Finite Group Databasebeta

← AGL(1,𝔽7)×D24 #72245524

Subgroups — AGL(1,𝔽7)×D24 #72245524

Order 2016.

326 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 77 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
1D12×AGL(1,𝔽7)C212yesmaximal, complemented
2D168⋊C3C211yesmaximal, characteristic, complemented
3ΓL(1,𝔽8)×D24C211yesmaximal, characteristic, complemented
4C3×C8×AGL(1,𝔽7)C211yesmaximal, characteristic, complemented
5#66562⋊C3C212yesmaximal, complemented
6D7×D24C311yesmaximal, characteristic, complemented
7D8×AGL(1,𝔽7)—31nomaximal
8C2×D3×AGL(1,𝔽7)—22no—
9C4⋊(C3⋊AGL(1,𝔽7))C2212yescomplemented
10ΓL(1,𝔽8)×D12C2212yescomplemented
11C3×C4×AGL(1,𝔽7)C2211yescharacteristic
12#4247⋊C3—22no—
13C3×C8×ΓL(1,𝔽8)C2211yescharacteristic, complemented
14C3×ΓL(1,𝔽8)⋊C8C2211yescharacteristic, complemented
15D7×D12C2×C312yescomplemented
16D4×AGL(1,𝔽7)—32no—
17D168C2×C311yescharacteristic, complemented
18C7×D24C2×C311yescharacteristic, complemented
19C3×C8×D7C2×C311yescharacteristic, complemented
20D8×ΓL(1,𝔽8)—31no—
21#66562C2×C312yescomplemented
22#1141⋊C3—32no—
23D56⋊C3—31no—
24C8×AGL(1,𝔽7)—21no—
25C8×AGL(1,𝔽7)D311yescharacteristic, complemented
26C2×C3×D24—71nomaximal, Hall
27C2×C3⋊AGL(1,𝔽7)—22no—
28C2×D3×ΓL(1,𝔽8)—22no—
29C2×C3×AGL(1,𝔽7)D411yescharacteristic
30D3×AGL(1,𝔽7)—44no—
31C3×C4×ΓL(1,𝔽8)C2311yescharacteristic
32C3×ΓL(1,𝔽8)⋊C4D411yescharacteristic
33D7×D8—31noHall
34C2×D3×D7—22no—
35C22×AGL(1,𝔽7)—62no—
36D84C22×C312yescomplemented
37C7×D12C22×C312yescomplemented
38C3×C4×D7C22×C311yescharacteristic
39D4×ΓL(1,𝔽8)—32no—
40#4247—22no—
41(C22⋊D7)⋊C3—62no—
42C4⋊AGL(1,𝔽7)—32no—
43C4×AGL(1,𝔽7)—21no—
44C4×AGL(1,𝔽7)C2×D311yescharacteristic
45C3×C7×C8C22×C311yescharacteristic, complemented
46C3×C7⋊C8C22×C311yescharacteristic, complemented
47C8×ΓL(1,𝔽8)—21no—
48C8×ΓL(1,𝔽8)C2×D311yescharacteristic, complemented
49ΓL(1,𝔽8)⋊C8—21no—
50ΓL(1,𝔽8)⋊C8C2×D311yescharacteristic, complemented
51C2×C3×D12—72no—
52C2×C32×C8—71no—
53C3×D24—72no—
54C3×D24—71no—
55C3×D24—141no—
56C2×C3×ΓL(1,𝔽8)C2×D411yescharacteristic
57C3⋊AGL(1,𝔽7)—42no—
58D3×ΓL(1,𝔽8)—42no—
59C3×AGL(1,𝔽7)D812yescomplemented
60D4×D7—32no—
61C7×D8—31no—
62#1141—32no—
63D56—31no—
64C8×D7C3×D311yescharacteristic, complemented
65C2×C3×D8—211no—
66C2×D24—71no—
67C2×D21—22no—
68C2×D3×C7—22no—
69C2×C3×D7C3×D411yescharacteristic
70C22×ΓL(1,𝔽8)—62no—
71D3×D7—44no—
72C2×AGL(1,𝔽7)—21no—
73C2×AGL(1,𝔽7)—62no—
74C2×AGL(1,𝔽7)—124no—
75C2×AGL(1,𝔽7)D1211yescharacteristic
76C3×C4×C7C23×C311yescharacteristic
77C3×Dic7C3×D411yescharacteristic
78C4×ΓL(1,𝔽8)—21no—
79C4×ΓL(1,𝔽8)C22×D311yescharacteristic
80ΓL(1,𝔽8)⋊C4—21no—
81ΓL(1,𝔽8)⋊C4D1211yescharacteristic
82C22×C3×D3—142no—
83C2×C32×C4—71no—
84C3×D12—72no—
85C3×D12—142no—
86C3×D12—142no—
87C32×C8—71no—
88C32×C8—141no—
89C3×ΓL(1,𝔽8)C2×D811yescharacteristic, complemented, Hall
90C22×D7—62no—
91C7×D4—32no—
92C22⋊D7—62no—
93D28—32no—
94C4×D7C2×C3×D311yescharacteristic
95C7×C8C2×C3×D311yescharacteristic, complemented
96C7⋊C8C2×C3×D311yescharacteristic, complemented
97C2×C3×D4—212no—
98C2×D12—72no—
99C3×D8—211no—
100C3×D8—211no—
101C3×D8—212no—
102C2×C3×C8—71no—
103C2×C3×C8—71no—
104C2×C3×C8—141no—
105D24—72no—
106D24—71no—
107D24AGL(1,𝔽7)11yescharacteristic, direct product factor
108C2×C3×C7C2×C3×D411yescharacteristic
109D21—42no—
110C3×D7C3×D812yescomplemented
111D3×C7—42no—
112C2×ΓL(1,𝔽8)—21no—
113C2×ΓL(1,𝔽8)—122no—
114C2×ΓL(1,𝔽8)C2×D1211yescharacteristic
115AGL(1,𝔽7)—22no—
116AGL(1,𝔽7)—122no—
117AGL(1,𝔽7)D2412yesdirect product factor
118C22×C32—71no—
119C2×C3×D3—142no—
120C2×C3×D3—282no—
121C2×C3×D3—284no—
122C32×C4—71no—
123C32×C4—141no—
124C2×D8—211noSylow
125C22×C7—62no—
126C2×D7—62no—
127C2×D7—124no—
128C2×D7C3×D1211yescharacteristic
129C4×C7C22×C3×D311yescharacteristic
130Dic7C3×D1211yescharacteristic
131C23×C3—422no—
132C22×D3—142no—
133C3×D4—212no—
134C3×D4—212no—
135C3×D4—422no—
136C2×C3×C4—71no—
137C2×C3×C4—71no—
138C2×C3×C4—141no—
139D12—72no—
140D12—142no—
141D12C2×AGL(1,𝔽7)12yescomplemented
142C3×C8—71no—
143C3×C8—71no—
144C3×C8—71no—
145C3×C8—141no—
146C3×C8—281no—
147C3×C8C2×AGL(1,𝔽7)11yescharacteristic, complemented
148C3×C7C2×C3×D811yescharacteristic, complemented
149ΓL(1,𝔽8)—21no—
150ΓL(1,𝔽8)C2×D2411yescharacteristic, complemented
151C2×C32—72no—
152C2×C32—141no—
153C3×D3—282no—
154C3×D3—562no—
155C2×D4—212no—
156D8—31no—
157D8—212no—
158D8—211no—
159C2×C8—71no—
160C2×C7—122no—
161C2×C7C2×C3×D1211yescharacteristic
162D7—122no—
163D7C3×D2412yescomplemented
164C22×C3—71no—
165C22×C3—71no—
166C22×C3—141no—
167C22×C3—422no—
168C22×C3—422no—
169C22×C3—844no—
170C2×D3—22no—
171C2×D3—142no—
172C2×D3—284no—
173C3×C4—71no—
174C3×C4—71no—
175C3×C4—71no—
176C3×C4—141no—
177C3×C4—281no—
178C3×C4C22×AGL(1,𝔽7)11yescharacteristic
179C32—141noSylow
180C23—422no—
181D4—32no—
182D4—212no—
183D4—422no—
184C2×C4—71no—
185C8—71no—
186C8D3×AGL(1,𝔽7)11yescharacteristic, complemented
187C7C2×C3×D2411yescharacteristic, complemented, Sylow
188C2×C3—72no—
189C2×C3—72no—
190C2×C3—71no—
191C2×C3—142no—
192C2×C3—281no—
193C2×C3—842no—
194C2×C3—842no—
195C2×C3D4×AGL(1,𝔽7)11yescharacteristic
196D3—42no—
197D3—282no—
198C22—62no—
199C22—71no—
200C22—422no—
201C22—844no—
202C4—71no—
203C4C2×D3×AGL(1,𝔽7)11yescharacteristic
204C3—71no—
205C3—281no—
206C3D8×AGL(1,𝔽7)11yescharacteristic, complemented
207C2—72no—
208C2—122no—
209C2—842no—
210C2D12×AGL(1,𝔽7)11yescharacteristic
subgroupG/H
centerC2D12×AGL(1,𝔽7)
commutator, cosocleC3×C4×C7C23×C3
FrattiniC4C2×D3×AGL(1,𝔽7)
FittingC3×C7×C8C22×C3
socleC2×C3×C7C2×C3×D4
hypercenterC8D3×AGL(1,𝔽7)
hypocenterC3×C7C2×C3×D8
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
1. D12×AGL(1,𝔽7)C2semidirect
2. D168⋊C3C2semidirect
3. ΓL(1,𝔽8)×D24C2semidirect
4. C3×C8×AGL(1,𝔽7)C2semidirect
5. #66562⋊C3C2semidirect
6. D7×D24C3semidirect
9. C4⋊(C3⋊AGL(1,𝔽7))C22semidirect
10. ΓL(1,𝔽8)×D12C22semidirect
13. C3×C8×ΓL(1,𝔽8)C22semidirect
14. C3×ΓL(1,𝔽8)⋊C8C22semidirect
15. D7×D12C2×C3semidirect
17. D168C2×C3semidirect
18. C7×D24C2×C3semidirect
19. C3×C8×D7C2×C3semidirect
21. #66562C2×C3semidirect
25. C8×AGL(1,𝔽7)D3semidirect
36. D84C22×C3semidirect
37. C7×D12C22×C3semidirect
45. C3×C7×C8C22×C3semidirect
46. C3×C7⋊C8C22×C3semidirect
48. C8×ΓL(1,𝔽8)C2×D3semidirect
50. ΓL(1,𝔽8)⋊C8C2×D3semidirect
59. C3×AGL(1,𝔽7)D8semidirect
64. C8×D7C3×D3semidirect
89. C3×ΓL(1,𝔽8)C2×D8semidirect
95. C7×C8C2×C3×D3semidirect
96. C7⋊C8C2×C3×D3semidirect
107. D24AGL(1,𝔽7)direct (N × H)
110. C3×D7C3×D8semidirect
117. AGL(1,𝔽7)D24direct (N × H)
141. D12C2×AGL(1,𝔽7)semidirect
147. C3×C8C2×AGL(1,𝔽7)semidirect
148. C3×C7C2×C3×D8semidirect
150. ΓL(1,𝔽8)C2×D24semidirect
163. D7C3×D24semidirect
186. C8D3×AGL(1,𝔽7)semidirect
187. C7C2×C3×D24semidirect
206. C3D8×AGL(1,𝔽7)semidirect

Maximal subgroups

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

subgroupindex
D12×AGL(1,𝔽7)2
D168⋊C32
ΓL(1,𝔽8)×D242
C3×C8×AGL(1,𝔽7)2
#66562⋊C32
D7×D243
D8×AGL(1,𝔽7)3
C2×C3×D247