Finite Group Databasebeta

← #13057445

Subgroups — #13057445

Order 1344.

504 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 162 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#6458⋊C3C213yesmaximal, complemented
3#6457⋊C3C213yesmaximal, complemented
4#6447⋊C3C213yesmaximal, complemented
5#6445⋊C3C213yesmaximal, complemented
6#10620712C211yesmaximal, characteristic, complemented
7#10620710C211yesmaximal, characteristic, complemented
8#90402C311yesmaximal, characteristic, complemented, Hall
9C23×AGL(1,𝔽7)C2211yescharacteristic
10C2×(C22⋊D7)⋊C3C2213yes—
11C2×C4⋊AGL(1,𝔽7)C2213yes—
12C2×C4⋊AGL(1,𝔽7)C2213yes—
13C2×C4⋊AGL(1,𝔽7)C2213yescomplemented
14C2×C4×AGL(1,𝔽7)C2213yescomplemented
15C4⋊C4×ΓL(1,𝔽8)C2213yescomplemented
16C22⋊C4×ΓL(1,𝔽8)C2213yes—
17C42×ΓL(1,𝔽8)C2211yescharacteristic, complemented
18(C2×AGL(1,𝔽7))⋊C4C2213yes—
19(C2×AGL(1,𝔽7))⋊C4C2213yes—
20(C2×AGL(1,𝔽7))⋊C4C2213yescomplemented
21(ΓL(1,𝔽8)⋊C4)⋊C4C2213yescomplemented
22C7×#126C2×C311yescharacteristic, complemented
23#6458C2×C313yescomplemented
24#6457C2×C313yescomplemented
25#6447C2×C313yescomplemented
26#6445C2×C313yescomplemented
27#6439C2×C311yescharacteristic, complemented
28#6437C2×C311yescharacteristic, complemented
29C3×#560—71nomaximal, Hall
30C23×ΓL(1,𝔽8)C2311yescharacteristic
31C22×AGL(1,𝔽7)—43no—
32C22×AGL(1,𝔽7)C2313yes—
33C22×AGL(1,𝔽7)C2311yescharacteristic
34C22×AGL(1,𝔽7)C2311yescharacteristic
35C2×C4×ΓL(1,𝔽8)C2313yes—
36C2×C4×ΓL(1,𝔽8)C2313yes—
37(C22⋊D7)⋊C3—43no—
38C2×ΓL(1,𝔽8)⋊C4C2313yes—
39C4⋊AGL(1,𝔽7)—43no—
40C4⋊AGL(1,𝔽7)—43no—
41C4⋊AGL(1,𝔽7)—43no—
42C4×AGL(1,𝔽7)—43no—
43C23×D7C22×C311yescharacteristic
44C2×C22⋊D7C22×C313yes—
45C2×D28C22×C313yes—
46C2×D28C22×C313yes—
47C2×D28C22×C313yescomplemented
48C2×C4×D7C22×C313yescomplemented
49C7×C4⋊C4C22×C313yescomplemented
50C7×C22⋊C4C22×C313yes—
51C42×C7C22×C311yescharacteristic, complemented
52(C2×D7)⋊C4C22×C313yes—
53(C2×D7)⋊C4C22×C313yes—
54(C2×D7)⋊C4C22×C313yescomplemented
55Dic7⋊C4C22×C313yescomplemented
56C3×#127—71no—
57C3×#126—71no—
58C3×#126—71no—
59C3×#123—73no—
60C3×#121—73no—
61C3×#121—73no—
62C3×#120—73no—
63C22×ΓL(1,𝔽8)—23no—
64C22×ΓL(1,𝔽8)C2411yescharacteristic
65C2×AGL(1,𝔽7)—23no—
66C2×AGL(1,𝔽7)—23no—
67C2×AGL(1,𝔽7)—23no—
68C2×AGL(1,𝔽7)—23no—
69C2×AGL(1,𝔽7)—23no—
70C2×AGL(1,𝔽7)—44no—
71C4×ΓL(1,𝔽8)—23no—
72C4×ΓL(1,𝔽8)—23no—
73ΓL(1,𝔽8)⋊C4—23no—
74#560—71noSylow
75C23×C7C23×C311yescharacteristic
76C22×D7—43no—
77C22×D7C23×C311yescharacteristic
78C22×D7C23×C311yescharacteristic
79C22×D7C23×C313yes—
80C2×C4×C7C23×C313yes—
81C2×C4×C7C23×C313yes—
82C22⋊D7—43no—
83C2×Dic7C23×C313yes—
84D28—43no—
85D28—43no—
86D28—43no—
87C4×D7—43no—
88C24×C3—71no—
89C2×C3×D4—73no—
90C2×C3×D4—73no—
91C2×C3×D4—73no—
92C2×C3×D4—73no—
93C22×C3×C4—73no—
94C3×C4⋊C4—73no—
95C3×C4⋊C4—73no—
96C3×C22⋊C4—73no—
97C3×C22⋊C4—73no—
98C3×C22⋊C4—73no—
99C3×C22⋊C4—73no—
100C3×C42—71no—
101C2×ΓL(1,𝔽8)—41no—
102C2×ΓL(1,𝔽8)H(5,𝔽2)13yes—
103AGL(1,𝔽7)—43no—
104AGL(1,𝔽7)—41no—
105AGL(1,𝔽7)—41no—
106#127—71no—
107#126—71no—
108#126AGL(1,𝔽7)11yescharacteristic, complemented
109#123—73no—
110#121—73no—
111#121—73no—
112#120—73no—
113C22×C7—23no—
114C22×C7C24×C311yescharacteristic
115C2×D7—23no—
116C2×D7—23no—
117C2×D7—23no—
118C2×D7—23no—
119C2×D7—23no—
120C2×D7—44no—
121C4×C7—23no—
122C4×C7—23no—
123Dic7—23no—
124C23×C3—71no—
125C23×C3—71no—
126C23×C3—73no—
127C23×C3—71no—
128C23×C3—283no—
129C3×D4—283no—
130C3×D4—283no—
131C3×D4—283no—
132C3×D4—283no—
133C2×C3×C4—73no—
134C2×C3×C4—73no—
135C2×C3×C4—73no—
136C2×C3×C4—283no—
137ΓL(1,𝔽8)#56011yescharacteristic, complemented, Hall
138C24—71no—
139C2×D4—73no—
140C2×D4—73no—
141C2×D4—73no—
142C2×D4—73no—
143C22×C4—73no—
144C4⋊C4—73no—
145C4⋊C4C2×AGL(1,𝔽7)13yescomplemented
146C22⋊C4—73no—
147C22⋊C4—73no—
148C22⋊C4—73no—
149C22⋊C4C2×AGL(1,𝔽7)13yes—
150C42C2×AGL(1,𝔽7)11yescharacteristic, complemented
151C2×C7—41no—
152C2×C7C3×H(5,𝔽2)13yes—
153D7—41no—
154D7—41no—
155D7—43no—
156C22×C3—71no—
157C22×C3—143no—
158C22×C3—143no—
159C22×C3—143no—
160C22×C3—143no—
161C22×C3—143no—
162C22×C3—143no—
163C22×C3—284no—
164C3×C4—143no—
165C3×C4—143no—
166C3×C4—143no—
167C23—71no—
168C23—71no—
169C23—73no—
170C23—283no—
171C23C22×AGL(1,𝔽7)11yescharacteristic
172D4—283no—
173D4—283no—
174D4—283no—
175D4—283no—
176C2×C4—73no—
177C2×C4—283no—
178C2×C4C22×AGL(1,𝔽7)13yes—
179C2×C4C22×AGL(1,𝔽7)13yes—
180C7C3×#56011yescharacteristic, complemented, Sylow
181C2×C3—73no—
182C2×C3—281no—
183C2×C3—281no—
184C2×C3—281no—
185C2×C3—283no—
186C22—23no—
187C22—143no—
188C22—143no—
189C22—143no—
190C22—143no—
191C22—143no—
192C22—284no—
193C22C23×AGL(1,𝔽7)11yescharacteristic
194C4—23no—
195C4—23no—
196C4—143no—
197C3—71noSylow
198C2—41no—
199C2—281no—
200C2—281no—
201C2—283no—
202C2#1062151813yes—
subgroupG/H
center, FrattiniC22C23×AGL(1,𝔽7)
commutator, socle, cosocleC22×C7C24×C3
FittingC7×#126C2×C3
hypercenter#126AGL(1,𝔽7)
hypocenterC7C3×#560
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. #6458⋊C3C2semidirect
3. #6457⋊C3C2semidirect
4. #6447⋊C3C2semidirect
5. #6445⋊C3C2semidirect
6. #10620712C2semidirect
7. #10620710C2semidirect
8. #90402C3semidirect
13. C2×C4⋊AGL(1,𝔽7)C22semidirect
14. C2×C4×AGL(1,𝔽7)C22semidirect
15. C4⋊C4×ΓL(1,𝔽8)C22semidirect
17. C42×ΓL(1,𝔽8)C22semidirect
20. (C2×AGL(1,𝔽7))⋊C4C22semidirect
21. (ΓL(1,𝔽8)⋊C4)⋊C4C22semidirect
22. C7×#126C2×C3semidirect
23. #6458C2×C3semidirect
24. #6457C2×C3semidirect
25. #6447C2×C3semidirect
26. #6445C2×C3semidirect
27. #6439C2×C3semidirect
28. #6437C2×C3semidirect
47. C2×D28C22×C3semidirect
48. C2×C4×D7C22×C3semidirect
49. C7×C4⋊C4C22×C3semidirect
51. C42×C7C22×C3semidirect
54. (C2×D7)⋊C4C22×C3semidirect
55. Dic7⋊C4C22×C3semidirect
108. #126AGL(1,𝔽7)semidirect
137. ΓL(1,𝔽8)#560semidirect
145. C4⋊C4C2×AGL(1,𝔽7)semidirect
150. C42C2×AGL(1,𝔽7)semidirect
180. C7C3×#560semidirect

Maximal subgroups

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

subgroupindex
ΓL(1,𝔽8)×#1262
#6458⋊C32
#6457⋊C32
#6447⋊C32
#6445⋊C32
#106207122
#106207102
#904023
C3×#5607