43C3×C7×C19C22×C3×C511yescharacteristic, complemented
44#87705C22×C3×C513yescomplemented
45C2×C5×D19C3×ΓL(1,𝔽8)11yescharacteristic, complemented, Hall
46C2×C3×C19⋊C3—71no—
47C3×D19⋊C3—72no—
48C3×C5×ΓL(1,𝔽8)—191noHall
49C3×C5×C19C22×ΓL(1,𝔽8)11yescharacteristic, complemented
50C5×C19⋊C3—73no—
51C2×C7×C19C2×C32×C511yescharacteristic, complemented
52C7×D19C2×C32×C512yescomplemented
53C22×C3×ΓL(1,𝔽8)—191noHall
54C2×C3×D19C5×ΓL(1,𝔽8)11yescharacteristic, complemented
55C2×D19⋊C3—73no—
56C2×C3×C5×C7—572no—
57C2×C3×C5×C7D19⋊C311yescharacteristic, complemented
58C2×C5×ΓL(1,𝔽8)—196no—
59C2×C5×ΓL(1,𝔽8)—193no—
60C2×C5×C19C2×C3×ΓL(1,𝔽8)11yescharacteristic, complemented
61C5×D19C2×C3×ΓL(1,𝔽8)12yescomplemented
62C22×C32×C5—1331noHall
63C3×C19⋊C3—71noHall
64C22×C5×C7—571noHall
65C7×C19C22×C32×C511yescharacteristic, complemented, Hall
66C2×C3×ΓL(1,𝔽8)—192no—
67C2×C3×ΓL(1,𝔽8)—191no—
68C2×C3×C19C2×C5×ΓL(1,𝔽8)11yescharacteristic, complemented
69C3×D19C2×C5×ΓL(1,𝔽8)12yescomplemented
70C2×C19⋊C3—73no—
71D19⋊C3—76no—
72C3×C5×C7C2×D19⋊C311yescharacteristic, complemented
73C5×ΓL(1,𝔽8)—193no—
74C5×C19C22×C3×ΓL(1,𝔽8)11yescharacteristic, complemented, Hall
75C2×C32×C5—1332no—
76C2×C32×C5—1331no—
77C22×C3×C7—571no—
78C22×ΓL(1,𝔽8)—193no—
79C2×D19C3×C5×ΓL(1,𝔽8)11yescharacteristic, complemented, Hall
80C2×C5×C7—572no—
81C2×C5×C7C3×D19⋊C311yescharacteristic, complemented
82C3×ΓL(1,𝔽8)—191noHall
83C22×C3×C5—571no—
84C22×C3×C5—1333no—
85C3×C19C22×C5×ΓL(1,𝔽8)11yescharacteristic, complemented
86C19⋊C3—73no—
87C32×C5—1331noHall
88C2×C3×C7—572no—
89C2×C3×C7C5×D19⋊C311yescharacteristic, complemented
90C2×ΓL(1,𝔽8)—196no—
91C2×ΓL(1,𝔽8)—193no—
92C2×C19C2×C3×C5×ΓL(1,𝔽8)11yescharacteristic, complemented
93D19C2×C3×C5×ΓL(1,𝔽8)12yescomplemented
94C22×C32—1331noHall
95C5×C7C2×C3×D19⋊C311yescharacteristic, complemented, Hall
96C2×C3×C5—572no—
97C2×C3×C5—1336no—
98C2×C3×C5—1333no—
99C2×C3×C5#1171632211yescharacteristic, direct product factor
100C22×C7—571noHall
101C3×C7C2×C5×D19⋊C311yescharacteristic, complemented
102ΓL(1,𝔽8)—193no—
103C22×C5—571noHall
104C19C22×C3×C5×ΓL(1,𝔽8)11yescharacteristic, complemented, Sylow
105C2×C32—1332no—
106C2×C32—1331no—
107C3×C5—1333no—
108C3×C5C2×#1171632211yescharacteristic, direct product factor
109C2×C7—572no—
110C2×C7C3×C5×D19⋊C311yescharacteristic, complemented
111C22×C3—571no—
112C22×C3—1333no—
113C2×C5—572no—
114C2×C5C3×#1171632211yescharacteristic, direct product factor
115C32—1331noSylow
116C7C2×C3×C5×D19⋊C311yescharacteristic, complemented, Sylow
117C2×C3—572no—
118C2×C3—1336no—
119C2×C3—1333no—
120C2×C3C5×#1171632211yescharacteristic, direct product factor
121C5C2×C3×#1171632211yescharacteristic, direct product factor, Sylow
122C22—571noSylow
123C3—1333no—
124C3C2×C5×#1171632211yescharacteristic, direct product factor
125C2—572no—
126C2C3×C5×#1171632211yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3×C5#11716322
commutator, cosocle, hypocenterC7×C19C22×C32×C5
FrattiniC1G
Fitting, socleC2×C3×C5×C7×C19C2×C3
LCSG ▷ C7×C19
UCSC1 ◁ C2×C3×C5

Complements

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

NH ≅ G/Ntype
1. C2×C3×C5×#87705C2semidirect
2. C3×C5×#11716322C2direct (N × H)
3. C2×C3×C5×C7×D19C3semidirect
4. C2×C5×#11716322C3direct (N × H)
5. C3×C5×#87705C22semidirect
6. C2×C3×#11716322C5direct (N × H)
7. C5×#11716322C2×C3direct (N × H)
8. C3×C5×C7×D19C2×C3semidirect
9. C2×C5×#87705C2×C3semidirect
10. C2×C3×C5×C7×C19C2×C3semidirect
12. C2×C5×C7×D19C32semidirect
13. C2×C3×#87705C2×C5semidirect
14. C3×#11716322C2×C5direct (N × H)
15. C3×C5×C7×C19C22×C3semidirect
16. C5×#87705C22×C3semidirect
19. C2×C3×C7×D19C3×C5semidirect
20. C2×#11716322C3×C5direct (N × H)
21. C2×C5×C7×C19C2×C32semidirect
22. C5×C7×D19C2×C32semidirect
24. C3×#87705C22×C5semidirect
25. C2×C3×C5×D19ΓL(1,𝔽8)semidirect
28. C2×C3×C7×C19C2×C3×C5semidirect
29. C3×C7×D19C2×C3×C5semidirect
30. C2×#87705C2×C3×C5semidirect
31. #11716322C2×C3×C5direct (N × H)
33. C5×C7×C19C22×C32semidirect
36. C2×C3×C5×C19C2×ΓL(1,𝔽8)semidirect
37. C3×C5×D19C2×ΓL(1,𝔽8)semidirect
40. C2×C7×D19C32×C5semidirect
43. C3×C7×C19C22×C3×C5semidirect
44. #87705C22×C3×C5semidirect
45. C2×C5×D19C3×ΓL(1,𝔽8)semidirect
49. C3×C5×C19C22×ΓL(1,𝔽8)semidirect
51. C2×C7×C19C2×C32×C5semidirect
52. C7×D19C2×C32×C5semidirect
54. C2×C3×D19C5×ΓL(1,𝔽8)semidirect
57. C2×C3×C5×C7D19⋊C3semidirect
60. C2×C5×C19C2×C3×ΓL(1,𝔽8)semidirect
61. C5×D19C2×C3×ΓL(1,𝔽8)semidirect
65. C7×C19C22×C32×C5semidirect
68. C2×C3×C19C2×C5×ΓL(1,𝔽8)semidirect
69. C3×D19C2×C5×ΓL(1,𝔽8)semidirect
72. C3×C5×C7C2×D19⋊C3semidirect
74. C5×C19C22×C3×ΓL(1,𝔽8)semidirect
79. C2×D19C3×C5×ΓL(1,𝔽8)semidirect
81. C2×C5×C7C3×D19⋊C3semidirect
85. C3×C19C22×C5×ΓL(1,𝔽8)semidirect
89. C2×C3×C7C5×D19⋊C3semidirect
92. C2×C19C2×C3×C5×ΓL(1,𝔽8)semidirect
93. D19C2×C3×C5×ΓL(1,𝔽8)semidirect
95. C5×C7C2×C3×D19⋊C3semidirect
99. C2×C3×C5#11716322direct (N × H)
101. C3×C7C2×C5×D19⋊C3semidirect
104. C19C22×C3×C5×ΓL(1,𝔽8)semidirect
108. C3×C5C2×#11716322direct (N × H)
110. C2×C7C3×C5×D19⋊C3semidirect
114. C2×C5C3×#11716322direct (N × H)
116. C7C2×C3×C5×D19⋊C3semidirect
120. C2×C3C5×#11716322direct (N × H)
121. C5C2×C3×#11716322direct (N × H)
124. C3C2×C5×#11716322direct (N × H)
126. C2C3×C5×#11716322direct (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×#877052
C3×C5×#117163222
C2×C3×C5×C7×D193
C2×C5×#117163223
C2×C3×#117163225
C2×C3×C5×D19⋊C37
C22×C3×C5×ΓL(1,𝔽8)19