Finite Group Databasebeta

← C2×C3⋊(C5⋊#11925895) #49137674

Subgroups — C2×C3⋊(C5⋊#11925895) #49137674

Order 36300.

352 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 49 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×#10597243C211yesmaximal, characteristic, complemented
2C3⋊(C5⋊#11925895)C212yesmaximal, direct product factor
3C2×C5⋊#11925895—31nomaximal, Hall
4C3×C5×#10597243C2211yescharacteristic, complemented, Hall
5C2×C11×D165C511yesmaximal, characteristic, complemented
6C2×C3⋊#11925895—51nomaximal
7C2×C5×#10597243D311yescharacteristic, complemented
8C5⋊#11925895—32no—
9C11×D165C2×C512yescomplemented
10C2×C3×#10597243—22no—
11C2×C3×#10597243D511yescharacteristic, complemented
12C3⋊#11925895—52no—
13C2×C3×C5×C112C2×C511yescharacteristic, complemented
14C2×D15×C11⋊C5—111nomaximal
15C2×C3⋊(C5⋊AGL(1,𝔽11))—111nomaximal
16C5×#10597243C2×D311yescharacteristic, complemented, Hall
17C2×C11×D55—31no—
18C2×#11925895—151no—
19C3×C5×C112C22×C511yescharacteristic, complemented
20C3×#10597243—22no—
21C3×#10597243C2×D511yescharacteristic, complemented
22C2×C3×C5×C11⋊C5—111no—
23C2×C3×C5×C11⋊C5—221no—
24C2×C3×C5×C11⋊C5—225no—
25C3⋊(C5⋊AGL(1,𝔽11))—112no—
26D15×C11⋊C5—112no—
27C2×C11×D33—51noHall
28C2×C5×C112C5×D311yescharacteristic, complemented
29C11×D55—32no—
30C2×#10597243—22no—
31C2×#10597243D1511yescharacteristic, complemented
32#11925895—152no—
33C2×C5⋊AGL(1,𝔽11)—331no—
34C2×D5×C11⋊C5—331no—
35C3×C5×C11⋊C5—111no—
36C3×C5×C11⋊C5—221no—
37C3×C5×C11⋊C5—225no—
38C2×C3×C112C5×D511yescharacteristic, complemented
39C11×D33—52no—
40C2×D165C11⋊C511yescharacteristic, complemented
41C2×C11×D15—111no—
42C2×C3⋊AGL(1,𝔽11)—551no—
43C2×D3×C11⋊C5—551no—
44C5×C112C2×C5×D311yescharacteristic, complemented
45#10597243—22no—
46#10597243C2×D1511yescharacteristic, complemented
47C2×C5×C11⋊C5—111no—
48C2×C5×C11⋊C5—111no—
49C2×C5×C11⋊C5—225no—
50C5⋊AGL(1,𝔽11)—332no—
51D5×C11⋊C5—332no—
52C2×C11×D11—151noHall
53C3×C112C2×C5×D511yescharacteristic, complemented, Hall
54C2×C3×C5×C11—25no—
55C2×C3×C5×C11AGL(1,𝔽11)11yescharacteristic, complemented
56C2×C3×C5×C11C2×C11⋊C511yescharacteristic, complemented
57D165C2×C11⋊C512yescomplemented
58C11×D15—112no—
59C2×C3×C11⋊C5—111no—
60C2×C3×C11⋊C5—2220no—
61C2×C3×C11⋊C5—225no—
62C2×C3×C11⋊C5—222no—
63C2×C3×C11⋊C5—222no—
64C2×C3×C11⋊C5—221no—
65C3⋊AGL(1,𝔽11)—552no—
66D3×C11⋊C5—552no—
67C2×C5×D15—1211noHall
68C5×C11⋊C5—111no—
69C5×C11⋊C5—111no—
70C5×C11⋊C5—225no—
71C2×C112C5×D1511yescharacteristic, complemented
72C11×D11—152no—
73C2×D55—31no—
74C2×D5×C11—331no—
75C22×C11⋊C5—1651no—
76C2×AGL(1,𝔽11)—1651no—
77C3×C5×C11—25no—
78C3×C5×C11C2×AGL(1,𝔽11)11yescharacteristic, complemented
79C3×C5×C11C22×C11⋊C511yescharacteristic, complemented
80C3×C11⋊C5—111no—
81C3×C11⋊C5—2220no—
82C3×C11⋊C5—225no—
83C3×C11⋊C5—222no—
84C3×C11⋊C5—222no—
85C3×C11⋊C5—221no—
86C2×C3×C52—2421no—
87C5×D15—1212no—
88C2×D33—51no—
89C2×D3×C11—551no—
90C112C2×C5×D1511yescharacteristic, complemented, Sylow
91C2×C5×C11—25no—
92C2×C5×C11D3×C11⋊C511yescharacteristic, complemented
93C2×C5×C11C3⋊AGL(1,𝔽11)11yescharacteristic, complemented
94D55—32no—
95D5×C11—332no—
96C2×C11⋊C5—111no—
97C2×C11⋊C5—2220no—
98C2×C11⋊C5—225no—
99C2×C11⋊C5—222no—
100C2×C11⋊C5—222no—
101C2×C11⋊C5—221no—
102C2×C11⋊C5—1652no—
103AGL(1,𝔽11)—1652no—
104C2×C5×D5—3631noHall
105C3×C52—2421noHall
106C2×C3×C11—25no—
107C2×C3×C11D5×C11⋊C511yescharacteristic, complemented
108C2×C3×C11C5⋊AGL(1,𝔽11)11yescharacteristic, complemented
109D33—52no—
110D3×C11—552no—
111C2×D15—111no—
112C2×C5×D3—6051no—
113C5×C11—25no—
114C5×C11C2×D3×C11⋊C511yescharacteristic, complemented
115C5×C11C2×C3⋊AGL(1,𝔽11)11yescharacteristic, complemented
116C11⋊C5—111no—
117C11⋊C5—2220no—
118C11⋊C5—225no—
119C11⋊C5—221no—
120C11⋊C5—222no—
121C11⋊C5—222no—
122C2×C52—2421no—
123C5×D5—3632no—
124C22×C11—1651no—
125C2×D11—151no—
126C3×C11—25no—
127C3×C11C2×D5×C11⋊C511yescharacteristic, complemented
128C3×C11C2×C5⋊AGL(1,𝔽11)11yescharacteristic, complemented
129C2×C3×C5—2421no—
130C2×C3×C5—2422no—
131C2×C3×C5#1192589511yescharacteristic, complemented
132D15—112no—
133C5×D3—6052no—
134C52—2421noSylow
135C2×C11—25no—
136C2×C11—1652no—
137C2×C11D15×C11⋊C511yescharacteristic, complemented
138C2×C11C3⋊(C5⋊AGL(1,𝔽11))11yescharacteristic, complemented
139D11—152no—
140C22×C5—18151no—
141C2×D5—331no—
142C3×C5—2421no—
143C3×C5—2422no—
144C3×C5C2×#1192589511yescharacteristic, complemented
145C2×D3—551noHall
146C11—25no—
147C11C2×C3⋊(C5⋊AGL(1,𝔽11))11yescharacteristic, complemented
148C11C2×D15×C11⋊C511yescharacteristic, complemented
149C2×C5—18152no—
150C2×C5—2421no—
151C2×C5—2422no—
152C2×C5C3⋊#1192589511yescharacteristic, complemented
153D5—332no—
154C2×C3C5⋊#1192589511yescharacteristic, complemented
155D3—552no—
156C5—2421no—
157C5—2422no—
158C5C2×C3⋊#1192589511yescharacteristic, complemented
159C22—1651noSylow
160C3C2×C5⋊#1192589511yescharacteristic, complemented, Sylow
161C2—1652no—
162C2C3⋊(C5⋊#11925895)11yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2C3⋊(C5⋊#11925895)
commutator, cosocle, hypocenterC3×C5×C112C22×C5
FrattiniC1G
Fitting, socleC2×C3×C5×C112C2×C5
LCSG ▷ C3×C5×C112
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C2×C3×C5×#10597243C2semidirect
2. C3⋊(C5⋊#11925895)C2direct (N × H)
4. C3×C5×#10597243C22semidirect
5. C2×C11×D165C5semidirect
7. C2×C5×#10597243D3semidirect
9. C11×D165C2×C5semidirect
11. C2×C3×#10597243D5semidirect
13. C2×C3×C5×C112C2×C5semidirect
16. C5×#10597243C2×D3semidirect
19. C3×C5×C112C22×C5semidirect
21. C3×#10597243C2×D5semidirect
28. C2×C5×C112C5×D3semidirect
31. C2×#10597243D15semidirect
38. C2×C3×C112C5×D5semidirect
40. C2×D165C11⋊C5semidirect
44. C5×C112C2×C5×D3semidirect
46. #10597243C2×D15semidirect
53. C3×C112C2×C5×D5semidirect
55. C2×C3×C5×C11AGL(1,𝔽11)semidirect
56. C2×C3×C5×C11C2×C11⋊C5semidirect
57. D165C2×C11⋊C5semidirect
71. C2×C112C5×D15semidirect
78. C3×C5×C11C2×AGL(1,𝔽11)semidirect
79. C3×C5×C11C22×C11⋊C5semidirect
90. C112C2×C5×D15semidirect
92. C2×C5×C11D3×C11⋊C5semidirect
93. C2×C5×C11C3⋊AGL(1,𝔽11)semidirect
107. C2×C3×C11D5×C11⋊C5semidirect
108. C2×C3×C11C5⋊AGL(1,𝔽11)semidirect
114. C5×C11C2×D3×C11⋊C5semidirect
115. C5×C11C2×C3⋊AGL(1,𝔽11)semidirect
127. C3×C11C2×D5×C11⋊C5semidirect
128. C3×C11C2×C5⋊AGL(1,𝔽11)semidirect
131. C2×C3×C5#11925895semidirect
137. C2×C11D15×C11⋊C5semidirect
138. C2×C11C3⋊(C5⋊AGL(1,𝔽11))semidirect
144. C3×C5C2×#11925895semidirect
147. C11C2×C3⋊(C5⋊AGL(1,𝔽11))semidirect
148. C11C2×D15×C11⋊C5semidirect
152. C2×C5C3⋊#11925895semidirect
154. C2×C3C5⋊#11925895semidirect
158. C5C2×C3⋊#11925895semidirect
160. C3C2×C5⋊#11925895semidirect
162. C2C3⋊(C5⋊#11925895)direct (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×#105972432
C3⋊(C5⋊#11925895)2
C2×C5⋊#119258953
C2×C11×D1655
C2×C3⋊#119258955
C2×D15×C11⋊C511
C2×C3⋊(C5⋊AGL(1,𝔽11))11