Finite Group Databasebeta

← C5×ΓL(1,𝔽16)×C37⋊C3 #45686294

Subgroups — C5×ΓL(1,𝔽16)×C37⋊C3 #45686294

Order 33300.

124 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 42 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
1C3×C5×D5×C37⋊C3C211yesmaximal, characteristic
2C5×C37×ΓL(1,𝔽16)C311yesmaximal, characteristic, complemented
3C5×AGL(1,𝔽5)×C37⋊C3—31nomaximal
4C3×C52×C37⋊C3C411yescharacteristic, complemented, Hall
5ΓL(1,𝔽16)×C37⋊C3C511yesmaximal, characteristic, direct product factor
6C5×Dic3×C37⋊C3—51nomaximal
7C3×C5×D5×C37C2×C311yescharacteristic
8C5×D5×C37⋊C3—21no—
9C5×D5×C37⋊C3D311yescharacteristic
10C5×AGL(1,𝔽5)×C37—31noHall
11C2×C3×C5×C37⋊C3—51no—
12C3×D5×C37⋊C3C2×C511yescharacteristic
13C52×C37⋊C3—21no—
14C52×C37⋊C3Dic311yescharacteristic, complemented
15C3×C52×C37C3×C411yescharacteristic, complemented
16C37×ΓL(1,𝔽16)C3×C511yescharacteristic, complemented
17AGL(1,𝔽5)×C37⋊C3—31no—
18C5×Dic3×C37—51no—
19C4×C5×C37⋊C3—151no—
20C5×D5×C37C3×D311yescharacteristic
21C3×C5×C37⋊C3—41no—
22C3×C5×C37⋊C3C4×C511yescharacteristic, complemented
23C3×C5×C37⋊C3AGL(1,𝔽5)11yescharacteristic, complemented
24Dic3×C37⋊C3—51noHall
25C2×C3×C5×C37—51no—
26C3×D5×C37C2×C3×C511yescharacteristic
27C2×C5×C37⋊C3—51no—
28C2×C5×C37⋊C3—101no—
29D5×C37⋊C3—21no—
30D5×C37⋊C3C5×D311yescharacteristic
31C52×C37C3×Dic311yescharacteristic, complemented, Hall
32C3×C5×ΓL(1,𝔽16)—371nomaximal, Hall
33AGL(1,𝔽5)×C37—31no—
34C4×C5×C37—151no—
35C2×C3×C37⋊C3—51no—
36C3×C5×C37—41no—
37C3×C5×C37C3×C4×C511yescharacteristic, complemented
38C3×C5×C37C3×AGL(1,𝔽5)11yescharacteristic, complemented
39C5×C37⋊C3—21no—
40C5×C37⋊C3—21no—
41C5×C37⋊C3—42no—
42C5×C37⋊C3—41no—
43C5×C37⋊C3C5×Dic311yescharacteristic, complemented
44C5×C37⋊C3ΓL(1,𝔽16)11yescharacteristic, direct product factor
45C32×C5×D5—371no—
46Dic3×C37—51no—
47C4×C37⋊C3—151no—
48C2×C5×C37—51no—
49D5×C37C3×C5×D311yescharacteristic
50C3×C37⋊C3C5×AGL(1,𝔽5)11yescharacteristic, complemented, Hall
51C5×ΓL(1,𝔽16)C37⋊C311yescharacteristic, direct product factor
52C3×C5×AGL(1,𝔽5)—1111no—
53C32×C52—371noHall
54C2×C3×C37—51no—
55C2×C37⋊C3—51no—
56C2×C37⋊C3—101no—
57C5×C37—41no—
58C5×C37C3×C5×Dic311yescharacteristic, complemented
59C5×C37C3×ΓL(1,𝔽16)11yescharacteristic, complemented
60C3×ΓL(1,𝔽16)—371no—
61C3×C5×Dic3—1851no—
62C3×C5×D5—371no—
63C3×C5×D5—741no—
64C3×C5×D5C2×C37⋊C311yescharacteristic
65C4×C37—151noHall
66C3×C37C3×C5×AGL(1,𝔽5)11yescharacteristic, complemented
67C37⋊C3—21no—
68C37⋊C3C5×ΓL(1,𝔽16)11yescharacteristic, direct product factor
69C5×AGL(1,𝔽5)—31noHall
70C2×C32×C5—1851no—
71C32×D5—371no—
72C3×C52—371no—
73C3×C52—741no—
74C3×C52C4×C37⋊C311yescharacteristic, complemented
75C2×C37—51no—
76ΓL(1,𝔽16)C5×C37⋊C311yescharacteristic, direct product factor
77C3×AGL(1,𝔽5)—1111no—
78C3×C4×C5—5551no—
79C5×Dic3—51no—
80C5×D5D3×C37⋊C311yescharacteristic
81C32×C5—371no—
82C32×C5—371no—
83C32×C5—1481no—
84C37C3×C5×ΓL(1,𝔽16)11yescharacteristic, complemented, Sylow
85C3×Dic3—1851noHall
86C2×C3×C5—51no—
87C2×C3×C5—3701no—
88C2×C3×C5—1851no—
89C3×D5—371no—
90C3×D5—741no—
91C3×D5C2×C5×C37⋊C311yescharacteristic
92C52Dic3×C37⋊C311yescharacteristic, complemented, Sylow
93AGL(1,𝔽5)—31no—
94C4×C5—151no—
95C2×C32—1851no—
96C3×C5—41no—
97C3×C5—371no—
98C3×C5—371no—
99C3×C5—741no—
100C3×C5—741no—
101C3×C5—1481no—
102C3×C5—1482no—
103C3×C5C4×C5×C37⋊C311yescharacteristic, complemented
104C3×C5AGL(1,𝔽5)×C37⋊C311yescharacteristic, complemented
105C3×C4—5551no—
106Dic3—51no—
107C2×C5—51no—
108D5C5×D3×C37⋊C311yescharacteristic
109C32—371noSylow
110C2×C3—51no—
111C2×C3—3701no—
112C2×C3—1851no—
113C5—41no—
114C5C5×Dic3×C37⋊C311yescharacteristic, complemented
115C5ΓL(1,𝔽16)×C37⋊C311yescharacteristic, direct product factor
116C4—151noSylow
117C3—371no—
118C3—741no—
119C3C5×AGL(1,𝔽5)×C37⋊C311yescharacteristic, complemented
120C2—51no—
subgroupG/H
center, hypercenterC5ΓL(1,𝔽16)×C37⋊C3
commutator, hypocenterC3×C5×C37C3×C4×C5
FrattiniC1G
Fitting, socleC3×C52×C37C3×C4
prime radicalC3×C5×D5×C37⋊C3C2
cosocleC3×D5×C37C2×C3×C5
LCSG ▷ C3×C5×C37
UCSC1 ◁ C5

Complements

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

NH ≅ G/Ntype
2. C5×C37×ΓL(1,𝔽16)C3semidirect
4. C3×C52×C37⋊C3C4semidirect
5. ΓL(1,𝔽16)×C37⋊C3C5direct (N × H)
14. C52×C37⋊C3Dic3semidirect
15. C3×C52×C37C3×C4semidirect
16. C37×ΓL(1,𝔽16)C3×C5semidirect
22. C3×C5×C37⋊C3C4×C5semidirect
23. C3×C5×C37⋊C3AGL(1,𝔽5)semidirect
31. C52×C37C3×Dic3semidirect
37. C3×C5×C37C3×C4×C5semidirect
38. C3×C5×C37C3×AGL(1,𝔽5)semidirect
43. C5×C37⋊C3C5×Dic3semidirect
44. C5×C37⋊C3ΓL(1,𝔽16)direct (N × H)
50. C3×C37⋊C3C5×AGL(1,𝔽5)semidirect
51. C5×ΓL(1,𝔽16)C37⋊C3direct (N × H)
58. C5×C37C3×C5×Dic3semidirect
59. C5×C37C3×ΓL(1,𝔽16)semidirect
66. C3×C37C3×C5×AGL(1,𝔽5)semidirect
68. C37⋊C3C5×ΓL(1,𝔽16)direct (N × H)
74. C3×C52C4×C37⋊C3semidirect
76. ΓL(1,𝔽16)C5×C37⋊C3direct (N × H)
84. C37C3×C5×ΓL(1,𝔽16)semidirect
92. C52Dic3×C37⋊C3semidirect
103. C3×C5C4×C5×C37⋊C3semidirect
104. C3×C5AGL(1,𝔽5)×C37⋊C3semidirect
114. C5C5×Dic3×C37⋊C3semidirect
115. C5ΓL(1,𝔽16)×C37⋊C3direct (N × H)
119. C3C5×AGL(1,𝔽5)×C37⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×C5×D5×C37⋊C32
C5×C37×ΓL(1,𝔽16)3
C5×AGL(1,𝔽5)×C37⋊C33
ΓL(1,𝔽16)×C37⋊C35
C5×Dic3×C37⋊C35
C3×C5×ΓL(1,𝔽16)37