Finite Group Databasebeta

← C2×C3×C5×#11925898 #49137632

Subgroups — C2×C3×C5×#11925898 #49137632

Order 36300.

328 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 116 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×#11925898C212yesmaximal, direct product factor
3C2×C5×#11925898C311yesmaximal, characteristic, direct product factor, Hall
4C3×C5×#10597244C2211yescharacteristic, complemented, Hall
5C2×C3×C5×C11×D11C511yesmaximal, characteristic, complemented
6C2×C3×#11925898C515yesmaximal, direct product factor
7C2×C5×#10597244C2×C311yescharacteristic, complemented
8C5×#11925898C2×C312yesdirect product factor
9C3×C5×C11×D11C2×C512yescomplemented
10C2×C3×#10597244C2×C515yescomplemented
11C3×#11925898C2×C5110yesdirect product factor
12C2×C3×C5×C112C2×C511yescharacteristic, complemented
13C22×C3×C5×C11⋊C5—111nomaximal
14C2×C3×C5×AGL(1,𝔽11)—111nomaximal
15C5×#10597244C22×C311yescharacteristic, complemented, Hall
16C2×C5×C11×D11C3×C511yescharacteristic, complemented
17C2×#11925898C3×C515yesdirect product factor
18C3×C5×C112C22×C511yescharacteristic, complemented
19C3×#10597244C22×C515yescomplemented
20C2×C3×C5×C11⋊C5—112no—
21C2×C3×C5×C11⋊C5—111no—
22C2×C3×C5×C11⋊C5—111no—
23C3×C5×AGL(1,𝔽11)—112no—
24C2×C3×C11×D11C5211yescharacteristic, complemented, Hall
25C2×C5×C112C2×C3×C511yescharacteristic, complemented
26C5×C11×D11C2×C3×C512yescomplemented
27C2×#10597244C2×C3×C515yescomplemented
28#11925898C2×C3×C5110yesdirect product factor
29C22×C5×C11⋊C5—111no—
30C2×C5×AGL(1,𝔽11)—111no—
31C3×C5×C11⋊C5—111no—
32C3×C5×C11⋊C5—111no—
33C2×C3×C112C2×C5211yescharacteristic, complemented
34C3×C11×D11C2×C5212yescomplemented
35C22×C3×C5×C11—551no—
36C2×C3×C5×D11C11⋊C511yescharacteristic, complemented
37C22×C3×C11⋊C5—115no—
38C2×C3×AGL(1,𝔽11)—115no—
39C5×C112C22×C3×C511yescharacteristic, complemented
40#10597244C22×C3×C515yescomplemented
41C2×C5×C11⋊C5—112no—
42C2×C5×C11⋊C5—111no—
43C2×C5×C11⋊C5—111no—
44C5×AGL(1,𝔽11)—112no—
45C2×C11×D11C3×C5211yescharacteristic, complemented, Hall
46C3×C112C22×C5211yescharacteristic, complemented, Hall
47C2×C3×C5×C11—101no—
48C2×C3×C5×C11—552no—
49C2×C3×C5×C11AGL(1,𝔽11)11yescharacteristic, complemented
50C2×C3×C5×C11C2×C11⋊C511yescharacteristic, complemented
51C3×C5×D11C2×C11⋊C512yescomplemented
52C2×C3×C11⋊C5—1110no—
53C2×C3×C11⋊C5—115no—
54C2×C3×C11⋊C5—115no—
55C3×AGL(1,𝔽11)—1110no—
56C22×C3×C52—1211noHall
57C5×C11⋊C5—111no—
58C5×C11⋊C5—111no—
59C2×C112C2×C3×C5211yescharacteristic, complemented
60C11×D11C2×C3×C5212yescomplemented
61C22×C5×C11—551no—
62C2×C5×D11C3×C11⋊C511yescharacteristic, complemented
63C22×C11⋊C5—115no—
64C2×AGL(1,𝔽11)—115no—
65C3×C5×C11—101no—
66C3×C5×C11C2×AGL(1,𝔽11)11yescharacteristic, complemented
67C3×C5×C11C22×C11⋊C511yescharacteristic, complemented
68C3×C11⋊C5—115no—
69C3×C11⋊C5—115no—
70C2×C3×C52—1212no—
71C2×C3×C52—1211no—
72C22×C3×C11—551no—
73C2×C3×D11C5×C11⋊C511yescharacteristic, complemented
74C112C22×C3×C5211yescharacteristic, complemented, Sylow
75C2×C5×C11—101no—
76C2×C5×C11—552no—
77C2×C5×C11C3×AGL(1,𝔽11)11yescharacteristic, complemented
78C2×C5×C11C2×C3×C11⋊C511yescharacteristic, complemented
79C5×D11C2×C3×C11⋊C512yescomplemented
80C2×C11⋊C5—1110no—
81C2×C11⋊C5—115no—
82C2×C11⋊C5—115no—
83AGL(1,𝔽11)—1110no—
84C22×C52—1211noHall
85C3×C52—1211noHall
86C2×C3×C11—101no—
87C2×C3×C11—552no—
88C2×C3×C11C5×AGL(1,𝔽11)11yescharacteristic, complemented
89C2×C3×C11C2×C5×C11⋊C511yescharacteristic, complemented
90C3×D11C2×C5×C11⋊C512yescomplemented
91C22×C3×C5—551no—
92C22×C3×C5—1215no—
93C5×C11—101no—
94C5×C11C2×C3×AGL(1,𝔽11)11yescharacteristic, complemented
95C5×C11C22×C3×C11⋊C511yescharacteristic, complemented
96C11⋊C5—115no—
97C11⋊C5—115no—
98C2×C52—1212no—
99C2×C52—1211no—
100C22×C11—551no—
101C2×D11C3×C5×C11⋊C511yescharacteristic, complemented
102C3×C11—101no—
103C3×C11C2×C5×AGL(1,𝔽11)11yescharacteristic, complemented
104C3×C11C22×C5×C11⋊C511yescharacteristic, complemented
105C2×C3×C5—552no—
106C2×C3×C5—12110no—
107C2×C3×C5—1215no—
108C2×C3×C5#1192589811yescharacteristic, direct product factor
109C52—1211noSylow
110C2×C11—101no—
111C2×C11—552no—
112C2×C11C3×C5×AGL(1,𝔽11)11yescharacteristic, complemented
113C2×C11C2×C3×C5×C11⋊C511yescharacteristic, complemented
114D11C2×C3×C5×C11⋊C512yescomplemented
115C22×C5—551no—
116C22×C5—1215no—
117C3×C5—1215no—
118C3×C5C2×#1192589811yescharacteristic, direct product factor
119C22×C3—551noHall
120C11—101no—
121C11C2×C3×C5×AGL(1,𝔽11)11yescharacteristic, complemented
122C11C22×C3×C5×C11⋊C511yescharacteristic, complemented
123C2×C5—552no—
124C2×C5—12110no—
125C2×C5—1215no—
126C2×C5C3×#1192589811yescharacteristic, direct product factor
127C2×C3—552no—
128C2×C3C5×#1192589811yescharacteristic, direct product factor
129C5—1215no—
130C5C2×C3×#1192589811yescharacteristic, direct product factor
131C22—551noSylow
132C3C2×C5×#1192589811yescharacteristic, direct product factor, Sylow
133C2—552no—
134C2C3×C5×#1192589811yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3×C5#11925898
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×#11925898C2direct (N × H)
3. C2×C5×#11925898C3direct (N × H)
4. C3×C5×#10597244C22semidirect
5. C2×C3×C5×C11×D11C5semidirect
6. C2×C3×#11925898C5direct (N × H)
7. C2×C5×#10597244C2×C3semidirect
8. C5×#11925898C2×C3direct (N × H)
9. C3×C5×C11×D11C2×C5semidirect
10. C2×C3×#10597244C2×C5semidirect
11. C3×#11925898C2×C5direct (N × H)
12. C2×C3×C5×C112C2×C5semidirect
15. C5×#10597244C22×C3semidirect
16. C2×C5×C11×D11C3×C5semidirect
17. C2×#11925898C3×C5direct (N × H)
18. C3×C5×C112C22×C5semidirect
19. C3×#10597244C22×C5semidirect
24. C2×C3×C11×D11C52semidirect
25. C2×C5×C112C2×C3×C5semidirect
26. C5×C11×D11C2×C3×C5semidirect
27. C2×#10597244C2×C3×C5semidirect
28. #11925898C2×C3×C5direct (N × H)
33. C2×C3×C112C2×C52semidirect
34. C3×C11×D11C2×C52semidirect
36. C2×C3×C5×D11C11⋊C5semidirect
39. C5×C112C22×C3×C5semidirect
40. #10597244C22×C3×C5semidirect
45. C2×C11×D11C3×C52semidirect
46. C3×C112C22×C52semidirect
49. C2×C3×C5×C11AGL(1,𝔽11)semidirect
50. C2×C3×C5×C11C2×C11⋊C5semidirect
51. C3×C5×D11C2×C11⋊C5semidirect
59. C2×C112C2×C3×C52semidirect
60. C11×D11C2×C3×C52semidirect
62. C2×C5×D11C3×C11⋊C5semidirect
66. C3×C5×C11C2×AGL(1,𝔽11)semidirect
67. C3×C5×C11C22×C11⋊C5semidirect
73. C2×C3×D11C5×C11⋊C5semidirect
74. C112C22×C3×C52semidirect
77. C2×C5×C11C3×AGL(1,𝔽11)semidirect
78. C2×C5×C11C2×C3×C11⋊C5semidirect
79. C5×D11C2×C3×C11⋊C5semidirect
88. C2×C3×C11C5×AGL(1,𝔽11)semidirect
89. C2×C3×C11C2×C5×C11⋊C5semidirect
90. C3×D11C2×C5×C11⋊C5semidirect
94. C5×C11C2×C3×AGL(1,𝔽11)semidirect
95. C5×C11C22×C3×C11⋊C5semidirect
101. C2×D11C3×C5×C11⋊C5semidirect
103. C3×C11C2×C5×AGL(1,𝔽11)semidirect
104. C3×C11C22×C5×C11⋊C5semidirect
108. C2×C3×C5#11925898direct (N × H)
112. C2×C11C3×C5×AGL(1,𝔽11)semidirect
113. C2×C11C2×C3×C5×C11⋊C5semidirect
114. D11C2×C3×C5×C11⋊C5semidirect
118. C3×C5C2×#11925898direct (N × H)
121. C11C2×C3×C5×AGL(1,𝔽11)semidirect
122. C11C22×C3×C5×C11⋊C5semidirect
126. C2×C5C3×#11925898direct (N × H)
128. C2×C3C5×#11925898direct (N × H)
130. C5C2×C3×#11925898direct (N × H)
132. C3C2×C5×#11925898direct (N × H)
134. C2C3×C5×#11925898direct (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×#119258982
C2×C5×#119258983
C2×C3×C5×C11×D115
C2×C3×#119258985
C22×C3×C5×C11⋊C511
C2×C3×C5×AGL(1,𝔽11)11