Finite Group Databasebeta

← C5×#20300024 #49137418

Subgroups — C5×#20300024 #49137418

Order 36300.

168 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 26 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
1C5×C3⋊#11925890C211yesmaximal, characteristic, complemented
2C5×C3⋊#11925890C211yesmaximal, characteristic, complemented
3C3×C5×#11925893C211yesmaximal, characteristic, complemented
4C5×#11925890⋊C2—31nomaximal, Hall
5C3×C5×#10597245C2211yescharacteristic, complemented, Hall
6C5×D33⋊D11—51nomaximal
7#20300024C511yesmaximal, characteristic, direct product factor
8C5×#11925890—31no—
9C5×#11925890—31no—
10C5×#11925893D311yescharacteristic, complemented
11C5×C11×D33—51no—
12C5×C11×D33—51no—
13C3×C5×C11⋊D11D511yescharacteristic, complemented
14C3⋊#11925890C2×C511yescharacteristic, complemented
15C3⋊#11925890C2×C511yescharacteristic, complemented
16C3×#11925893—21no—
17C3×#11925893—21no—
18C3×#11925893C2×C511yescharacteristic, complemented
19C5×#10597245C2×D311yescharacteristic, complemented, Hall
20C5×D112—151no—
21#11925890⋊C2—31no—
22C3×C5×C112C2×D511yescharacteristic, complemented
23C3×#10597245—21no—
24C3×#10597245—21no—
25C3×#10597245C22×C511yescharacteristic, complemented
26C3×C5×AGL(1,𝔽11)—221no—
27D33⋊D11—51noHall
28C5×C11⋊D11D1511yescharacteristic, complemented
29C5×C11×D11—151no—
30C5×C11×D11—151no—
31#11925893—21no—
32#11925893—21no—
33#11925893C5×D311yescharacteristic, complemented
34#11925890—31no—
35#11925890—31no—
36C3×C5×C11⋊C5—441no—
37C11×D33—51no—
38C11×D33—51no—
39C3×C11⋊D11C5×D511yescharacteristic, complemented
40C5×D3×D11—551no—
41C5×D3×D11—551no—
42C5×C112C2×D1511yescharacteristic, complemented
43#10597245—21no—
44#10597245—21no—
45#10597245C2×C5×D311yescharacteristic, complemented
46C5×AGL(1,𝔽11)—221no—
47D112—151noHall
48C3×C112C2×C5×D511yescharacteristic, complemented, Hall
49C5×D33—51no—
50C5×D33—51no—
51C5×D3×C11—551no—
52C5×D3×C11—551no—
53C3×C5×D11—221no—
54C3×C5×D11—1101no—
55C3×C5×D11—1101no—
56C3×AGL(1,𝔽11)—221no—
57C3×AGL(1,𝔽11)—221no—
58C3×AGL(1,𝔽11)—221no—
59C3×AGL(1,𝔽11)—222no—
60C2×C5×D15—1211nomaximal, Hall
61C5×C11⋊C5—441no—
62C11⋊D11C5×D1511yescharacteristic, complemented
63C11×D11—151no—
64C11×D11—151no—
65C2×C5×D11—1651no—
66C2×C5×D11—1651no—
67C3×C5×C11—21no—
68C3×C5×C11—101no—
69C3×C5×C11—101no—
70C3×C11⋊C5—221no—
71C3×C11⋊C5—221no—
72C3×C11⋊C5—441no—
73C3×C11⋊C5—442no—
74C2×C3×C52—1211no—
75C5×D15—1211no—
76C5×D15—1211no—
77D3×D11—551no—
78D3×D11—551no—
79C112C2×C5×D1511yescharacteristic, complemented, Sylow
80C2×C5×C11—1651no—
81C2×C5×C11—1651no—
82C5×D11—151no—
83C5×D11—151no—
84C5×D11—221no—
85C5×D11—1101no—
86C5×D11—1101no—
87AGL(1,𝔽11)—221no—
88AGL(1,𝔽11)—221no—
89AGL(1,𝔽11)—221no—
90AGL(1,𝔽11)—222no—
91C2×C5×D5—3631noHall
92C3×C52—4841noHall
93D33—51no—
94D33—51no—
95C3×D11—221no—
96C3×D11—1101no—
97C3×D11—1101no—
98D3×C11—551no—
99D3×C11—551no—
100C2×D15—1211no—
101C2×C5×D3—6051no—
102C5×C11—21no—
103C5×C11—101no—
104C5×C11—101no—
105C11⋊C5—441no—
106C11⋊C5—441no—
107C11⋊C5—442no—
108C11⋊C5—441no—
109C2×C52—1211no—
110C5×D5—3631no—
111C5×D5—3631no—
112C2×D11—1651no—
113C2×D11—1651no—
114C3×C11—21no—
115C3×C11—101no—
116C3×C11—101no—
117C2×C3×C5—1211no—
118C2×C3×C5—1211no—
119C2×C3×C5—2421no—
120C2×C3×C5—2421no—
121D15—2421no—
122D15—2421no—
123C5×D3—551no—
124C5×D3—551no—
125C52—1211noSylow
126C2×C11—1651no—
127C2×C11—1651no—
128D11—151no—
129D11—151no—
130D11—221no—
131D11—551no—
132D11—1101no—
133C22×C5—18151no—
134C2×D5—3631no—
135C3×C5—4841no—
136C3×C5—4841no—
137C3×C5—4841no—
138C3×C5#11925890⋊C211yescharacteristic, complemented
139C2×D3—6051noHall
140C11—21no—
141C11—51no—
142C11—101no—
143C2×C5—1211no—
144C2×C5—1211no—
145C2×C5—1651no—
146C2×C5—1651no—
147C2×C5—2421no—
148C2×C5—2421no—
149D5—3631no—
150D5—7261no—
151C2×C3—1211no—
152D3—551no—
153D3—551no—
154C5—4841no—
155C5—4841no—
156C5—4841no—
157C5#2030002411yescharacteristic, direct product factor
158C22—18151noSylow
159C3C5×#11925890⋊C211yescharacteristic, complemented, Sylow
160C2—1211no—
161C2—1651no—
162C2—1651no—
subgroupG/H
center, hypercenterC5#20300024
commutator, cosocle, hypocenterC3×#10597245C22×C5
FrattiniC1G
Fitting, socleC3×C5×C112C2×D5
LCSG ▷ C3×#10597245
UCSC1 ◁ C5

Complements

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

NH ≅ G/Ntype
1. C5×C3⋊#11925890C2semidirect
2. C5×C3⋊#11925890C2semidirect
3. C3×C5×#11925893C2semidirect
5. C3×C5×#10597245C22semidirect
7. #20300024C5direct (N × H)
10. C5×#11925893D3semidirect
13. C3×C5×C11⋊D11D5semidirect
14. C3⋊#11925890C2×C5semidirect
15. C3⋊#11925890C2×C5semidirect
18. C3×#11925893C2×C5semidirect
19. C5×#10597245C2×D3semidirect
22. C3×C5×C112C2×D5semidirect
25. C3×#10597245C22×C5semidirect
28. C5×C11⋊D11D15semidirect
33. #11925893C5×D3semidirect
39. C3×C11⋊D11C5×D5semidirect
42. C5×C112C2×D15semidirect
45. #10597245C2×C5×D3semidirect
48. C3×C112C2×C5×D5semidirect
62. C11⋊D11C5×D15semidirect
79. C112C2×C5×D15semidirect
138. C3×C5#11925890⋊C2semidirect
157. C5#20300024direct (N × H)
159. C3C5×#11925890⋊C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C5×C3⋊#119258902
C3×C5×#119258932
C5×#11925890⋊C23
C5×D33⋊D115
#203000245
C2×C5×D15121