Finite Group Databasebeta

← C2×#10621175 #13060777

Subgroups — C2×#10621175 #13060777

Order 1344.

132 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 132 conjugacy classes of subgroups — too many to draw. Only its 29 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#10621175C212yesmaximal, direct product factor
2C2×C7⋊(C42⋊C3)C211yesmaximal, characteristic, complemented
3C2×C7×#126C311yesmaximal, characteristic, complemented, Hall
4C22×C22⋊ΓL(1,𝔽8)—41nomaximal
5C7⋊(C42⋊C3)C2211yescharacteristic, complemented
6C7×#126—32no—
7C7×#126C2×C312yescomplemented
8C2×C7×C4⋊C4—31no—
9C2×C7×C22⋊C4—31no—
10C2×C42×C7C2×C311yescharacteristic, complemented
11C2×#126⋊C3—71nomaximal, Hall
12C2×C22⋊ΓL(1,𝔽8)—41no—
13C2×C22⋊ΓL(1,𝔽8)—42no—
14C24×C7A411yescharacteristic
15C22×C4×C7—31no—
16C22×C4×C7—31no—
17C7×C4⋊C4—32no—
18C7×C4⋊C4—32no—
19C7×C22⋊C4—32no—
20C7×C22⋊C4—32no—
21C42×C7—31no—
22C42×C7C22×C311yescharacteristic, complemented
23#126⋊C3—72no—
24C2×C42⋊C3—71no—
25C22⋊ΓL(1,𝔽8)—41no—
26C22×ΓL(1,𝔽8)—161no—
27C2×#126ΓL(1,𝔽8)11yescharacteristic, complemented, Sylow
28C23×C7—61no—
29C23×C7—61no—
30C23×C7C2×A411yescharacteristic
31C23×C7C2×A412yes—
32C2×C4×C7—31no—
33C2×C4×C7—31no—
34C2×C4×C7—32no—
35C2×C4×C7—61no—
36C2×C4×C7—61no—
37C2×C4×C7—61no—
38C2×C4×C7—61no—
39C22×A4—281no—
40C42⋊C3—71no—
41C2×ΓL(1,𝔽8)—162no—
42C2×ΓL(1,𝔽8)—161no—
43#126—32no—
44#126C2×ΓL(1,𝔽8)12yescomplemented
45C2×C4⋊C4—31no—
46C2×C22⋊C4—31no—
47C2×C42C2×ΓL(1,𝔽8)11yescharacteristic, complemented
48C22×C7—31no—
49C22×C7—31no—
50C22×C7—41no—
51C22×C7—62no—
52C22×C7—121no—
53C22×C7C22×A411yescharacteristic
54C4×C7—61no—
55C4×C7—61no—
56C4×C7—62no—
57C2×A4—281no—
58C2×A4—282no—
59ΓL(1,𝔽8)—161noHall
60C24C22⋊ΓL(1,𝔽8)11yescharacteristic
61C22×C4—31no—
62C22×C4—31no—
63C4⋊C4—32no—
64C4⋊C4—32no—
65C22⋊C4—32no—
66C22⋊C4—32no—
67C42—31no—
68C42C22×ΓL(1,𝔽8)11yescharacteristic, complemented
69C2×C7—31no—
70C2×C7—31no—
71C2×C7—42no—
72C2×C7#126⋊C311yescharacteristic, complemented
73C22×C3—1121no—
74A4—281no—
75C23—61no—
76C23—61no—
77C23C2×C22⋊ΓL(1,𝔽8)11yescharacteristic
78C23C2×C22⋊ΓL(1,𝔽8)12yes—
79C2×C4—31no—
80C2×C4—31no—
81C2×C4—32no—
82C2×C4—61no—
83C2×C4—61no—
84C2×C4—61no—
85C2×C4—61no—
86C7C2×#126⋊C311yescharacteristic, complemented, Sylow
87C2×C3—1122no—
88C2×C3—1121no—
89C22—31no—
90C22—31no—
91C22—41no—
92C22—62no—
93C22—121no—
94C22C22×C22⋊ΓL(1,𝔽8)11yescharacteristic
95C4—61no—
96C4—61no—
97C4—62no—
98C3—1121noSylow
99C2—31no—
100C2—31no—
101C2—42no—
102C2#1062117511yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2#10621175
commutator, cosocle, hypocenterC42×C7C22×C3
FrattiniC22C22×C22⋊ΓL(1,𝔽8)
FittingC2×C7×#126C3
socleC23×C7C2×A4
LCSG ▷ C42×C7
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. #10621175C2direct (N × H)
2. C2×C7⋊(C42⋊C3)C2semidirect
3. C2×C7×#126C3semidirect
5. C7⋊(C42⋊C3)C22semidirect
7. C7×#126C2×C3semidirect
10. C2×C42×C7C2×C3semidirect
22. C42×C7C22×C3semidirect
27. C2×#126ΓL(1,𝔽8)semidirect
44. #126C2×ΓL(1,𝔽8)semidirect
47. C2×C42C2×ΓL(1,𝔽8)semidirect
68. C42C22×ΓL(1,𝔽8)semidirect
72. C2×C7#126⋊C3semidirect
86. C7C2×#126⋊C3semidirect
102. C2#10621175direct (N × H)

Maximal subgroups

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

subgroupindex
#106211752
C2×C7⋊(C42⋊C3)2
C2×C7×#1263
C22×C22⋊ΓL(1,𝔽8)4
C2×#126⋊C37