Finite Group Databasebeta

← #126×AGL(1,𝔽7) #13057438

Subgroups — #126×AGL(1,𝔽7) #13057438

Order 1344.

492 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 182 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
1ΓL(1,𝔽8)×#126C211yesmaximal, characteristic, complemented
2#6461⋊C3C213yesmaximal, complemented
3C4⋊C4×AGL(1,𝔽7)C213yesmaximal, complemented
4C22⋊C4×AGL(1,𝔽7)C213yesmaximal
5#6442⋊C3C213yesmaximal, complemented
6#10620712C211yesmaximal, characteristic, complemented
7C42×AGL(1,𝔽7)C211yesmaximal, characteristic, complemented
8D7×#126C311yesmaximal, characteristic, complemented, Hall
9C23×AGL(1,𝔽7)C2211yescharacteristic
10C2×C4×AGL(1,𝔽7)C2213yes—
11C2×C4×AGL(1,𝔽7)C2213yes—
12C4⋊C4×ΓL(1,𝔽8)C2213yescomplemented
13C22⋊C4×ΓL(1,𝔽8)C2213yes—
14C42×ΓL(1,𝔽8)C2211yescharacteristic, complemented
15(C22⋊Dic7)⋊C3C2213yes—
16(C2×AGL(1,𝔽7))⋊C4C2213yes—
17(C2×AGL(1,𝔽7))⋊C4C2213yes—
18C4⋊(ΓL(1,𝔽8)⋊C4)C2213yescomplemented
19(ΓL(1,𝔽8)⋊C4)⋊C4C2213yescomplemented
20(ΓL(1,𝔽8)⋊C4)⋊C4C2213yescomplemented
21C4×ΓL(1,𝔽8)⋊C4C2213yescomplemented
22C7×#126C2×C311yescharacteristic, complemented
23#6461C2×C313yescomplemented
24D7×C4⋊C4C2×C313yescomplemented
25D7×C22⋊C4C2×C313yes—
26#6442C2×C313yescomplemented
27#6439C2×C311yescharacteristic, complemented
28C42×D7C2×C311yescharacteristic, complemented
29C2×C3×#126—71nomaximal, Hall
30C23×ΓL(1,𝔽8)C2311yescharacteristic
31C22×AGL(1,𝔽7)—26no—
32C22×AGL(1,𝔽7)C2311yescharacteristic
33C22×AGL(1,𝔽7)C2311yescharacteristic
34C2×C4×ΓL(1,𝔽8)C2313yes—
35C2×C4×ΓL(1,𝔽8)C2313yes—
36C2×ΓL(1,𝔽8)⋊C4C2313yes—
37C2×ΓL(1,𝔽8)⋊C4C2313yes—
38C4×AGL(1,𝔽7)—26no—
39C4×AGL(1,𝔽7)—26no—
40C23×D7C22×C311yescharacteristic
41C2×C4×D7C22×C313yes—
42C2×C4×D7C22×C313yes—
43C7×C4⋊C4C22×C313yescomplemented
44C7×C22⋊C4C22×C313yes—
45C42×C7C22×C311yescharacteristic, complemented
46C22⋊Dic7C22×C313yes—
47(C2×D7)⋊C4C22×C313yes—
48(C2×D7)⋊C4C22×C313yes—
49C4⋊Dic7C22×C313yescomplemented
50Dic7⋊C4C22×C313yescomplemented
51Dic7⋊C4C22×C313yescomplemented
52C4×Dic7C22×C313yescomplemented
53C3×#126—71no—
54C3×#126—73no—
55C3×#126—73no—
56C3×#126—71no—
57C2×C3×C4⋊C4—73no—
58C2×C3×C22⋊C4—73no—
59C2×C3×C42—71no—
60C22×ΓL(1,𝔽8)—23no—
61C22×ΓL(1,𝔽8)C2411yescharacteristic
62C2×AGL(1,𝔽7)—23no—
63C2×AGL(1,𝔽7)—44no—
64C2×AGL(1,𝔽7)#4116yes—
65C4×ΓL(1,𝔽8)—23no—
66C4×ΓL(1,𝔽8)—23no—
67ΓL(1,𝔽8)⋊C4—23no—
68ΓL(1,𝔽8)⋊C4—23no—
69C2×#126—71noSylow
70C23×C7C23×C311yescharacteristic
71C22×D7—26no—
72C22×D7C23×C311yescharacteristic
73C22×D7C23×C311yescharacteristic
74C2×C4×C7C23×C313yes—
75C2×C4×C7C23×C313yes—
76C2×Dic7C23×C313yes—
77C2×Dic7C23×C313yes—
78C4×D7—26no—
79C4×D7—26no—
80C24×C3—71no—
81C22×C3×C4—73no—
82C22×C3×C4—73no—
83C3×C4⋊C4—73no—
84C3×C4⋊C4—73no—
85C3×C4⋊C4—73no—
86C3×C4⋊C4—73no—
87C3×C22⋊C4—73no—
88C3×C22⋊C4—73no—
89C3×C22⋊C4—73no—
90C3×C22⋊C4—73no—
91C3×C42—73no—
92C3×C42—71no—
93C2×ΓL(1,𝔽8)—41no—
94C2×ΓL(1,𝔽8)C2×#4113yes—
95AGL(1,𝔽7)—41no—
96AGL(1,𝔽7)#12614yesdirect product factor
97#126—73no—
98#126—73no—
99#126—71no—
100#126AGL(1,𝔽7)11yescharacteristic, direct product factor
101C2×C4⋊C4—73no—
102C2×C22⋊C4—73no—
103C2×C42—71no—
104C22×C7—23no—
105C22×C7C24×C311yescharacteristic
106C2×D7—23no—
107C2×D7—44no—
108C2×D7C3×#4116yes—
109C4×C7—23no—
110C4×C7—23no—
111Dic7—23no—
112Dic7—23no—
113C23×C3—71no—
114C23×C3—71no—
115C23×C3—71no—
116C23×C3—146no—
117C2×C3×C4—73no—
118C2×C3×C4—73no—
119C2×C3×C4—73no—
120C2×C3×C4—73no—
121C2×C3×C4—146no—
122C2×C3×C4—146no—
123ΓL(1,𝔽8)C2×#12611yescharacteristic, complemented, Hall
124C24—71no—
125C22×C4—73no—
126C22×C4—73no—
127C4⋊C4—73no—
128C4⋊C4—73no—
129C4⋊C4—73no—
130C4⋊C4C2×AGL(1,𝔽7)13yescomplemented
131C22⋊C4—73no—
132C22⋊C4—73no—
133C22⋊C4—73no—
134C22⋊C4C2×AGL(1,𝔽7)13yes—
135C42—73no—
136C42C2×AGL(1,𝔽7)11yescharacteristic, complemented
137C2×C7—41no—
138C2×C7C2×C3×#4113yes—
139D7—41no—
140D7C3×#12614yescomplemented
141C22×C3—76no—
142C22×C3—71no—
143C22×C3—143no—
144C22×C3—143no—
145C22×C3—284no—
146C3×C4—143no—
147C3×C4—143no—
148C3×C4—143no—
149C3×C4—143no—
150C23—71no—
151C23—71no—
152C23—146no—
153C23C22×AGL(1,𝔽7)11yescharacteristic
154C2×C4—73no—
155C2×C4—73no—
156C2×C4—146no—
157C2×C4—146no—
158C2×C4C22×AGL(1,𝔽7)13yes—
159C2×C4C22×AGL(1,𝔽7)13yes—
160C7C2×C3×#12611yescharacteristic, complemented, Sylow
161C2×C3—74no—
162C2×C3—73no—
163C2×C3—281no—
164C2×C3—281no—
165C22—23no—
166C22—76no—
167C22—143no—
168C22—284no—
169C22C23×AGL(1,𝔽7)11yescharacteristic
170C4—23no—
171C4—23no—
172C4—143no—
173C4—143no—
174C3—71noSylow
175C2—41no—
176C2—74no—
177C2—281no—
178C2#41×AGL(1,𝔽7)13yes—
subgroupG/H
center, FrattiniC22C23×AGL(1,𝔽7)
commutator, socle, cosocleC22×C7C24×C3
FittingC7×#126C2×C3
prime radicalC23×AGL(1,𝔽7)C22
hypercenter#126AGL(1,𝔽7)
hypocenterC7C2×C3×#126
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ #126

Complements

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

NH ≅ G/Ntype
1. ΓL(1,𝔽8)×#126C2semidirect
2. #6461⋊C3C2semidirect
3. C4⋊C4×AGL(1,𝔽7)C2semidirect
5. #6442⋊C3C2semidirect
6. #10620712C2semidirect
7. C42×AGL(1,𝔽7)C2semidirect
8. D7×#126C3semidirect
12. C4⋊C4×ΓL(1,𝔽8)C22semidirect
14. C42×ΓL(1,𝔽8)C22semidirect
18. C4⋊(ΓL(1,𝔽8)⋊C4)C22semidirect
19. (ΓL(1,𝔽8)⋊C4)⋊C4C22semidirect
20. (ΓL(1,𝔽8)⋊C4)⋊C4C22semidirect
21. C4×ΓL(1,𝔽8)⋊C4C22semidirect
22. C7×#126C2×C3semidirect
23. #6461C2×C3semidirect
24. D7×C4⋊C4C2×C3semidirect
26. #6442C2×C3semidirect
27. #6439C2×C3semidirect
28. C42×D7C2×C3semidirect
43. C7×C4⋊C4C22×C3semidirect
45. C42×C7C22×C3semidirect
49. C4⋊Dic7C22×C3semidirect
50. Dic7⋊C4C22×C3semidirect
51. Dic7⋊C4C22×C3semidirect
52. C4×Dic7C22×C3semidirect
96. AGL(1,𝔽7)#126direct (N × H)
100. #126AGL(1,𝔽7)direct (N × H)
123. ΓL(1,𝔽8)C2×#126semidirect
130. C4⋊C4C2×AGL(1,𝔽7)semidirect
136. C42C2×AGL(1,𝔽7)semidirect
140. D7C3×#126semidirect
160. C7C2×C3×#126semidirect

Maximal subgroups

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

subgroupindex
ΓL(1,𝔽8)×#1262
#6461⋊C32
C4⋊C4×AGL(1,𝔽7)2
C22⋊C4×AGL(1,𝔽7)2
#6442⋊C32
#106207122
C42×AGL(1,𝔽7)2
D7×#1263
C2×C3×#1267