Finite Group Databasebeta

← C2×C3×C5×#11716325 #37561441

Subgroups — C2×C3×C5×#11716325 #37561441

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×#87704C211yesmaximal, characteristic, complemented
2C3×C5×#11716325C212yesmaximal, direct product factor
3C2×C3×C5×D7×C19C311yesmaximal, characteristic, complemented
4C2×C5×#11716325C313yesmaximal, direct product factor
5C3×C5×#87704C2211yescharacteristic, complemented, Hall
6C2×C3×#11716325C511yesmaximal, characteristic, direct product factor, Hall
7C5×#11716325C2×C316yesdirect product factor
8C3×C5×D7×C19C2×C312yescomplemented
9C2×C5×#87704C2×C313yescomplemented
10C2×C3×C5×C7×C19C2×C311yescharacteristic, complemented
11C22×C3×C5×C19⋊C3—71nomaximal, Hall
12C2×C5×D7×C19C3211yescharacteristic, complemented, Hall
13C2×C3×#87704C2×C511yescharacteristic, complemented
14C3×#11716325C2×C512yesdirect product factor
15C3×C5×C7×C19C22×C311yescharacteristic, complemented
16C5×#87704C22×C313yescomplemented
17C2×C3×C5×C19⋊C3—72no—
18C2×C3×C5×C19⋊C3—71no—
19C2×C3×D7×C19C3×C511yescharacteristic, complemented
20C2×#11716325C3×C513yesdirect product factor
21C2×C5×C7×C19C2×C3211yescharacteristic, complemented
22C5×D7×C19C2×C3212yescomplemented
23C2×C3×C5×AGL(1,𝔽7)—191nomaximal, Hall
24C3×#87704C22×C511yescharacteristic, complemented, Hall
25C22×C3×C5×C19—71no—
26C22×C5×C19⋊C3—73no—
27C3×C5×C19⋊C3—71noHall
28C2×C3×C7×C19C2×C3×C511yescharacteristic, complemented
29C3×D7×C19C2×C3×C512yescomplemented
30C2×#87704C2×C3×C513yescomplemented
31#11716325C2×C3×C516yesdirect product factor
32C22×C3×C19⋊C3—71noHall
33C5×C7×C19C22×C3211yescharacteristic, complemented, Hall
34C2×C3×C5×ΓL(1,𝔽8)—191no—
35C3×C5×AGL(1,𝔽7)—192no—
36C2×C3×C5×C19—212no—
37C2×C3×C5×C19AGL(1,𝔽7)11yescharacteristic, complemented
38C2×C5×C19⋊C3—76no—
39C2×C5×C19⋊C3—73no—
40C2×D7×C19C32×C511yescharacteristic, complemented, Hall
41C2×C3×C5×D7C19⋊C311yescharacteristic, complemented
42C2×C5×AGL(1,𝔽7)—193no—
43C3×C7×C19C22×C3×C511yescharacteristic, complemented
44#87704C22×C3×C513yescomplemented
45C22×C5×C19—71noHall
46C2×C3×C19⋊C3—72no—
47C2×C3×C19⋊C3—71no—
48C3×C5×ΓL(1,𝔽8)—191noHall
49C3×C5×C19C2×AGL(1,𝔽7)11yescharacteristic, complemented
50C5×C19⋊C3—73no—
51C2×C7×C19C2×C32×C511yescharacteristic, complemented
52D7×C19C2×C32×C512yescomplemented
53C2×C3×AGL(1,𝔽7)—191noHall
54C22×C3×C19—211no—
55C22×C19⋊C3—73no—
56C2×C3×C5×C7C2×C19⋊C311yescharacteristic, complemented
57C3×C5×D7C2×C19⋊C312yescomplemented
58C2×C5×ΓL(1,𝔽8)—193no—
59C5×AGL(1,𝔽7)—196no—
60C2×C5×C19—212no—
61C2×C5×C19C3×AGL(1,𝔽7)11yescharacteristic, complemented
62C22×C32×C5—1331noHall
63C3×C19⋊C3—71noHall
64C2×C5×D7C3×C19⋊C311yescharacteristic, complemented, Hall
65C7×C19C22×C32×C511yescharacteristic, complemented, Hall
66C2×C3×ΓL(1,𝔽8)—191no—
67C3×AGL(1,𝔽7)—192no—
68C2×C3×C19—212no—
69C2×C3×C19C5×AGL(1,𝔽7)11yescharacteristic, complemented
70C2×C19⋊C3—76no—
71C2×C19⋊C3—73no—
72C3×C5×C7C22×C19⋊C311yescharacteristic, complemented
73C5×ΓL(1,𝔽8)—193no—
74C5×C19C2×C3×AGL(1,𝔽7)11yescharacteristic, complemented, Hall
75C2×C32×C5—1332no—
76C2×C32×C5—1331no—
77C2×C3×D7C5×C19⋊C311yescharacteristic, complemented
78C2×AGL(1,𝔽7)—193no—
79C22×C19—211noHall
80C2×C5×C7C2×C3×C19⋊C311yescharacteristic, complemented
81C5×D7C2×C3×C19⋊C312yescomplemented
82C3×ΓL(1,𝔽8)—191noHall
83C22×C3×C5—71no—
84C22×C3×C5—1333no—
85C3×C19C2×C5×AGL(1,𝔽7)11yescharacteristic, complemented
86C19⋊C3—73no—
87C32×C5—1331noHall
88C2×C3×C7C2×C5×C19⋊C311yescharacteristic, complemented
89C3×D7C2×C5×C19⋊C312yescomplemented
90C2×ΓL(1,𝔽8)—193no—
91AGL(1,𝔽7)—196no—
92C2×C19—212no—
93C2×C19C3×C5×AGL(1,𝔽7)11yescharacteristic, complemented
94C22×C32—1331noHall
95C5×C7C22×C3×C19⋊C311yescharacteristic, complemented, Hall
96C2×C3×C5—212no—
97C2×C3×C5—1336no—
98C2×C3×C5—1333no—
99C2×C3×C5#1171632511yescharacteristic, direct product factor
100C2×D7C3×C5×C19⋊C311yescharacteristic, complemented, Hall
101C3×C7C22×C5×C19⋊C311yescharacteristic, complemented
102ΓL(1,𝔽8)—193no—
103C22×C5—71noHall
104C19C2×C3×C5×AGL(1,𝔽7)11yescharacteristic, complemented, Sylow
105C2×C32—1332no—
106C2×C32—1331no—
107C3×C5—1333no—
108C3×C5C2×#1171632511yescharacteristic, direct product factor
109C2×C7C2×C3×C5×C19⋊C311yescharacteristic, complemented
110D7C2×C3×C5×C19⋊C312yescomplemented
111C22×C3—211no—
112C22×C3—1333no—
113C2×C5—212no—
114C2×C5C3×#1171632511yescharacteristic, direct product factor
115C32—1331noSylow
116C7C22×C3×C5×C19⋊C311yescharacteristic, complemented, Sylow
117C2×C3—212no—
118C2×C3—1336no—
119C2×C3—1333no—
120C2×C3C5×#1171632511yescharacteristic, direct product factor
121C5C2×C3×#1171632511yescharacteristic, direct product factor, Sylow
122C22—211noSylow
123C3—1333no—
124C3C2×C5×#1171632511yescharacteristic, direct product factor
125C2—212no—
126C2C3×C5×#1171632511yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3×C5#11716325
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×#87704C2semidirect
2. C3×C5×#11716325C2direct (N × H)
3. C2×C3×C5×D7×C19C3semidirect
4. C2×C5×#11716325C3direct (N × H)
5. C3×C5×#87704C22semidirect
6. C2×C3×#11716325C5direct (N × H)
7. C5×#11716325C2×C3direct (N × H)
8. C3×C5×D7×C19C2×C3semidirect
9. C2×C5×#87704C2×C3semidirect
10. C2×C3×C5×C7×C19C2×C3semidirect
12. C2×C5×D7×C19C32semidirect
13. C2×C3×#87704C2×C5semidirect
14. C3×#11716325C2×C5direct (N × H)
15. C3×C5×C7×C19C22×C3semidirect
16. C5×#87704C22×C3semidirect
19. C2×C3×D7×C19C3×C5semidirect
20. C2×#11716325C3×C5direct (N × H)
21. C2×C5×C7×C19C2×C32semidirect
22. C5×D7×C19C2×C32semidirect
24. C3×#87704C22×C5semidirect
28. C2×C3×C7×C19C2×C3×C5semidirect
29. C3×D7×C19C2×C3×C5semidirect
30. C2×#87704C2×C3×C5semidirect
31. #11716325C2×C3×C5direct (N × H)
33. C5×C7×C19C22×C32semidirect
37. C2×C3×C5×C19AGL(1,𝔽7)semidirect
40. C2×D7×C19C32×C5semidirect
41. C2×C3×C5×D7C19⋊C3semidirect
43. C3×C7×C19C22×C3×C5semidirect
44. #87704C22×C3×C5semidirect
49. C3×C5×C19C2×AGL(1,𝔽7)semidirect
51. C2×C7×C19C2×C32×C5semidirect
52. D7×C19C2×C32×C5semidirect
56. C2×C3×C5×C7C2×C19⋊C3semidirect
57. C3×C5×D7C2×C19⋊C3semidirect
61. C2×C5×C19C3×AGL(1,𝔽7)semidirect
64. C2×C5×D7C3×C19⋊C3semidirect
65. C7×C19C22×C32×C5semidirect
69. C2×C3×C19C5×AGL(1,𝔽7)semidirect
72. C3×C5×C7C22×C19⋊C3semidirect
74. C5×C19C2×C3×AGL(1,𝔽7)semidirect
77. C2×C3×D7C5×C19⋊C3semidirect
80. C2×C5×C7C2×C3×C19⋊C3semidirect
81. C5×D7C2×C3×C19⋊C3semidirect
85. C3×C19C2×C5×AGL(1,𝔽7)semidirect
88. C2×C3×C7C2×C5×C19⋊C3semidirect
89. C3×D7C2×C5×C19⋊C3semidirect
93. C2×C19C3×C5×AGL(1,𝔽7)semidirect
95. C5×C7C22×C3×C19⋊C3semidirect
99. C2×C3×C5#11716325direct (N × H)
100. C2×D7C3×C5×C19⋊C3semidirect
101. C3×C7C22×C5×C19⋊C3semidirect
104. C19C2×C3×C5×AGL(1,𝔽7)semidirect
108. C3×C5C2×#11716325direct (N × H)
109. C2×C7C2×C3×C5×C19⋊C3semidirect
110. D7C2×C3×C5×C19⋊C3semidirect
114. C2×C5C3×#11716325direct (N × H)
116. C7C22×C3×C5×C19⋊C3semidirect
120. C2×C3C5×#11716325direct (N × H)
121. C5C2×C3×#11716325direct (N × H)
124. C3C2×C5×#11716325direct (N × H)
126. C2C3×C5×#11716325direct (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×#877042
C3×C5×#117163252
C2×C3×C5×D7×C193
C2×C5×#117163253
C2×C3×#117163255
C22×C3×C5×C19⋊C37
C2×C3×C5×AGL(1,𝔽7)19