Finite Group Databasebeta

← C13×AGL(1,𝔽7)×D33 #49114724

Subgroups — C13×AGL(1,𝔽7)×D33 #49114724

Order 36036.

176 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 54 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
1C13×ΓL(1,𝔽8)×D33C211yesmaximal, characteristic, complemented
2C13×C3⋊(C11⋊AGL(1,𝔽7))C211yesmaximal, characteristic, complemented
3C3×C11×C13×AGL(1,𝔽7)C211yesmaximal, characteristic, complemented
4C13×D7×D33C311yesmaximal, characteristic, complemented
5C13×D11×AGL(1,𝔽7)—31nomaximal
6C3×C11×C13×ΓL(1,𝔽8)C2211yescharacteristic, complemented, Hall
7C13×C11⋊AGL(1,𝔽7)—31no—
8C11×C13×AGL(1,𝔽7)—21no—
9C11×C13×AGL(1,𝔽7)D311yescharacteristic, complemented
10C13×ΓL(1,𝔽8)×D11—31no—
11C13×D231C2×C311yescharacteristic, complemented
12C7×C13×D33C2×C311yescharacteristic, complemented
13C3×C11×C13×D7C2×C311yescharacteristic, complemented
14C2×C3×C13×D33—71nomaximal, Hall
15C13×D7×D11—31noHall
16D3×C13×AGL(1,𝔽7)—111nomaximal, Hall
17C11×C13×ΓL(1,𝔽8)—21no—
18C11×C13×ΓL(1,𝔽8)C2×D311yescharacteristic, complemented
19C3×C7×C11×C13C22×C311yescharacteristic, complemented
20AGL(1,𝔽7)×D33C1311yesmaximal, characteristic, direct product factor, Hall
21C3×C13×D33—71no—
22C3×C13×D33—71no—
23C2×C32×C11×C13—71no—
24C13×D77—31no—
25C11×C13×D7C3×D311yescharacteristic, complemented
26C7×C13×D11—31no—
27C2×C13×D33—71no—
28C2×C3×C13×D11—211no—
29C13×C3⋊AGL(1,𝔽7)—111no—
30D3×C13×ΓL(1,𝔽8)—111no—
31C3×C13×AGL(1,𝔽7)D1111yescharacteristic, complemented
32C3⋊(C11⋊AGL(1,𝔽7))C2×C1311yescharacteristic, complemented
33ΓL(1,𝔽8)×D33C2×C1311yescharacteristic, complemented
34C3×C11×AGL(1,𝔽7)C2×C1311yescharacteristic, complemented
35C32×C11×C13—71noHall
36D3×C13×D7—111no—
37C2×C13×AGL(1,𝔽7)—331no—
38C7×C11×C13C2×C3×D311yescharacteristic, complemented, Hall
39D7×D33C3×C1311yescharacteristic, complemented
40D11×AGL(1,𝔽7)—31no—
41C2×C3×C11×C13—71no—
42C2×C3×C11×C13—71no—
43C2×C3×C11×C13—141no—
44C13×D33—71no—
45C13×D33AGL(1,𝔽7)11yescharacteristic, direct product factor
46C3×C13×D11—211no—
47C3×C13×D11—211no—
48C3×C13×ΓL(1,𝔽8)C2×D1111yescharacteristic, complemented, Hall
49C3×C11×ΓL(1,𝔽8)C22×C1311yescharacteristic, complemented, Hall
50C2×C13×D11—211noHall
51C13×D21—111no—
52D3×C7×C13—111no—
53C3×C13×D7C3×D1111yescharacteristic, complemented
54C2×C13×ΓL(1,𝔽8)—331no—
55C13×AGL(1,𝔽7)—21no—
56C13×AGL(1,𝔽7)—331no—
57C13×AGL(1,𝔽7)D3311yescharacteristic, direct product factor
58C2×C3×D3×C13—771noHall
59D231C2×C3×C1311yescharacteristic, complemented
60C7×D33C2×C3×C1311yescharacteristic, complemented
61C3×C11×D7C2×C3×C1311yescharacteristic, complemented
62C11⋊AGL(1,𝔽7)—31no—
63C11×AGL(1,𝔽7)—21no—
64C11×AGL(1,𝔽7)D3×C1311yescharacteristic, complemented
65ΓL(1,𝔽8)×D11—31no—
66C3×C11×C13—71no—
67C3×C11×C13—141no—
68C3×C11×C13C2×AGL(1,𝔽7)11yescharacteristic, complemented
69C2×C3×D33—71noHall
70C2×C13×D7—331noHall
71D7×D11—31noHall
72C2×C11×C13—71no—
73C13×D11—31no—
74C13×D11—211no—
75C3×C7×C13C2×C3×D1111yescharacteristic, complemented
76C13×ΓL(1,𝔽8)—21no—
77C13×ΓL(1,𝔽8)C2×D3311yescharacteristic, complemented
78D3×AGL(1,𝔽7)—111noHall
79C2×C32×C13—71no—
80C3×D3×C13—771no—
81C3×D3×C13—771no—
82C3×C7×C11C22×C3×C1311yescharacteristic, complemented
83C11×ΓL(1,𝔽8)—21no—
84C11×ΓL(1,𝔽8)C2×D3×C1311yescharacteristic, complemented
85C2×C32×C11—71no—
86C3×D33—71no—
87C3×D33—71no—
88C2×C7×C13—331no—
89C13×D7—331no—
90C13×D7C3×D3311yescharacteristic, complemented
91C22×C3×C13—2311no—
92C2×D3×C13—771no—
93D77—31no—
94C7×D11—31no—
95C11×D7C3×D3×C1311yescharacteristic, complemented
96C11×C13D3×AGL(1,𝔽7)11yescharacteristic, complemented, Hall
97C2×D33—71no—
98C2×C3×D11—211no—
99C3⋊AGL(1,𝔽7)—111no—
100D3×ΓL(1,𝔽8)—111no—
101C3×AGL(1,𝔽7)C13×D1111yescharacteristic, complemented
102C32×C13—71noHall
103C32×C11—71noHall
104C7×C13C2×C3×D3311yescharacteristic, complemented, Hall
105D3×D7—111no—
106C2×AGL(1,𝔽7)—331no—
107C2×C3×C13—71no—
108C2×C3×C13—71no—
109C2×C3×C13—141no—
110C2×C3×C13—2311no—
111C2×C3×C13—2311no—
112D3×C13—111no—
113D3×C13—771no—
114C7×C11C2×C3×D3×C1311yescharacteristic, complemented, Hall
115C2×C3×C11—71no—
116C2×C3×C11—71no—
117C2×C3×C11—141no—
118D33—71no—
119D33C13×AGL(1,𝔽7)11yescharacteristic, direct product factor
120C3×D11—211no—
121C3×D11—211no—
122C3×ΓL(1,𝔽8)C2×C13×D1111yescharacteristic, complemented, Hall
123C22×C13—2311noHall
124C2×D11—211noHall
125D21—111no—
126C3×D7C3×C13×D1111yescharacteristic, complemented
127D3×C7—111no—
128C2×ΓL(1,𝔽8)—331no—
129AGL(1,𝔽7)—21no—
130AGL(1,𝔽7)—331no—
131AGL(1,𝔽7)C13×D3311yescharacteristic, direct product factor
132C3×C13—71no—
133C3×C13—141no—
134C3×C13D11×AGL(1,𝔽7)11yescharacteristic, complemented
135C2×C3×D3—771noHall
136C3×C11—71no—
137C3×C11—141no—
138C3×C11C2×C13×AGL(1,𝔽7)11yescharacteristic, complemented
139C2×D7—331noHall
140C2×C13—71no—
141C2×C13—331no—
142C2×C13—2311no—
143C2×C11—71no—
144D11—31no—
145D11—211no—
146C3×C7C2×C3×C13×D1111yescharacteristic, complemented
147ΓL(1,𝔽8)—21no—
148ΓL(1,𝔽8)C2×C13×D3311yescharacteristic, complemented
149C2×C32—71no—
150C3×D3—771no—
151C3×D3—771no—
152C2×C7—331no—
153D7—331no—
154D7C3×C13×D3311yescharacteristic, complemented
155C13AGL(1,𝔽7)×D3311yescharacteristic, direct product factor, Sylow
156C22×C3—2311no—
157C2×D3—771no—
158C11D3×C13×AGL(1,𝔽7)11yescharacteristic, complemented, Sylow
159C32—71noSylow
160C7C2×C3×C13×D3311yescharacteristic, complemented, Sylow
161C2×C3—71no—
162C2×C3—71no—
163C2×C3—141no—
164C2×C3—2311no—
165C2×C3—2311no—
166D3—111no—
167D3—771no—
168C22—2311noSylow
169C3—71no—
170C3—141no—
171C3C13×D11×AGL(1,𝔽7)11yescharacteristic, complemented
172C2—71no—
173C2—331no—
174C2—2311no—
subgroupG/H
center, hypercenterC13AGL(1,𝔽7)×D33
commutator, cosocle, hypocenterC3×C7×C11C22×C3×C13
FrattiniC1G
Fitting, socleC3×C7×C11×C13C22×C3
LCSG ▷ C3×C7×C11
UCSC1 ◁ C13

Complements

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

NH ≅ G/Ntype
1. C13×ΓL(1,𝔽8)×D33C2semidirect
2. C13×C3⋊(C11⋊AGL(1,𝔽7))C2semidirect
3. C3×C11×C13×AGL(1,𝔽7)C2semidirect
4. C13×D7×D33C3semidirect
6. C3×C11×C13×ΓL(1,𝔽8)C22semidirect
9. C11×C13×AGL(1,𝔽7)D3semidirect
11. C13×D231C2×C3semidirect
12. C7×C13×D33C2×C3semidirect
13. C3×C11×C13×D7C2×C3semidirect
18. C11×C13×ΓL(1,𝔽8)C2×D3semidirect
19. C3×C7×C11×C13C22×C3semidirect
20. AGL(1,𝔽7)×D33C13direct (N × H)
25. C11×C13×D7C3×D3semidirect
31. C3×C13×AGL(1,𝔽7)D11semidirect
32. C3⋊(C11⋊AGL(1,𝔽7))C2×C13semidirect
33. ΓL(1,𝔽8)×D33C2×C13semidirect
34. C3×C11×AGL(1,𝔽7)C2×C13semidirect
38. C7×C11×C13C2×C3×D3semidirect
39. D7×D33C3×C13semidirect
45. C13×D33AGL(1,𝔽7)direct (N × H)
48. C3×C13×ΓL(1,𝔽8)C2×D11semidirect
49. C3×C11×ΓL(1,𝔽8)C22×C13semidirect
53. C3×C13×D7C3×D11semidirect
57. C13×AGL(1,𝔽7)D33direct (N × H)
59. D231C2×C3×C13semidirect
60. C7×D33C2×C3×C13semidirect
61. C3×C11×D7C2×C3×C13semidirect
64. C11×AGL(1,𝔽7)D3×C13semidirect
68. C3×C11×C13C2×AGL(1,𝔽7)semidirect
75. C3×C7×C13C2×C3×D11semidirect
77. C13×ΓL(1,𝔽8)C2×D33semidirect
82. C3×C7×C11C22×C3×C13semidirect
84. C11×ΓL(1,𝔽8)C2×D3×C13semidirect
90. C13×D7C3×D33semidirect
95. C11×D7C3×D3×C13semidirect
96. C11×C13D3×AGL(1,𝔽7)semidirect
101. C3×AGL(1,𝔽7)C13×D11semidirect
104. C7×C13C2×C3×D33semidirect
114. C7×C11C2×C3×D3×C13semidirect
119. D33C13×AGL(1,𝔽7)direct (N × H)
122. C3×ΓL(1,𝔽8)C2×C13×D11semidirect
126. C3×D7C3×C13×D11semidirect
131. AGL(1,𝔽7)C13×D33direct (N × H)
134. C3×C13D11×AGL(1,𝔽7)semidirect
138. C3×C11C2×C13×AGL(1,𝔽7)semidirect
146. C3×C7C2×C3×C13×D11semidirect
148. ΓL(1,𝔽8)C2×C13×D33semidirect
154. D7C3×C13×D33semidirect
155. C13AGL(1,𝔽7)×D33direct (N × H)
158. C11D3×C13×AGL(1,𝔽7)semidirect
160. C7C2×C3×C13×D33semidirect
171. C3C13×D11×AGL(1,𝔽7)semidirect

Maximal subgroups

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

subgroupindex
C13×ΓL(1,𝔽8)×D332
C13×C3⋊(C11⋊AGL(1,𝔽7))2
C3×C11×C13×AGL(1,𝔽7)2
C13×D7×D333
C13×D11×AGL(1,𝔽7)3
C2×C3×C13×D337
D3×C13×AGL(1,𝔽7)11
AGL(1,𝔽7)×D3313