Finite Group Databasebeta

← C2×C3×C5×#11925894 #49137633

Subgroups — C2×C3×C5×#11925894 #49137633

Order 36300.

360 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 104 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×#10597244C211yesmaximal, characteristic, complemented
2C3×C5×#11925894C212yesmaximal, direct product factor
3C2×C5×#11925894C311yesmaximal, characteristic, direct product factor, Hall
4C3×C5×#10597244C2211yescharacteristic, complemented, Hall
5C2×C3×C5×C11⋊D11C511yesmaximal, characteristic, complemented
6C2×C3×#11925894C515yesmaximal, direct product factor
7C2×C5×#10597244C2×C311yescharacteristic, complemented
8C5×#11925894C2×C312yesdirect product factor
9C3×C5×C11⋊D11C2×C512yescomplemented
10C2×C3×#10597244C2×C515yescomplemented
11C3×#11925894C2×C5110yesdirect product factor
12C2×C3×C5×C112C2×C511yescharacteristic, complemented
13C2×C3×C5×AGL(1,𝔽11)—112nomaximal
14C5×#10597244C22×C311yescharacteristic, complemented, Hall
15C2×C5×C11⋊D11C3×C511yescharacteristic, complemented
16C2×#11925894C3×C515yesdirect product factor
17C3×C5×C112C22×C511yescharacteristic, complemented
18C3×#10597244C22×C515yescomplemented
19C2×C3×C5×C11⋊C5—112no—
20C3×C5×AGL(1,𝔽11)—114no—
21C2×C3×C11⋊D11C5211yescharacteristic, complemented, Hall
22C2×C5×C112C2×C3×C511yescharacteristic, complemented
23C5×C11⋊D11C2×C3×C512yescomplemented
24C2×#10597244C2×C3×C515yescomplemented
25#11925894C2×C3×C5110yesdirect product factor
26C2×C5×AGL(1,𝔽11)—112no—
27C3×C5×C11⋊C5—112no—
28C2×C3×C112C2×C5211yescharacteristic, complemented
29C3×C11⋊D11C2×C5212yescomplemented
30C2×C3×C5×D11—552no—
31C2×C3×C5×D11—552no—
32C2×C3×AGL(1,𝔽11)—1110no—
33C5×C112C22×C3×C511yescharacteristic, complemented
34#10597244C22×C3×C515yescomplemented
35C2×C5×C11⋊C5—112no—
36C5×AGL(1,𝔽11)—114no—
37C2×C11⋊D11C3×C5211yescharacteristic, complemented, Hall
38C3×C112C22×C5211yescharacteristic, complemented, Hall
39C2×C3×C5×C11—52no—
40C2×C3×C5×C11AGL(1,𝔽11)12yescomplemented
41C3×C5×D11—554no—
42C3×C5×D11—554no—
43C2×C3×C11⋊C5—1110no—
44C3×AGL(1,𝔽11)—1120no—
45C22×C3×C52—1211noHall
46C5×C11⋊C5—112no—
47C2×C112C2×C3×C5211yescharacteristic, complemented
48C11⋊D11C2×C3×C5212yescomplemented
49C2×C5×D11—552no—
50C2×C5×D11—552no—
51C2×AGL(1,𝔽11)—1110no—
52C3×C5×C11—52no—
53C3×C5×C11C2×AGL(1,𝔽11)12yescomplemented
54C3×C11⋊C5—1110no—
55C2×C3×C52—1212no—
56C2×C3×C52—1211no—
57C2×C3×D11—552no—
58C2×C3×D11—552no—
59C112C22×C3×C5211yescharacteristic, complemented, Sylow
60C2×C5×C11—52no—
61C2×C5×C11C3×AGL(1,𝔽11)12yescomplemented
62C5×D11—554no—
63C5×D11—554no—
64C2×C11⋊C5—1110no—
65AGL(1,𝔽11)—1120no—
66C22×C52—1211noHall
67C3×C52—1211noHall
68C2×C3×C11—52no—
69C2×C3×C11C5×AGL(1,𝔽11)12yescomplemented
70C3×D11—554no—
71C3×D11—554no—
72C22×C3×C5—6051no—
73C22×C3×C5—1215no—
74C5×C11—52no—
75C5×C11C2×C3×AGL(1,𝔽11)12yescomplemented
76C11⋊C5—1110no—
77C2×C52—1212no—
78C2×C52—1211no—
79C2×D11—552no—
80C2×D11—552no—
81C3×C11—52no—
82C3×C11C2×C5×AGL(1,𝔽11)12yescomplemented
83C2×C3×C5—6052no—
84C2×C3×C5—12110no—
85C2×C3×C5—1215no—
86C2×C3×C5#1192589411yescharacteristic, direct product factor
87C52—1211noSylow
88C2×C11—52no—
89C2×C11C3×C5×AGL(1,𝔽11)12yescomplemented
90D11—554no—
91D11—554no—
92C22×C5—6051no—
93C22×C5—1215no—
94C3×C5—1215no—
95C3×C5C2×#1192589411yescharacteristic, direct product factor
96C22×C3—6051noHall
97C11—52no—
98C11C2×C3×C5×AGL(1,𝔽11)12yescomplemented
99C2×C5—6052no—
100C2×C5—12110no—
101C2×C5—1215no—
102C2×C5C3×#1192589411yescharacteristic, direct product factor
103C2×C3—6052no—
104C2×C3C5×#1192589411yescharacteristic, direct product factor
105C5—1215no—
106C5C2×C3×#1192589411yescharacteristic, direct product factor
107C22—6051noSylow
108C3C2×C5×#1192589411yescharacteristic, direct product factor, Sylow
109C2—6052no—
110C2C3×C5×#1192589411yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3×C5#11925894
commutator, cosocle, hypocenterC112C22×C3×C52
FrattiniC1G
Fitting, socleC2×C3×C5×C112C2×C5
LCSG ▷ C112
UCSC1 ◁ C2×C3×C5

Complements

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

NH ≅ G/Ntype
1. C2×C3×C5×#10597244C2semidirect
2. C3×C5×#11925894C2direct (N × H)
3. C2×C5×#11925894C3direct (N × H)
4. C3×C5×#10597244C22semidirect
5. C2×C3×C5×C11⋊D11C5semidirect
6. C2×C3×#11925894C5direct (N × H)
7. C2×C5×#10597244C2×C3semidirect
8. C5×#11925894C2×C3direct (N × H)
9. C3×C5×C11⋊D11C2×C5semidirect
10. C2×C3×#10597244C2×C5semidirect
11. C3×#11925894C2×C5direct (N × H)
12. C2×C3×C5×C112C2×C5semidirect
14. C5×#10597244C22×C3semidirect
15. C2×C5×C11⋊D11C3×C5semidirect
16. C2×#11925894C3×C5direct (N × H)
17. C3×C5×C112C22×C5semidirect
18. C3×#10597244C22×C5semidirect
21. C2×C3×C11⋊D11C52semidirect
22. C2×C5×C112C2×C3×C5semidirect
23. C5×C11⋊D11C2×C3×C5semidirect
24. C2×#10597244C2×C3×C5semidirect
25. #11925894C2×C3×C5direct (N × H)
28. C2×C3×C112C2×C52semidirect
29. C3×C11⋊D11C2×C52semidirect
33. C5×C112C22×C3×C5semidirect
34. #10597244C22×C3×C5semidirect
37. C2×C11⋊D11C3×C52semidirect
38. C3×C112C22×C52semidirect
40. C2×C3×C5×C11AGL(1,𝔽11)semidirect
47. C2×C112C2×C3×C52semidirect
48. C11⋊D11C2×C3×C52semidirect
53. C3×C5×C11C2×AGL(1,𝔽11)semidirect
59. C112C22×C3×C52semidirect
61. C2×C5×C11C3×AGL(1,𝔽11)semidirect
69. C2×C3×C11C5×AGL(1,𝔽11)semidirect
75. C5×C11C2×C3×AGL(1,𝔽11)semidirect
82. C3×C11C2×C5×AGL(1,𝔽11)semidirect
86. C2×C3×C5#11925894direct (N × H)
89. C2×C11C3×C5×AGL(1,𝔽11)semidirect
95. C3×C5C2×#11925894direct (N × H)
98. C11C2×C3×C5×AGL(1,𝔽11)semidirect
102. C2×C5C3×#11925894direct (N × H)
104. C2×C3C5×#11925894direct (N × H)
106. C5C2×C3×#11925894direct (N × H)
108. C3C2×C5×#11925894direct (N × H)
110. C2C3×C5×#11925894direct (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×#105972442
C3×C5×#119258942
C2×C5×#119258943
C2×C3×C5×C11⋊D115
C2×C3×#119258945
C2×C3×C5×AGL(1,𝔽11)11