Finite Group Databasebeta

← D3×C11×C13×AGL(1,𝔽7) #49114712

Subgroups — D3×C11×C13×AGL(1,𝔽7) #49114712

Order 36036.

176 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 68 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
1D3×C11×C13×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
2C11×C13×C3⋊AGL(1,𝔽7)C211yesmaximal, characteristic, complemented
3C3×C11×C13×AGL(1,𝔽7)C211yesmaximal, characteristic, complemented
4D3×C11×C13×D7C311yesmaximal, characteristic, complemented
5C2×C11×C13×AGL(1,𝔽7)—31nomaximal
6C3×C11×C13×ΓL(1,𝔽8)C2211yescharacteristic, complemented, Hall
7C11×C13×AGL(1,𝔽7)—21no—
8C11×C13×AGL(1,𝔽7)—31no—
9C11×C13×AGL(1,𝔽7)D311yescharacteristic, direct product factor
10C11×C13×D21C2×C311yescharacteristic, complemented
11D3×C7×C11×C13C2×C311yescharacteristic, complemented
12C3×C11×C13×D7C2×C311yescharacteristic, complemented
13C2×C11×C13×ΓL(1,𝔽8)—31no—
14C2×C3×D3×C11×C13—71nomaximal, Hall
15C2×C11×C13×D7—31noHall
16D3×C13×AGL(1,𝔽7)C1111yesmaximal, characteristic, direct product factor, Hall
17C11×C13×ΓL(1,𝔽8)—21no—
18C11×C13×ΓL(1,𝔽8)C2×D311yescharacteristic, complemented
19C3×C7×C11×C13C22×C311yescharacteristic, complemented
20D3×C11×AGL(1,𝔽7)C1311yesmaximal, characteristic, direct product factor, Hall
21C3×D3×C11×C13—71no—
22C3×D3×C11×C13—71no—
23C2×C32×C11×C13—71no—
24C11×C13×D7—31no—
25C11×C13×D7C3×D311yescharacteristic, complemented
26C2×C7×C11×C13—31no—
27C22×C3×C11×C13—211no—
28C2×D3×C11×C13—71no—
29C13×C3⋊AGL(1,𝔽7)C2×C1111yescharacteristic, complemented
30D3×C13×ΓL(1,𝔽8)C2×C1111yescharacteristic, complemented
31C3×C13×AGL(1,𝔽7)C2×C1111yescharacteristic, complemented
32C11×C3⋊AGL(1,𝔽7)C2×C1311yescharacteristic, complemented
33D3×C11×ΓL(1,𝔽8)C2×C1311yescharacteristic, complemented
34C3×C11×AGL(1,𝔽7)C2×C1311yescharacteristic, complemented
35C32×C11×C13—71noHall
36D3×C13×D7C3×C1111yescharacteristic, complemented
37C2×C13×AGL(1,𝔽7)—31no—
38C7×C11×C13C2×C3×D311yescharacteristic, complemented, Hall
39D3×C11×D7C3×C1311yescharacteristic, complemented
40C2×C11×AGL(1,𝔽7)—31no—
41C2×C3×C11×C13—71no—
42C2×C3×C11×C13—71no—
43C2×C3×C11×C13—141no—
44C2×C3×C11×C13—211no—
45C2×C3×C11×C13—211no—
46D3×C11×C13—71no—
47D3×C11×C13AGL(1,𝔽7)11yescharacteristic, direct product factor
48C3×C13×ΓL(1,𝔽8)C22×C1111yescharacteristic, complemented, Hall
49C3×C11×ΓL(1,𝔽8)C22×C1311yescharacteristic, complemented, Hall
50C22×C11×C13—211noHall
51C13×D21C2×C3×C1111yescharacteristic, complemented
52D3×C7×C13C2×C3×C1111yescharacteristic, complemented
53C3×C13×D7C2×C3×C1111yescharacteristic, complemented
54C2×C13×ΓL(1,𝔽8)—31no—
55C13×AGL(1,𝔽7)—21no—
56C13×AGL(1,𝔽7)—31no—
57C13×AGL(1,𝔽7)D3×C1111yescharacteristic, direct product factor
58C2×C3×D3×C13—71noHall
59C11×D21C2×C3×C1311yescharacteristic, complemented
60D3×C7×C11C2×C3×C1311yescharacteristic, complemented
61C3×C11×D7C2×C3×C1311yescharacteristic, complemented
62C2×C11×ΓL(1,𝔽8)—31no—
63C11×AGL(1,𝔽7)—21no—
64C11×AGL(1,𝔽7)—31no—
65C11×AGL(1,𝔽7)D3×C1311yescharacteristic, direct product factor
66C3×C11×C13—71no—
67C3×C11×C13—141no—
68C3×C11×C13C2×AGL(1,𝔽7)11yescharacteristic, complemented
69C2×C3×D3×C11—71noHall
70C2×C13×D7—31noHall
71C2×C11×D7—31noHall
72C2×C11×C13—31no—
73C2×C11×C13—71no—
74C2×C11×C13—211no—
75C3×C7×C13C22×C3×C1111yescharacteristic, complemented
76C13×ΓL(1,𝔽8)—21no—
77C13×ΓL(1,𝔽8)C2×D3×C1111yescharacteristic, complemented
78D3×AGL(1,𝔽7)C11×C1311yescharacteristic, direct product factor, Hall
79C2×C32×C13—71no—
80C3×D3×C13—71no—
81C3×D3×C13—71no—
82C3×C7×C11C22×C3×C1311yescharacteristic, complemented
83C11×ΓL(1,𝔽8)—21no—
84C11×ΓL(1,𝔽8)C2×D3×C1311yescharacteristic, complemented
85C2×C32×C11—71no—
86C3×D3×C11—71no—
87C3×D3×C11—71no—
88C2×C7×C13—31no—
89C13×D7—31no—
90C13×D7C3×D3×C1111yescharacteristic, complemented
91C22×C3×C13—211no—
92C2×D3×C13—71no—
93C2×C7×C11—31no—
94C11×D7—31no—
95C11×D7C3×D3×C1311yescharacteristic, complemented
96C11×C13D3×AGL(1,𝔽7)11yescharacteristic, direct product factor, Hall
97C22×C3×C11—211no—
98C2×D3×C11—71no—
99C3⋊AGL(1,𝔽7)C2×C11×C1311yescharacteristic, complemented
100D3×ΓL(1,𝔽8)C2×C11×C1311yescharacteristic, complemented
101C3×AGL(1,𝔽7)C2×C11×C1311yescharacteristic, complemented
102C32×C13—71noHall
103C32×C11—71noHall
104C7×C13C2×C3×D3×C1111yescharacteristic, complemented, Hall
105D3×D7C3×C11×C1311yescharacteristic, complemented
106C2×AGL(1,𝔽7)—31no—
107C2×C3×C13—71no—
108C2×C3×C13—71no—
109C2×C3×C13—141no—
110C2×C3×C13—211no—
111C2×C3×C13—211no—
112D3×C13—71no—
113D3×C13C11×AGL(1,𝔽7)11yescharacteristic, direct product factor
114C7×C11C2×C3×D3×C1311yescharacteristic, complemented, Hall
115C2×C3×C11—71no—
116C2×C3×C11—71no—
117C2×C3×C11—141no—
118C2×C3×C11—211no—
119C2×C3×C11—211no—
120D3×C11—71no—
121D3×C11C13×AGL(1,𝔽7)11yescharacteristic, direct product factor
122C3×ΓL(1,𝔽8)C22×C11×C1311yescharacteristic, complemented, Hall
123C22×C13—211noHall
124C22×C11—211noHall
125D21C2×C3×C11×C1311yescharacteristic, complemented
126C3×D7C2×C3×C11×C1311yescharacteristic, complemented
127D3×C7C2×C3×C11×C1311yescharacteristic, complemented
128C2×ΓL(1,𝔽8)—31no—
129AGL(1,𝔽7)—21no—
130AGL(1,𝔽7)—31no—
131AGL(1,𝔽7)D3×C11×C1311yescharacteristic, direct product factor
132C3×C13—71no—
133C3×C13—141no—
134C3×C13C2×C11×AGL(1,𝔽7)11yescharacteristic, complemented
135C2×C3×D3—71noHall
136C3×C11—71no—
137C3×C11—141no—
138C3×C11C2×C13×AGL(1,𝔽7)11yescharacteristic, complemented
139C2×D7—31noHall
140C2×C13—31no—
141C2×C13—71no—
142C2×C13—211no—
143C2×C11—31no—
144C2×C11—71no—
145C2×C11—211no—
146C3×C7C22×C3×C11×C1311yescharacteristic, complemented
147ΓL(1,𝔽8)—21no—
148ΓL(1,𝔽8)C2×D3×C11×C1311yescharacteristic, complemented
149C2×C32—71no—
150C3×D3—71no—
151C3×D3—71no—
152C2×C7—31no—
153D7—31no—
154D7C3×D3×C11×C1311yescharacteristic, complemented
155C13D3×C11×AGL(1,𝔽7)11yescharacteristic, direct product factor, Sylow
156C22×C3—211no—
157C2×D3—71no—
158C11D3×C13×AGL(1,𝔽7)11yescharacteristic, direct product factor, Sylow
159C32—71noSylow
160C7C2×C3×D3×C11×C1311yescharacteristic, complemented, Sylow
161C2×C3—71no—
162C2×C3—71no—
163C2×C3—141no—
164C2×C3—211no—
165C2×C3—211no—
166D3—71no—
167D3C11×C13×AGL(1,𝔽7)11yescharacteristic, direct product factor
168C22—211noSylow
169C3—71no—
170C3—141no—
171C3C2×C11×C13×AGL(1,𝔽7)11yescharacteristic, complemented
172C2—31no—
173C2—71no—
174C2—211no—
subgroupG/H
center, hypercenterC11×C13D3×AGL(1,𝔽7)
commutator, cosocle, hypocenterC3×C7C22×C3×C11×C13
FrattiniC1G
Fitting, socleC3×C7×C11×C13C22×C3
LCSG ▷ C3×C7
UCSC1 ◁ C11×C13

Complements

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

NH ≅ G/Ntype
1. D3×C11×C13×ΓL(1,𝔽8)C2semidirect
2. C11×C13×C3⋊AGL(1,𝔽7)C2semidirect
3. C3×C11×C13×AGL(1,𝔽7)C2semidirect
4. D3×C11×C13×D7C3semidirect
6. C3×C11×C13×ΓL(1,𝔽8)C22semidirect
9. C11×C13×AGL(1,𝔽7)D3direct (N × H)
10. C11×C13×D21C2×C3semidirect
11. D3×C7×C11×C13C2×C3semidirect
12. C3×C11×C13×D7C2×C3semidirect
16. D3×C13×AGL(1,𝔽7)C11direct (N × H)
18. C11×C13×ΓL(1,𝔽8)C2×D3semidirect
19. C3×C7×C11×C13C22×C3semidirect
20. D3×C11×AGL(1,𝔽7)C13direct (N × H)
25. C11×C13×D7C3×D3semidirect
29. C13×C3⋊AGL(1,𝔽7)C2×C11semidirect
30. D3×C13×ΓL(1,𝔽8)C2×C11semidirect
31. C3×C13×AGL(1,𝔽7)C2×C11semidirect
32. C11×C3⋊AGL(1,𝔽7)C2×C13semidirect
33. D3×C11×ΓL(1,𝔽8)C2×C13semidirect
34. C3×C11×AGL(1,𝔽7)C2×C13semidirect
36. D3×C13×D7C3×C11semidirect
38. C7×C11×C13C2×C3×D3semidirect
39. D3×C11×D7C3×C13semidirect
47. D3×C11×C13AGL(1,𝔽7)direct (N × H)
48. C3×C13×ΓL(1,𝔽8)C22×C11semidirect
49. C3×C11×ΓL(1,𝔽8)C22×C13semidirect
51. C13×D21C2×C3×C11semidirect
52. D3×C7×C13C2×C3×C11semidirect
53. C3×C13×D7C2×C3×C11semidirect
57. C13×AGL(1,𝔽7)D3×C11direct (N × H)
59. C11×D21C2×C3×C13semidirect
60. D3×C7×C11C2×C3×C13semidirect
61. C3×C11×D7C2×C3×C13semidirect
65. C11×AGL(1,𝔽7)D3×C13direct (N × H)
68. C3×C11×C13C2×AGL(1,𝔽7)semidirect
75. C3×C7×C13C22×C3×C11semidirect
77. C13×ΓL(1,𝔽8)C2×D3×C11semidirect
78. D3×AGL(1,𝔽7)C11×C13direct (N × H)
82. C3×C7×C11C22×C3×C13semidirect
84. C11×ΓL(1,𝔽8)C2×D3×C13semidirect
90. C13×D7C3×D3×C11semidirect
95. C11×D7C3×D3×C13semidirect
96. C11×C13D3×AGL(1,𝔽7)direct (N × H)
99. C3⋊AGL(1,𝔽7)C2×C11×C13semidirect
100. D3×ΓL(1,𝔽8)C2×C11×C13semidirect
101. C3×AGL(1,𝔽7)C2×C11×C13semidirect
104. C7×C13C2×C3×D3×C11semidirect
105. D3×D7C3×C11×C13semidirect
113. D3×C13C11×AGL(1,𝔽7)direct (N × H)
114. C7×C11C2×C3×D3×C13semidirect
121. D3×C11C13×AGL(1,𝔽7)direct (N × H)
122. C3×ΓL(1,𝔽8)C22×C11×C13semidirect
125. D21C2×C3×C11×C13semidirect
126. C3×D7C2×C3×C11×C13semidirect
127. D3×C7C2×C3×C11×C13semidirect
131. AGL(1,𝔽7)D3×C11×C13direct (N × H)
134. C3×C13C2×C11×AGL(1,𝔽7)semidirect
138. C3×C11C2×C13×AGL(1,𝔽7)semidirect
146. C3×C7C22×C3×C11×C13semidirect
148. ΓL(1,𝔽8)C2×D3×C11×C13semidirect
154. D7C3×D3×C11×C13semidirect
155. C13D3×C11×AGL(1,𝔽7)direct (N × H)
158. C11D3×C13×AGL(1,𝔽7)direct (N × H)
160. C7C2×C3×D3×C11×C13semidirect
167. D3C11×C13×AGL(1,𝔽7)direct (N × H)
171. C3C2×C11×C13×AGL(1,𝔽7)semidirect

Maximal subgroups

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

subgroupindex
D3×C11×C13×ΓL(1,𝔽8)2
C11×C13×C3⋊AGL(1,𝔽7)2
C3×C11×C13×AGL(1,𝔽7)2
D3×C11×C13×D73
C2×C11×C13×AGL(1,𝔽7)3
C2×C3×D3×C11×C137
D3×C13×AGL(1,𝔽7)11
D3×C11×AGL(1,𝔽7)13