Finite Group Databasebeta

← #13057328

Subgroups — #13057328

Order 1344.

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

normal subgroups only

This group has 520 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)×#120C211yesmaximal, characteristic, complemented
2#10620789C212yesmaximal, complemented
3#6500⋊C3C213yesmaximal, complemented
4#6499⋊C3C213yesmaximal, complemented
5#6448⋊C3C216yesmaximal, complemented
6#90285C311yesmaximal, characteristic, complemented, Hall
7C24×ΓL(1,𝔽8)C2211yescharacteristic, complemented
8C23×AGL(1,𝔽7)C2211yescharacteristic, complemented
9C2×D4×ΓL(1,𝔽8)C2213yes—
10C2×(C22⋊D7)⋊C3C2216yes—
11C2×ΓL(1,𝔽8)⋊Q8C2213yescomplemented
12C22⋊C4×ΓL(1,𝔽8)C2213yescomplemented
13(C22⋊Dic7)⋊C3C2216yescomplemented
14(C22⋊Dic7)⋊C3C2213yescomplemented
15(C2×AGL(1,𝔽7))⋊C4C2216yescomplemented
16C4×ΓL(1,𝔽8)⋊C4C2213yescomplemented
17C7×#120C2×C311yescharacteristic, complemented
18#6516C2×C312yescomplemented
19#6500C2×C313yescomplemented
20#6499C2×C313yescomplemented
21#6448C2×C316yescomplemented
22C3×H(3,𝔽4)—71nomaximal, Hall
23C23×ΓL(1,𝔽8)—43no—
24C23×ΓL(1,𝔽8)C2313yes—
25C23×ΓL(1,𝔽8)C2311yescharacteristic
26C22×AGL(1,𝔽7)—43no—
27C22×AGL(1,𝔽7)C2312yes—
28D4×ΓL(1,𝔽8)—43no—
29C2×C4×ΓL(1,𝔽8)C2313yes—
30(C22⋊D7)⋊C3—46no—
31C2×ΓL(1,𝔽8)⋊C4C2316yes—
32ΓL(1,𝔽8)⋊Q8—43no—
33C24×C7C22×C311yescharacteristic, complemented
34C23×D7C22×C311yescharacteristic, complemented
35C2×C7×D4C22×C313yes—
36C2×C22⋊D7C22×C316yes—
37C2×Dic14C22×C313yescomplemented
38C7×C22⋊C4C22×C313yescomplemented
39C22⋊Dic7C22×C316yescomplemented
40C22⋊Dic7C22×C313yescomplemented
41(C2×D7)⋊C4C22×C316yescomplemented
42C4×Dic7C22×C313yescomplemented
43C3×#124—76no—
44C3×#124—73no—
45C3×#120—73no—
46C3×#120—72no—
47C3×#120—71no—
48C22×ΓL(1,𝔽8)—26no—
49C22×ΓL(1,𝔽8)—23no—
50C22×ΓL(1,𝔽8)—23no—
51C22×ΓL(1,𝔽8)—44no—
52C22×ΓL(1,𝔽8)C2411yescharacteristic
53C2×AGL(1,𝔽7)—26no—
54C2×AGL(1,𝔽7)—44no—
55C4×ΓL(1,𝔽8)—23no—
56ΓL(1,𝔽8)⋊C4—26no—
57H(3,𝔽4)—71noSylow
58C23×C7—43no—
59C23×C7C23×C311yescharacteristic
60C23×C7C23×C313yes—
61C22×D7—43no—
62C22×D7C23×C312yes—
63C7×D4—43no—
64C2×C4×C7C23×C313yes—
65C22⋊D7—46no—
66C2×Dic7C23×C316yes—
67Dic14—43no—
68C24×C3—71no—
69C24×C3—71no—
70C2×C3×Q8—73no—
71C2×C3×D4—76no—
72C2×C3×D4—73no—
73C3×C22⋊C4—76no—
74C3×C22⋊C4—73no—
75C3×C22⋊C4—76no—
76C3×C22⋊C4—73no—
77C3×C42—73no—
78C2×ΓL(1,𝔽8)—41no—
79C2×ΓL(1,𝔽8)—43no—
80C2×ΓL(1,𝔽8)H(5,𝔽2)13yes—
81AGL(1,𝔽7)—42no—
82#124—73no—
83#124—76no—
84#120—72no—
85#120—73no—
86#120AGL(1,𝔽7)11yescharacteristic, complemented
87C22×C7—26no—
88C22×C7—23no—
89C22×C7—23no—
90C22×C7—44no—
91C22×C7C24×C311yescharacteristic
92C2×D7—26no—
93C2×D7—44no—
94C4×C7—23no—
95Dic7—26no—
96C23×C3—72no—
97C23×C3—71no—
98C23×C3—73no—
99C23×C3—283no—
100C23×C3—283no—
101C3×Q8—283no—
102C3×D4—283no—
103C3×D4—286no—
104C2×C3×C4—76no—
105C2×C3×C4—73no—
106ΓL(1,𝔽8)H(3,𝔽4)11yescharacteristic, complemented, Hall
107C24—71no—
108C24C2×AGL(1,𝔽7)11yescharacteristic, complemented
109C2×Q8—73no—
110C2×D4—76no—
111C2×D4C2×AGL(1,𝔽7)13yes—
112C22⋊C4—73no—
113C22⋊C4—76no—
114C22⋊C4—76no—
115C22⋊C4C2×AGL(1,𝔽7)13yescomplemented
116C42—73no—
117C2×C7—41no—
118C2×C7—43no—
119C2×C7C3×H(5,𝔽2)13yes—
120D7—42no—
121C22×C3—71no—
122C22×C3—143no—
123C22×C3—143no—
124C22×C3—146no—
125C22×C3—146no—
126C22×C3—284no—
127C22×C3—284no—
128C3×C4—143no—
129C3×C4—146no—
130C23—43no—
131C23—72no—
132C23—283no—
133C23C22×AGL(1,𝔽7)11yescharacteristic
134C23C22×AGL(1,𝔽7)13yes—
135Q8—283no—
136D4—43no—
137D4—286no—
138C2×C4—76no—
139C2×C4C22×AGL(1,𝔽7)13yes—
140C7C3×H(3,𝔽4)11yescharacteristic, complemented, Sylow
141C2×C3—73no—
142C2×C3—283no—
143C2×C3—281no—
144C2×C3—282no—
145C22—23no—
146C22—23no—
147C22—26no—
148C22—44no—
149C22—146no—
150C22—284no—
151C22C23×AGL(1,𝔽7)11yescharacteristic
152C4—23no—
153C4—146no—
154C3—71noSylow
155C2—43no—
156C2—41no—
157C2—282no—
158C2#6548⋊C313yes—
subgroupG/H
center, FrattiniC22C23×AGL(1,𝔽7)
commutator, socle, cosocleC22×C7C24×C3
FittingC7×#120C2×C3
hypercenter#120AGL(1,𝔽7)
hypocenterC7C3×H(3,𝔽4)
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. ΓL(1,𝔽8)×#120C2semidirect
2. #10620789C2semidirect
3. #6500⋊C3C2semidirect
4. #6499⋊C3C2semidirect
5. #6448⋊C3C2semidirect
6. #90285C3semidirect
7. C24×ΓL(1,𝔽8)C22semidirect
8. C23×AGL(1,𝔽7)C22semidirect
11. C2×ΓL(1,𝔽8)⋊Q8C22semidirect
12. C22⋊C4×ΓL(1,𝔽8)C22semidirect
13. (C22⋊Dic7)⋊C3C22semidirect
14. (C22⋊Dic7)⋊C3C22semidirect
15. (C2×AGL(1,𝔽7))⋊C4C22semidirect
16. C4×ΓL(1,𝔽8)⋊C4C22semidirect
17. C7×#120C2×C3semidirect
18. #6516C2×C3semidirect
19. #6500C2×C3semidirect
20. #6499C2×C3semidirect
21. #6448C2×C3semidirect
33. C24×C7C22×C3semidirect
34. C23×D7C22×C3semidirect
37. C2×Dic14C22×C3semidirect
38. C7×C22⋊C4C22×C3semidirect
39. C22⋊Dic7C22×C3semidirect
40. C22⋊Dic7C22×C3semidirect
41. (C2×D7)⋊C4C22×C3semidirect
42. C4×Dic7C22×C3semidirect
86. #120AGL(1,𝔽7)semidirect
106. ΓL(1,𝔽8)H(3,𝔽4)semidirect
108. C24C2×AGL(1,𝔽7)semidirect
115. C22⋊C4C2×AGL(1,𝔽7)semidirect
140. C7C3×H(3,𝔽4)semidirect

Maximal subgroups

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

subgroupindex
ΓL(1,𝔽8)×#1202
#106207892
#6500⋊C32
#6499⋊C32
#6448⋊C32
#902853
C3×H(3,𝔽4)7