Finite Group Databasebeta

← C2×C3×C5⋊#11925893 #49137651

Subgroups — C2×C3×C5⋊#11925893 #49137651

Order 36300.

232 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 52 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×#10597245C211yesmaximal, characteristic, complemented
2C3×C5⋊#11925893C212yesmaximal, direct product factor
3C2×C5⋊#11925893C311yesmaximal, characteristic, direct product factor, Hall
4C3×C5×#10597245C2211yescharacteristic, complemented, Hall
5C2×C3×C5⋊(C11⋊D11)C511yesmaximal, characteristic, complemented
6C2×C3×#11925893—51nomaximal
7C2×C5×#10597245C2×C311yescharacteristic, complemented
8C5⋊#11925893C2×C312yesdirect product factor
9C3×C5⋊(C11⋊D11)C2×C512yescomplemented
10C2×C3×#10597245—22no—
11C2×C3×#10597245D511yescharacteristic, complemented
12C3×#11925893—52no—
13C2×C3×C5×C112C2×C511yescharacteristic, complemented
14C2×C3×C5⋊AGL(1,𝔽11)—112nomaximal
15C5×#10597245C22×C311yescharacteristic, complemented, Hall
16C2×C5⋊(C11⋊D11)C3×C511yescharacteristic, complemented
17C2×#11925893—51no—
18C3×C5×C112C22×C511yescharacteristic, complemented
19C3×#10597245—22no—
20C3×#10597245C2×D511yescharacteristic, complemented
21C2×C3×C5×C11⋊C5—222no—
22C3×C5⋊AGL(1,𝔽11)—114no—
23C2×C3×C11⋊D11—51noHall
24C2×C5×C112C2×C3×C511yescharacteristic, complemented
25C5⋊(C11⋊D11)C2×C3×C512yescomplemented
26C2×#10597245—22no—
27C2×#10597245C3×D511yescharacteristic, complemented
28#11925893—52no—
29C2×C5⋊AGL(1,𝔽11)—112no—
30C3×C5×C11⋊C5—222no—
31C2×C3×C112C5×D511yescharacteristic, complemented
32C3×C11⋊D11—52no—
33C2×C3×D55—112no—
34C2×C3×D55—552no—
35C2×C3×AGL(1,𝔽11)—552no—
36C5×C112C22×C3×C511yescharacteristic, complemented
37#10597245—22no—
38#10597245C2×C3×D511yescharacteristic, complemented
39C2×C5×C11⋊C5—112no—
40C5⋊AGL(1,𝔽11)—114no—
41C2×C11⋊D11—51noHall
42C3×C112C2×C5×D511yescharacteristic, complemented, Hall
43C2×C3×C5×C11—52no—
44C2×C3×C5×C11AGL(1,𝔽11)12yescomplemented
45C3×D55—114no—
46C3×D55—554no—
47C2×C3×C11⋊C5—222no—
48C2×C3×C11⋊C5—224no—
49C3×AGL(1,𝔽11)—554no—
50C2×C3×C5×D5—1211noHall
51C5×C11⋊C5—112no—
52C2×C112C3×C5×D511yescharacteristic, complemented
53C11⋊D11—52no—
54C2×D55—112no—
55C2×D55—552no—
56C2×AGL(1,𝔽11)—552no—
57C3×C5×C11—52no—
58C3×C5×C11C2×AGL(1,𝔽11)12yescomplemented
59C3×C11⋊C5—222no—
60C3×C11⋊C5—224no—
61C2×C3×C52—2421no—
62C3×C5×D5—1212no—
63C2×C3×D11—2752no—
64C2×C3×D11—552no—
65C112C2×C3×C5×D511yescharacteristic, complemented, Sylow
66C2×C5×C11—52no—
67C2×C5×C11C3×AGL(1,𝔽11)12yescomplemented
68D55—114no—
69D55—554no—
70C2×C11⋊C5—224no—
71C2×C11⋊C5—222no—
72AGL(1,𝔽11)—554no—
73C2×C5×D5—1211noHall
74C3×C52—2421noHall
75C2×C3×C11—52no—
76C2×C3×C11C5⋊AGL(1,𝔽11)12yescomplemented
77C3×D11—2754no—
78C3×D11—554no—
79C22×C3×C5—6051no—
80C2×C3×D5—1211no—
81C5×C11—52no—
82C5×C11C2×C3×AGL(1,𝔽11)12yescomplemented
83C11⋊C5—222no—
84C11⋊C5—224no—
85C2×C52—2421no—
86C5×D5—1212no—
87C2×D11—2752no—
88C2×D11—552no—
89C3×C11—52no—
90C3×C11C2×C5⋊AGL(1,𝔽11)12yescomplemented
91C2×C3×C5—6052no—
92C2×C3×C5—1211no—
93C2×C3×C5—2422no—
94C2×C3×C5#1192589311yescharacteristic, complemented
95C3×D5—1212no—
96C52—2421noSylow
97C2×C11—52no—
98C2×C11C3×C5⋊AGL(1,𝔽11)12yescomplemented
99D11—2754no—
100D11—554no—
101C22×C5—6051no—
102C2×D5—1211no—
103C3×C5—1211no—
104C3×C5—2422no—
105C3×C5C2×#1192589311yescharacteristic, complemented
106C22×C3—6051noHall
107C11—52no—
108C11C2×C3×C5⋊AGL(1,𝔽11)12yescomplemented
109C2×C5—6052no—
110C2×C5—2421no—
111C2×C5—2422no—
112C2×C5C3×#1192589311yescharacteristic, complemented
113D5—1212no—
114C2×C3—6052no—
115C2×C3C5⋊#1192589311yescharacteristic, direct product factor
116C5—2421no—
117C5—2422no—
118C5C2×C3×#1192589311yescharacteristic, complemented
119C22—6051noSylow
120C3C2×C5⋊#1192589311yescharacteristic, direct product factor, Sylow
121C2—6052no—
122C2C3×C5⋊#1192589311yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3C5⋊#11925893
commutator, cosocle, hypocenterC5×C112C22×C3×C5
FrattiniC1G
Fitting, socleC2×C3×C5×C112C2×C5
LCSG ▷ C5×C112
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
1. C2×C3×C5×#10597245C2semidirect
2. C3×C5⋊#11925893C2direct (N × H)
3. C2×C5⋊#11925893C3direct (N × H)
4. C3×C5×#10597245C22semidirect
5. C2×C3×C5⋊(C11⋊D11)C5semidirect
7. C2×C5×#10597245C2×C3semidirect
8. C5⋊#11925893C2×C3direct (N × H)
9. C3×C5⋊(C11⋊D11)C2×C5semidirect
11. C2×C3×#10597245D5semidirect
13. C2×C3×C5×C112C2×C5semidirect
15. C5×#10597245C22×C3semidirect
16. C2×C5⋊(C11⋊D11)C3×C5semidirect
18. C3×C5×C112C22×C5semidirect
20. C3×#10597245C2×D5semidirect
24. C2×C5×C112C2×C3×C5semidirect
25. C5⋊(C11⋊D11)C2×C3×C5semidirect
27. C2×#10597245C3×D5semidirect
31. C2×C3×C112C5×D5semidirect
36. C5×C112C22×C3×C5semidirect
38. #10597245C2×C3×D5semidirect
42. C3×C112C2×C5×D5semidirect
44. C2×C3×C5×C11AGL(1,𝔽11)semidirect
52. C2×C112C3×C5×D5semidirect
58. C3×C5×C11C2×AGL(1,𝔽11)semidirect
65. C112C2×C3×C5×D5semidirect
67. C2×C5×C11C3×AGL(1,𝔽11)semidirect
76. C2×C3×C11C5⋊AGL(1,𝔽11)semidirect
82. C5×C11C2×C3×AGL(1,𝔽11)semidirect
90. C3×C11C2×C5⋊AGL(1,𝔽11)semidirect
94. C2×C3×C5#11925893semidirect
98. C2×C11C3×C5⋊AGL(1,𝔽11)semidirect
105. C3×C5C2×#11925893semidirect
108. C11C2×C3×C5⋊AGL(1,𝔽11)semidirect
112. C2×C5C3×#11925893semidirect
115. C2×C3C5⋊#11925893direct (N × H)
118. C5C2×C3×#11925893semidirect
120. C3C2×C5⋊#11925893direct (N × H)
122. C2C3×C5⋊#11925893direct (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×#105972452
C3×C5⋊#119258932
C2×C5⋊#119258933
C2×C3×C5⋊(C11⋊D11)5
C2×C3×#119258935
C2×C3×C5⋊AGL(1,𝔽11)11