Finite Group Databasebeta

← #49137617

Subgroups — #49137617

Order 36300.

160 conjugacy classes of subgroups; only the 28 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 28 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⋊D5×C112⋊C3C211yesmaximal, characteristic
2C11×C11⋊#1043—31nomaximal, Hall
3C52×C112⋊C3C411yescharacteristic, complemented, Hall
4#20300037—51nomaximal
5#20300037—53nomaximal
6#20300037—52nomaximal
7C112×C5⋊D5D311yescharacteristic
8D5×C112⋊C3—51no—
9D5×C112⋊C3—53no—
10D5×C112⋊C3—52no—
11C52×C112Dic311yescharacteristic, complemented, Hall
12C11×C11⋊AGL(1,𝔽5)—151no—
13C11×C11⋊AGL(1,𝔽5)—153no—
14C11×C11⋊AGL(1,𝔽5)—152no—
15C5×C112⋊C3AGL(1,𝔽5)11yescharacteristic, complemented
16C5×C112⋊C3AGL(1,𝔽5)13yescomplemented
17C5×C112⋊C3AGL(1,𝔽5)12yescomplemented
18#13089712—251noHall
19D5×C112—51no—
20D5×C112—53no—
21D5×C112—52no—
22C11⋊#1043—31no—
23C11×#1043—331no—
24C2×C112⋊C3—251no—
25C5×C112ΓL(1,𝔽16)11yescharacteristic, complemented
26C5×C112ΓL(1,𝔽16)13yescomplemented
27C5×C112ΓL(1,𝔽16)12yescomplemented
28C11×C5⋊D5—31no—
29C11×C5⋊D5—61no—
30C11×C5⋊D5—61no—
31C11×Dic11—751noHall
32C112⋊C3#104311yescharacteristic, complemented, Hall
33C3⋊#1043—1211nomaximal, Hall
34C52×C11—31no—
35C52×C11—61no—
36C52×C11—61no—
37C2×C112—251no—
38C11⋊AGL(1,𝔽5)—151no—
39C11⋊AGL(1,𝔽5)—153no—
40C11⋊AGL(1,𝔽5)—152no—
41C11×AGL(1,𝔽5)—1651no—
42C11×AGL(1,𝔽5)—1653no—
43C11×AGL(1,𝔽5)—1652no—
44C3×C5⋊D5—2421no—
45C112C3⋊#104311yescharacteristic, complemented, Sylow
46D5×C11—151no—
47D5×C11—153no—
48D5×C11—301no—
49D5×C11—303no—
50D5×C11—302no—
51D5×C11—301no—
52D5×C11—303no—
53D5×C11—302no—
54D5×C11—302no—
55#1043—331noHall
56C3×C52—2421noHall
57ΓL(1,𝔽16)—6051no—
58ΓL(1,𝔽16)—6053no—
59ΓL(1,𝔽16)—6052no—
60C5×C11—31no—
61C5×C11—33no—
62C5×C11—61no—
63C5×C11—63no—
64C5×C11—62no—
65C5×C11—61no—
66C5×C11—63no—
67C5×C11—62no—
68C5×C11—62no—
69C5⋊D5#1062439111yescharacteristic
70C4×C11—8251no—
71Dic11—751no—
72C3×D5—6051no—
73C3×D5—12103no—
74C3×D5—12102no—
75C52#1308971211yescharacteristic, complemented, Sylow
76C2×C11—1501no—
77C2×C11—1501no—
78C2×C11—1501no—
79AGL(1,𝔽5)—1651no—
80AGL(1,𝔽5)—1653no—
81AGL(1,𝔽5)—1652no—
82C3×C5—1211no—
83C3×C5—2423no—
84C3×C5—2422no—
85Dic3—30251noHall
86C11—61no—
87C11—61no—
88C11—61no—
89D5—51no—
90D5—53no—
91D5—52no—
92C2×C3—60501no—
93C5#2030003711yescharacteristic, complemented
94C5#2030003713yescomplemented
95C5#2030003712yescomplemented
96C4—8251noSylow
97C3—2421noSylow
98C2—251no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, hypocenterC52×C112⋊C3C4
Fitting, socleC52×C112Dic3
prime radical, cosocleC5⋊D5×C112⋊C3C2
LCSG ▷ C52×C112⋊C3
UCSC1

Complements

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

NH ≅ G/Ntype
3. C52×C112⋊C3C4semidirect
11. C52×C112Dic3semidirect
15. C5×C112⋊C3AGL(1,𝔽5)semidirect
16. C5×C112⋊C3AGL(1,𝔽5)semidirect
17. C5×C112⋊C3AGL(1,𝔽5)semidirect
25. C5×C112ΓL(1,𝔽16)semidirect
26. C5×C112ΓL(1,𝔽16)semidirect
27. C5×C112ΓL(1,𝔽16)semidirect
32. C112⋊C3#1043semidirect
45. C112C3⋊#1043semidirect
75. C52#13089712semidirect
93. C5#20300037semidirect
94. C5#20300037semidirect
95. C5#20300037semidirect

Maximal subgroups

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

subgroupindex
C5⋊D5×C112⋊C32
C11×C11⋊#10433
#203000375
C3⋊#1043121