Finite Group Databasebeta

← C2×C3×C5×#11759817 #44478151

Subgroups — C2×C3×C5×#11759817 #44478151

Order 30420.

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

normal subgroups only

This group has 384 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×#92788C211yesmaximal, characteristic, complemented
2C3×C5×#11759817C212yesmaximal, direct product factor
3C2×C3×C5×C13×D13C311yesmaximal, characteristic, complemented
4C2×C5×#11759817C313yesmaximal, direct product factor
5C3×C5×#92788C2211yescharacteristic, complemented, Hall
6C2×C3×#11759817C511yesmaximal, characteristic, direct product factor, Hall
7C3×C5×C13×D13C2×C312yescomplemented
8C2×C5×#92788C2×C313yescomplemented
9C5×#11759817C2×C316yesdirect product factor
10C2×C3×C5×C132C2×C311yescharacteristic, complemented
11C2×C5×C13×D13C3211yescharacteristic, complemented, Hall
12C2×C3×#92788C2×C511yescharacteristic, complemented
13C3×#11759817C2×C512yesdirect product factor
14C5×#92788C22×C313yescomplemented
15C3×C5×C132C22×C311yescharacteristic, complemented
16C22×C3×C5×C13⋊C3—131nomaximal
17C2×C3×C5×D13⋊C3—131nomaximal
18C2×C3×C13×D13C3×C511yescharacteristic, complemented
19C2×#11759817C3×C513yesdirect product factor
20C2×C5×C132C2×C3211yescharacteristic, complemented
21C5×C13×D13C2×C3212yescomplemented
22C3×#92788C22×C511yescharacteristic, complemented, Hall
23C2×C3×C5×C13⋊C3—132no—
24C2×C3×C5×C13⋊C3—131no—
25C2×C3×C5×C13⋊C3—131no—
26C2×C3×C5×C13⋊C3—266no—
27C3×C5×D13⋊C3—132no—
28C2×C3×C132C2×C3×C511yescharacteristic, complemented
29C3×C13×D13C2×C3×C512yescomplemented
30C2×#92788C2×C3×C513yescomplemented
31#11759817C2×C3×C516yesdirect product factor
32C5×C132C22×C3211yescharacteristic, complemented, Hall
33C22×C3×C5×C13—391no—
34C2×C3×C5×D13C13⋊C311yescharacteristic, complemented
35C22×C5×C13⋊C3—133no—
36C2×C5×D13⋊C3—133no—
37C2×C13×D13C32×C511yescharacteristic, complemented, Hall
38C3×C5×C13⋊C3—131no—
39C3×C5×C13⋊C3—131no—
40C3×C5×C13⋊C3—266no—
41C3×C132C22×C3×C511yescharacteristic, complemented
42#92788C22×C3×C513yescomplemented
43C22×C3×C13⋊C3—131no—
44C2×C3×D13⋊C3—131no—
45C2×C3×C5×C13—26no—
46C2×C3×C5×C13—392no—
47C2×C3×C5×C13D13⋊C311yescharacteristic, complemented
48C2×C3×C5×C13C2×C13⋊C311yescharacteristic, complemented
49C3×C5×D13C2×C13⋊C312yescomplemented
50C2×C5×C13⋊C3—136no—
51C2×C5×C13⋊C3—133no—
52C2×C5×C13⋊C3—133no—
53C2×C5×C13⋊C3—2618no—
54C5×D13⋊C3—136no—
55C2×C132C2×C32×C511yescharacteristic, complemented
56C13×D13C2×C32×C512yescomplemented
57C22×C5×C13—391no—
58C2×C5×D13C3×C13⋊C311yescharacteristic, complemented
59C2×C3×C13⋊C3—132no—
60C2×C3×C13⋊C3—131no—
61C2×C3×C13⋊C3—131no—
62C2×C3×C13⋊C3—266no—
63C3×D13⋊C3—132no—
64C3×C5×C13—26no—
65C3×C5×C13C2×D13⋊C311yescharacteristic, complemented
66C3×C5×C13C22×C13⋊C311yescharacteristic, complemented
67C5×C13⋊C3—133no—
68C5×C13⋊C3—133no—
69C5×C13⋊C3—2618no—
70C22×C32×C5—1691noHall
71C132C22×C32×C511yescharacteristic, complemented, Sylow
72C22×C3×C13—391no—
73C2×C3×D13C5×C13⋊C311yescharacteristic, complemented
74C22×C13⋊C3—133no—
75C2×D13⋊C3—133no—
76C2×C5×C13—26no—
77C2×C5×C13—392no—
78C2×C5×C13C3×D13⋊C311yescharacteristic, complemented
79C2×C5×C13C2×C3×C13⋊C311yescharacteristic, complemented
80C5×D13C2×C3×C13⋊C312yescomplemented
81C3×C13⋊C3—131no—
82C3×C13⋊C3—131no—
83C3×C13⋊C3—266no—
84C2×C32×C5—1692no—
85C2×C32×C5—1691no—
86C2×C3×C13—26no—
87C2×C3×C13—392no—
88C2×C3×C13C5×D13⋊C311yescharacteristic, complemented
89C2×C3×C13C2×C5×C13⋊C311yescharacteristic, complemented
90C3×D13C2×C5×C13⋊C312yescomplemented
91C2×C13⋊C3—136no—
92C2×C13⋊C3—133no—
93C2×C13⋊C3—133no—
94C2×C13⋊C3—2618no—
95D13⋊C3—136no—
96C5×C13—26no—
97C5×C13C2×C3×D13⋊C311yescharacteristic, complemented
98C5×C13C22×C3×C13⋊C311yescharacteristic, complemented
99C22×C3×C5—391no—
100C22×C3×C5—1693no—
101C22×C13—391no—
102C2×D13C3×C5×C13⋊C311yescharacteristic, complemented
103C32×C5—1691noHall
104C3×C13—26no—
105C3×C13C2×C5×D13⋊C311yescharacteristic, complemented
106C3×C13C22×C5×C13⋊C311yescharacteristic, complemented
107C13⋊C3—133no—
108C13⋊C3—133no—
109C13⋊C3—2618no—
110C22×C32—1691noHall
111C2×C3×C5—392no—
112C2×C3×C5—1696no—
113C2×C3×C5—1693no—
114C2×C3×C5#1175981711yescharacteristic, direct product factor
115C2×C13—26no—
116C2×C13—392no—
117C2×C13C3×C5×D13⋊C311yescharacteristic, complemented
118C2×C13C2×C3×C5×C13⋊C311yescharacteristic, complemented
119D13C2×C3×C5×C13⋊C312yescomplemented
120C22×C5—391noHall
121C2×C32—1692no—
122C2×C32—1691no—
123C3×C5—1693no—
124C3×C5C2×#1175981711yescharacteristic, direct product factor
125C13—26no—
126C13C2×C3×C5×D13⋊C311yescharacteristic, complemented
127C13C22×C3×C5×C13⋊C311yescharacteristic, complemented
128C22×C3—391no—
129C22×C3—1693no—
130C2×C5—392no—
131C2×C5C3×#1175981711yescharacteristic, direct product factor
132C32—1691noSylow
133C2×C3—392no—
134C2×C3—1696no—
135C2×C3—1693no—
136C2×C3C5×#1175981711yescharacteristic, direct product factor
137C5C2×C3×#1175981711yescharacteristic, direct product factor, Sylow
138C22—391noSylow
139C3—1693no—
140C3C2×C5×#1175981711yescharacteristic, direct product factor
141C2—392no—
142C2C3×C5×#1175981711yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3×C5#11759817
commutator, cosocle, hypocenterC132C22×C32×C5
FrattiniC1G
Fitting, socleC2×C3×C5×C132C2×C3
LCSG ▷ C132
UCSC1 ◁ C2×C3×C5

Complements

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

NH ≅ G/Ntype
1. C2×C3×C5×#92788C2semidirect
2. C3×C5×#11759817C2direct (N × H)
3. C2×C3×C5×C13×D13C3semidirect
4. C2×C5×#11759817C3direct (N × H)
5. C3×C5×#92788C22semidirect
6. C2×C3×#11759817C5direct (N × H)
7. C3×C5×C13×D13C2×C3semidirect
8. C2×C5×#92788C2×C3semidirect
9. C5×#11759817C2×C3direct (N × H)
10. C2×C3×C5×C132C2×C3semidirect
11. C2×C5×C13×D13C32semidirect
12. C2×C3×#92788C2×C5semidirect
13. C3×#11759817C2×C5direct (N × H)
14. C5×#92788C22×C3semidirect
15. C3×C5×C132C22×C3semidirect
18. C2×C3×C13×D13C3×C5semidirect
19. C2×#11759817C3×C5direct (N × H)
20. C2×C5×C132C2×C32semidirect
21. C5×C13×D13C2×C32semidirect
22. C3×#92788C22×C5semidirect
28. C2×C3×C132C2×C3×C5semidirect
29. C3×C13×D13C2×C3×C5semidirect
30. C2×#92788C2×C3×C5semidirect
31. #11759817C2×C3×C5direct (N × H)
32. C5×C132C22×C32semidirect
34. C2×C3×C5×D13C13⋊C3semidirect
37. C2×C13×D13C32×C5semidirect
41. C3×C132C22×C3×C5semidirect
42. #92788C22×C3×C5semidirect
47. C2×C3×C5×C13D13⋊C3semidirect
48. C2×C3×C5×C13C2×C13⋊C3semidirect
49. C3×C5×D13C2×C13⋊C3semidirect
55. C2×C132C2×C32×C5semidirect
56. C13×D13C2×C32×C5semidirect
58. C2×C5×D13C3×C13⋊C3semidirect
65. C3×C5×C13C2×D13⋊C3semidirect
66. C3×C5×C13C22×C13⋊C3semidirect
71. C132C22×C32×C5semidirect
73. C2×C3×D13C5×C13⋊C3semidirect
78. C2×C5×C13C3×D13⋊C3semidirect
79. C2×C5×C13C2×C3×C13⋊C3semidirect
80. C5×D13C2×C3×C13⋊C3semidirect
88. C2×C3×C13C5×D13⋊C3semidirect
89. C2×C3×C13C2×C5×C13⋊C3semidirect
90. C3×D13C2×C5×C13⋊C3semidirect
97. C5×C13C2×C3×D13⋊C3semidirect
98. C5×C13C22×C3×C13⋊C3semidirect
102. C2×D13C3×C5×C13⋊C3semidirect
105. C3×C13C2×C5×D13⋊C3semidirect
106. C3×C13C22×C5×C13⋊C3semidirect
114. C2×C3×C5#11759817direct (N × H)
117. C2×C13C3×C5×D13⋊C3semidirect
118. C2×C13C2×C3×C5×C13⋊C3semidirect
119. D13C2×C3×C5×C13⋊C3semidirect
124. C3×C5C2×#11759817direct (N × H)
126. C13C2×C3×C5×D13⋊C3semidirect
127. C13C22×C3×C5×C13⋊C3semidirect
131. C2×C5C3×#11759817direct (N × H)
136. C2×C3C5×#11759817direct (N × H)
137. C5C2×C3×#11759817direct (N × H)
140. C3C2×C5×#11759817direct (N × H)
142. C2C3×C5×#11759817direct (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×#927882
C3×C5×#117598172
C2×C3×C5×C13×D133
C2×C5×#117598173
C2×C3×#117598175
C22×C3×C5×C13⋊C313
C2×C3×C5×D13⋊C313