Finite Group Databasebeta

← C4⋊(C7⋊#226) #13057156

Subgroups — C4⋊(C7⋊#226) #13057156

Order 1344.

134 conjugacy classes of subgroups; only the 31 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 31 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
1C2×C7⋊#226C212yesmaximal
2C4×C7×SL(2,𝔽3)C211yesmaximal, characteristic
3#89803—31nomaximal, Hall
4C2×C7×SL(2,𝔽3)C2211yescharacteristic
5C7⋊#226—22no—
6C4⋊Dic21—41nomaximal
7C4×C7×Q8D311yescharacteristic
8C2×#1144—32no—
9#6432—31no—
10#6384—32no—
11C4⋊(C7⋊C8)—31no—
12C4⋊#226—71nomaximal, Hall
13C2×C3×C4×C7—41no—
14C2×Dic21—42no—
15C7×SL(2,𝔽3)D411yescharacteristic
16C2×C7×Q8C2×D311yescharacteristic
17C2×Dic14—32no—
18C7×C4⋊C4—31no—
19C42×C7—31no—
20#1144—62no—
21#1144—62no—
22C4⋊Dic7—61no—
23C4⋊Dic7—61no—
24C2×C7⋊C8—32no—
25C2×#226—72no—
26C4×SL(2,𝔽3)D711yescharacteristic
27C22×C3×C7—41no—
28C3×C4×C7—42no—
29Dic21—82no—
30#462—211noSylow
31C7×Q8—31no—
32C7×Q8D1211yescharacteristic
33C2×C4×C7—31no—
34C2×C4×C7—31no—
35C2×C4×C7S411yescharacteristic
36C2×Dic7—62no—
37Dic14—62no—
38Dic14—62no—
39C7⋊C8—62no—
40C2×SL(2,𝔽3)C2×D711yescharacteristic
41#226—142no—
42C4⋊Dic3—281no—
43C2×C3×C7—41no—
44C2×C3×C7—41no—
45C2×C3×C7—41no—
46C2×Q16—212no—
47C4⋊Q8—211no—
48C4×Q8D2111yescharacteristic
49C4⋊C8—211no—
50#103—212no—
51C22×C7C2×S411yescharacteristic
52C4×C7—31no—
53C4×C7—31no—
54C4×C7—61no—
55C4×C7#22612yescomplemented
56Dic7—122no—
57C2×C3×C4—41no—
58C2×Dic3—282no—
59SL(2,𝔽3)D2811yescharacteristic
60C3×C7—41noHall
61C2×Q8—212no—
62C2×Q8C2×D2111yescharacteristic
63Q16—422no—
64Q16—422no—
65C2×C8—212no—
66C4⋊C4—31no—
67C4⋊C4—421no—
68C4⋊C4—421no—
69C42—31no—
70C2×C7C4⋊S411yescharacteristic
71C2×C7C2×#22611yescharacteristic
72C2×C7#98411yescharacteristic
73C22×C3—41no—
74C3×C4—42no—
75Dic3—562no—
76Q8—31no—
77Q8—422no—
78Q8—422no—
79Q8D8411yescharacteristic
80C2×C4—31no—
81C2×C4—31no—
82C2×C4—422no—
83C2×C4C7⋊S411yescharacteristic
84C8—422no—
85C7C4⋊#22611yescharacteristic, complemented, Sylow
86C2×C3—41no—
87C2×C3—41no—
88C2×C3—41no—
89C22C2×C7⋊S411yescharacteristic
90C4—31no—
91C4—31no—
92C4—61no—
93C4—842no—
94C4C7⋊#22612yescomplemented
95C3—41noSylow
96C2C4⋊(C7⋊S4)11yescharacteristic
97C2C2×C7⋊#22611yescharacteristic
98C2#1062150211yescharacteristic
subgroupG/H
center, FrattiniC22C2×C7⋊S4
commutator, prime radical, cosocleC2×C7×SL(2,𝔽3)C22
FittingC4×C7×Q8D3
socleC22×C7C2×S4
hypercenterC2×C4C7⋊S4
hypocenterC7×SL(2,𝔽3)D4
LCSG ▷ C2×C7×SL(2,𝔽3) ▷ C7×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
55. C4×C7#226semidirect
85. C7C4⋊#226semidirect
94. C4C7⋊#226semidirect

Maximal subgroups

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

subgroupindex
C2×C7⋊#2262
C4×C7×SL(2,𝔽3)2
#898033
C4⋊Dic214
C4⋊#2267