Finite Group Databasebeta

← C5×C11⋊C5×C3⋊AGL(1,𝔽7) #46831543

Subgroups — C5×C11⋊C5×C3⋊AGL(1,𝔽7) #46831543

Order 34650.

288 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 80 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×ΓL(1,𝔽8)×C11⋊C5C211yesmaximal, characteristic, complemented, Hall
2C5×D21×C11⋊C5C311yesmaximal, characteristic, complemented
3C5×AGL(1,𝔽7)×C11⋊C5—31nomaximal
4C5×C11×C3⋊AGL(1,𝔽7)C511yesmaximal, characteristic, complemented
5C11⋊C5×C3⋊AGL(1,𝔽7)C515yesmaximal, direct product factor
6C3×C5×C7×C11⋊C5C2×C311yescharacteristic, complemented
7C5×ΓL(1,𝔽8)×C11⋊C5—21no—
8C5×ΓL(1,𝔽8)×C11⋊C5D311yescharacteristic, complemented
9C3×C5×D3×C11⋊C5—71nomaximal, Hall
10C5×D7×C11⋊C5—31noHall
11C3×C5×C11×ΓL(1,𝔽8)C2×C511yescharacteristic, complemented
12C3×ΓL(1,𝔽8)×C11⋊C5C2×C515yescomplemented
13C52×C3⋊AGL(1,𝔽7)—111nomaximal, Hall
14C32×C5×C11⋊C5—71noHall
15AGL(1,𝔽7)×C11⋊C5—35no—
16C5×C11×AGL(1,𝔽7)—31no—
17D21×C11⋊C5C3×C515yescomplemented
18C5×C11×D21C3×C511yescharacteristic, complemented
19C5×C7×C11⋊C5C3×D311yescharacteristic, complemented, Hall
20C2×C3×C5×C11⋊C5—211no—
21C5×D3×C11⋊C5—71no—
22C3×C52×ΓL(1,𝔽8)—111noHall
23C11×C3⋊AGL(1,𝔽7)C5211yescharacteristic, complemented, Hall
24C3×C5×C7×C11C2×C3×C511yescharacteristic, complemented
25C5×C11×ΓL(1,𝔽8)—21no—
26C5×C11×ΓL(1,𝔽8)C5×D311yescharacteristic, complemented
27C3×C7×C11⋊C5C2×C3×C515yescomplemented
28ΓL(1,𝔽8)×C11⋊C5—25no—
29ΓL(1,𝔽8)×C11⋊C5C5×D315yescomplemented
30C52×D21—111no—
31C52×AGL(1,𝔽7)—331no—
32C3×C5×D3×C11—71no—
33C3×D3×C11⋊C5—75no—
34C3×C5×C11⋊C5—71no—
35C3×C5×C11⋊C5—141no—
36C3×C5×C11⋊C5AGL(1,𝔽7)11yescharacteristic, complemented
37C5×C11×D7—31no—
38D7×C11⋊C5—35no—
39C3×C11×ΓL(1,𝔽8)C2×C5211yescharacteristic, complemented, Hall
40C5×C3⋊AGL(1,𝔽7)—115no—
41C5×C3⋊AGL(1,𝔽7)C11⋊C511yescharacteristic, direct product factor
42C2×C5×C11⋊C5—211noHall
43C3×C52×C7—111no—
44C52×ΓL(1,𝔽8)—111no—
45C52×ΓL(1,𝔽8)—221no—
46C32×C5×C11—71no—
47C32×C11⋊C5—75no—
48C11×D21C3×C5211yescharacteristic, complemented
49C11×AGL(1,𝔽7)—31no—
50C3×C52×D3—771noHall
51C5×C7×C11C3×C5×D311yescharacteristic, complemented
52C7×C11⋊C5C3×C5×D315yescomplemented
53C52×D7—331noHall
54C2×C3×C5×C11—211no—
55C5×D3×C11—71no—
56C2×C3×C11⋊C5—215no—
57D3×C11⋊C5—75no—
58C3×C5×ΓL(1,𝔽8)—115no—
59C3×C5×ΓL(1,𝔽8)C2×C11⋊C511yescharacteristic, complemented
60C5×C11⋊C5C3⋊AGL(1,𝔽7)11yescharacteristic, direct product factor, Hall
61C3×C7×C11C2×C3×C5211yescharacteristic, complemented
62C11×ΓL(1,𝔽8)—21no—
63C11×ΓL(1,𝔽8)C52×D311yescharacteristic, complemented
64C32×C52—771noHall
65C5×D21—115no—
66C5×D21C3×C11⋊C511yescharacteristic, complemented
67C5×AGL(1,𝔽7)—31no—
68C5×AGL(1,𝔽7)—335no—
69C3×D3×C11—71noHall
70C52×C7—111noHall
71C3×C5×C11—71no—
72C3×C5×C11—141no—
73C3×C5×C11C5×AGL(1,𝔽7)11yescharacteristic, complemented
74C3×C11⋊C5—75no—
75C3×C11⋊C5—145no—
76C3×C11⋊C5C5×AGL(1,𝔽7)15yescomplemented
77C11×D7—31noHall
78C2×C3×C52—2311no—
79C52×D3—771no—
80C3⋊AGL(1,𝔽7)C5×C11⋊C511yescharacteristic, direct product factor, Hall
81C2×C5×C11—211no—
82C2×C11⋊C5—215no—
83C3×C5×C7—115no—
84C3×C5×C7C2×C3×C11⋊C511yescharacteristic, complemented
85C5×ΓL(1,𝔽8)—21no—
86C5×ΓL(1,𝔽8)—115no—
87C5×ΓL(1,𝔽8)—225no—
88C5×ΓL(1,𝔽8)D3×C11⋊C511yescharacteristic, complemented
89C32×C11—71noHall
90C3×C5×D3—71no—
91C3×C5×D3—775no—
92C7×C11C3×C52×D311yescharacteristic, complemented, Hall
93C3×C52—111no—
94C3×C52—771no—
95C3×C52—1541no—
96C5×D7—31no—
97C5×D7—335no—
98C2×C3×C11—211no—
99D3×C11—71no—
100C3×ΓL(1,𝔽8)C2×C5×C11⋊C511yescharacteristic, complemented, Hall
101C5×C11C5×C3⋊AGL(1,𝔽7)11yescharacteristic, complemented
102C11⋊C5C5×C3⋊AGL(1,𝔽7)15yesdirect product factor
103C2×C52—2311noHall
104C32×C5—71no—
105C32×C5—775no—
106D21C3×C5×C11⋊C511yescharacteristic, complemented
107AGL(1,𝔽7)—31no—
108C5×C7—115no—
109C5×C7C3×D3×C11⋊C511yescharacteristic, complemented
110C3×C11—71no—
111C3×C11—141no—
112C3×C11C52×AGL(1,𝔽7)11yescharacteristic, complemented
113C2×C3×C5—211no—
114C2×C3×C5—2315no—
115C5×D3—71no—
116C5×D3—775no—
117C52—111noSylow
118C2×C11—211noHall
119C3×C7C2×C3×C5×C11⋊C511yescharacteristic, complemented
120ΓL(1,𝔽8)—21no—
121ΓL(1,𝔽8)C5×D3×C11⋊C511yescharacteristic, complemented
122C3×D3—71noHall
123C3×C5—71no—
124C3×C5—115no—
125C3×C5—141no—
126C3×C5—775no—
127C3×C5—1545no—
128C3×C5AGL(1,𝔽7)×C11⋊C511yescharacteristic, complemented
129D7—31noHall
130C11C52×C3⋊AGL(1,𝔽7)11yescharacteristic, complemented, Sylow
131C2×C5—211no—
132C2×C5—2315no—
133C32—71noSylow
134C7C3×C5×D3×C11⋊C511yescharacteristic, complemented, Sylow
135C2×C3—211no—
136D3—71no—
137C5—115no—
138C5C11⋊C5×C3⋊AGL(1,𝔽7)11yescharacteristic, direct product factor
139C3—71no—
140C3—141no—
141C3C5×AGL(1,𝔽7)×C11⋊C511yescharacteristic, complemented
142C2—211noSylow
subgroupG/H
center, hypercenterC5C11⋊C5×C3⋊AGL(1,𝔽7)
commutator, cosocle, hypocenterC3×C7×C11C2×C3×C52
FrattiniC1G
Fitting, socleC3×C5×C7×C11C2×C3×C5
LCSG ▷ C3×C7×C11
UCSC1 ◁ C5

Complements

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

NH ≅ G/Ntype
1. C3×C5×ΓL(1,𝔽8)×C11⋊C5C2semidirect
2. C5×D21×C11⋊C5C3semidirect
4. C5×C11×C3⋊AGL(1,𝔽7)C5semidirect
5. C11⋊C5×C3⋊AGL(1,𝔽7)C5direct (N × H)
6. C3×C5×C7×C11⋊C5C2×C3semidirect
8. C5×ΓL(1,𝔽8)×C11⋊C5D3semidirect
11. C3×C5×C11×ΓL(1,𝔽8)C2×C5semidirect
12. C3×ΓL(1,𝔽8)×C11⋊C5C2×C5semidirect
17. D21×C11⋊C5C3×C5semidirect
18. C5×C11×D21C3×C5semidirect
19. C5×C7×C11⋊C5C3×D3semidirect
23. C11×C3⋊AGL(1,𝔽7)C52semidirect
24. C3×C5×C7×C11C2×C3×C5semidirect
26. C5×C11×ΓL(1,𝔽8)C5×D3semidirect
27. C3×C7×C11⋊C5C2×C3×C5semidirect
29. ΓL(1,𝔽8)×C11⋊C5C5×D3semidirect
36. C3×C5×C11⋊C5AGL(1,𝔽7)semidirect
39. C3×C11×ΓL(1,𝔽8)C2×C52semidirect
41. C5×C3⋊AGL(1,𝔽7)C11⋊C5direct (N × H)
48. C11×D21C3×C52semidirect
51. C5×C7×C11C3×C5×D3semidirect
52. C7×C11⋊C5C3×C5×D3semidirect
59. C3×C5×ΓL(1,𝔽8)C2×C11⋊C5semidirect
60. C5×C11⋊C5C3⋊AGL(1,𝔽7)direct (N × H)
61. C3×C7×C11C2×C3×C52semidirect
63. C11×ΓL(1,𝔽8)C52×D3semidirect
66. C5×D21C3×C11⋊C5semidirect
73. C3×C5×C11C5×AGL(1,𝔽7)semidirect
76. C3×C11⋊C5C5×AGL(1,𝔽7)semidirect
80. C3⋊AGL(1,𝔽7)C5×C11⋊C5direct (N × H)
84. C3×C5×C7C2×C3×C11⋊C5semidirect
88. C5×ΓL(1,𝔽8)D3×C11⋊C5semidirect
92. C7×C11C3×C52×D3semidirect
100. C3×ΓL(1,𝔽8)C2×C5×C11⋊C5semidirect
101. C5×C11C5×C3⋊AGL(1,𝔽7)semidirect
102. C11⋊C5C5×C3⋊AGL(1,𝔽7)direct (N × H)
106. D21C3×C5×C11⋊C5semidirect
109. C5×C7C3×D3×C11⋊C5semidirect
112. C3×C11C52×AGL(1,𝔽7)semidirect
119. C3×C7C2×C3×C5×C11⋊C5semidirect
121. ΓL(1,𝔽8)C5×D3×C11⋊C5semidirect
128. C3×C5AGL(1,𝔽7)×C11⋊C5semidirect
130. C11C52×C3⋊AGL(1,𝔽7)semidirect
134. C7C3×C5×D3×C11⋊C5semidirect
138. C5C11⋊C5×C3⋊AGL(1,𝔽7)direct (N × H)
141. C3C5×AGL(1,𝔽7)×C11⋊C5semidirect

Maximal subgroups

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

subgroupindex
C3×C5×ΓL(1,𝔽8)×C11⋊C52
C5×D21×C11⋊C53
C5×AGL(1,𝔽7)×C11⋊C53
C5×C11×C3⋊AGL(1,𝔽7)5
C11⋊C5×C3⋊AGL(1,𝔽7)5
C3×C5×D3×C11⋊C57
C52×C3⋊AGL(1,𝔽7)11