Finite Group Databasebeta

← #560⋊ΓL(1,𝔽8) #13060785

Subgroups — #560⋊ΓL(1,𝔽8) #13060785

Order 1344.

134 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 27 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
1#127⋊ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
2#10621175C211yesmaximal, characteristic, complemented
3#10621174C211yesmaximal, characteristic, complemented
4C7×#560C311yesmaximal, characteristic, complemented, Hall
5C22×C22⋊ΓL(1,𝔽8)—41nomaximal
6C7⋊(C42⋊C3)C2211yescharacteristic, complemented
7C7×#127C2×C311yescharacteristic, complemented
8C7×#126C2×C311yescharacteristic, complemented
9C7×#126C2×C311yescharacteristic, complemented
10C7×#123—31no—
11C7×#121—31no—
12C7×#121—31no—
13C7×#120—31no—
14#560⋊C3—71nomaximal, Hall
15C2×C22⋊ΓL(1,𝔽8)—41no—
16C2×C22⋊ΓL(1,𝔽8)—41no—
17C2×C22⋊ΓL(1,𝔽8)—41no—
18C24×C7A411yescharacteristic
19C2×C7×D4—31no—
20C2×C7×D4—31no—
21C2×C7×D4—31no—
22C2×C7×D4—31no—
23C22×C4×C7—31no—
24C7×C4⋊C4—31no—
25C7×C4⋊C4—31no—
26C7×C22⋊C4—31no—
27C7×C22⋊C4—31no—
28C7×C22⋊C4—31no—
29C7×C22⋊C4—31no—
30C42×C7C22×C311yescharacteristic, complemented
31#127⋊C3—71no—
32#126⋊C3—71no—
33#126⋊C3—71no—
34C22⋊ΓL(1,𝔽8)—41no—
35C22×ΓL(1,𝔽8)—161no—
36#560ΓL(1,𝔽8)11yescharacteristic, complemented, Sylow
37C23×C7—31no—
38C23×C7—121no—
39C23×C7C2×A411yescharacteristic
40C23×C7C2×A411yescharacteristic
41C23×C7C2×A411yescharacteristic
42C7×D4—121no—
43C7×D4—121no—
44C7×D4—121no—
45C7×D4—121no—
46C2×C4×C7—31no—
47C2×C4×C7—31no—
48C2×C4×C7—31no—
49C2×C4×C7—121no—
50C22×A4—281no—
51C42⋊C3—71no—
52C2×ΓL(1,𝔽8)—161no—
53C2×ΓL(1,𝔽8)—161no—
54C2×ΓL(1,𝔽8)—161no—
55#127C2×ΓL(1,𝔽8)11yescharacteristic, complemented
56#126C2×ΓL(1,𝔽8)11yescharacteristic, complemented
57#126C2×ΓL(1,𝔽8)11yescharacteristic, complemented
58#123—31no—
59#121—31no—
60#121—31no—
61#120—31no—
62C22×C7—41no—
63C22×C7—61no—
64C22×C7—61no—
65C22×C7—61no—
66C22×C7—61no—
67C22×C7—61no—
68C22×C7—61no—
69C22×C7—121no—
70C22×C7C22×A411yescharacteristic
71C4×C7—61no—
72C4×C7—61no—
73C4×C7—61no—
74C2×A4—281no—
75C2×A4—281no—
76C2×A4—281no—
77ΓL(1,𝔽8)—161noHall
78C24C22⋊ΓL(1,𝔽8)11yescharacteristic
79C2×D4—31no—
80C2×D4—31no—
81C2×D4—31no—
82C2×D4—31no—
83C22×C4—31no—
84C4⋊C4—31no—
85C4⋊C4—31no—
86C22⋊C4—31no—
87C22⋊C4—31no—
88C22⋊C4—31no—
89C22⋊C4—31no—
90C42C22×ΓL(1,𝔽8)11yescharacteristic, complemented
91C2×C7—31no—
92C2×C7—41no—
93C2×C7—41no—
94C2×C7—41no—
95C2×C7—121no—
96C22×C3—1121no—
97A4—281no—
98C23—31no—
99C23—121no—
100C23C2×C22⋊ΓL(1,𝔽8)11yescharacteristic
101C23C2×C22⋊ΓL(1,𝔽8)11yescharacteristic
102C23C2×C22⋊ΓL(1,𝔽8)11yescharacteristic
103D4—121no—
104D4—121no—
105D4—121no—
106D4—121no—
107C2×C4—31no—
108C2×C4—31no—
109C2×C4—31no—
110C2×C4—121no—
111C7#560⋊C311yescharacteristic, complemented, Sylow
112C2×C3—1121no—
113C2×C3—1121no—
114C2×C3—1121no—
115C22—41no—
116C22—61no—
117C22—61no—
118C22—61no—
119C22—61no—
120C22—61no—
121C22—61no—
122C22—121no—
123C22C22×C22⋊ΓL(1,𝔽8)11yescharacteristic
124C4—61no—
125C4—61no—
126C4—61no—
127C3—1121noSylow
128C2—31no—
129C2—41no—
130C2—41no—
131C2—41no—
132C2—121no—
subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterC42×C7C22×C3
FrattiniC22C22×C22⋊ΓL(1,𝔽8)
FittingC7×#560C3
socleC22×C7C22×A4
LCSG ▷ C42×C7
UCSC1

Complements

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

NH ≅ G/Ntype
1. #127⋊ΓL(1,𝔽8)C2semidirect
2. #10621175C2semidirect
3. #10621174C2semidirect
4. C7×#560C3semidirect
6. C7⋊(C42⋊C3)C22semidirect
7. C7×#127C2×C3semidirect
8. C7×#126C2×C3semidirect
9. C7×#126C2×C3semidirect
30. C42×C7C22×C3semidirect
36. #560ΓL(1,𝔽8)semidirect
55. #127C2×ΓL(1,𝔽8)semidirect
56. #126C2×ΓL(1,𝔽8)semidirect
57. #126C2×ΓL(1,𝔽8)semidirect
90. C42C22×ΓL(1,𝔽8)semidirect
111. C7#560⋊C3semidirect

Maximal subgroups

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

subgroupindex
#127⋊ΓL(1,𝔽8)2
#106211752
#106211742
C7×#5603
C22×C22⋊ΓL(1,𝔽8)4
#560⋊C37