Finite Group Databasebeta

← C7×#5930 #13062127

Subgroups — C7×#5930 #13062127

Order 1344.

128 conjugacy classes of subgroups; only the 18 normal subgroups are shown.

normal subgroups only

This group has 128 conjugacy classes of subgroups — too many to draw. Only its 18 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
1C7×C22⋊SL(2,𝔽3)C211yesmaximal, characteristic
2C7×#451—31nomaximal, Hall
3C7×SL(2,ℤ/4)—41nomaximal
4C7×#226—42nomaximal
5C22×C7×Q8D311yescharacteristic
6C2×C7×Q16—32no—
7C7×C22⋊Q8—31no—
8C7×#103—32no—
9C7×C22⋊C8—31no—
10#5930C711yesmaximal, characteristic, direct product factor, Hall
11C2×C7×A4—41no—
12C7×SL(2,𝔽3)—42no—
13C7×SL(2,𝔽3)—81no—
14C2×C7×Q8—32no—
15C2×C7×Q8—31no—
16C2×C7×Q8—61no—
17C22×C4×C7—31no—
18C7×Q16—62no—
19C7×Q16—62no—
20C2×C7×C8—32no—
21C7×C4⋊C4—31no—
22C7×C4⋊C4—61no—
23C7×C22⋊C4—61no—
24C22⋊SL(2,𝔽3)C2×C711yescharacteristic
25C7×A4—41no—
26C7×Dic3—161no—
27#451—31noSylow
28C23×C7S411yescharacteristic
29C7×Q8—21no—
30C7×Q8—32no—
31C7×Q8—61no—
32C7×Q8—62no—
33C7×Q8S412yes—
34C2×C4×C7—31no—
35C2×C4×C7—31no—
36C2×C4×C7—62no—
37C2×C4×C7—61no—
38C2×C4×C7—61no—
39C7×C8—62no—
40SL(2,ℤ/4)—41no—
41#226—42no—
42C2×C3×C7—161no—
43C22×Q8D3×C711yescharacteristic
44C2×Q16—32no—
45C22⋊Q8—31no—
46#103—32no—
47C22⋊C8—31no—
48C22×C7—31no—
49C22×C7—31no—
50C22×C7#22611yescharacteristic, complemented
51C4×C7—32no—
52C4×C7—61no—
53C4×C7—121no—
54C4×C7—121no—
55C2×A4—41no—
56SL(2,𝔽3)—42no—
57SL(2,𝔽3)—81no—
58C3×C7—161noHall
59C2×Q8—32no—
60C2×Q8—31no—
61C2×Q8—61no—
62C22×C4—31no—
63Q16—62no—
64Q16—62no—
65C2×C8—32no—
66C4⋊C4—31no—
67C4⋊C4—61no—
68C22⋊C4—61no—
69C2×C7—31no—
70C2×C7—31no—
71C2×C7#102011yescharacteristic
72A4—41no—
73Dic3—161no—
74C23C7×S411yescharacteristic
75Q8—21no—
76Q8—32no—
77Q8—61no—
78Q8—62no—
79Q8C7×S412yes—
80C2×C4—31no—
81C2×C4—31no—
82C2×C4—62no—
83C2×C4—61no—
84C2×C4—61no—
85C8—62no—
86C7#593011yescharacteristic, direct product factor, Sylow
87C2×C3—161no—
88C22—31no—
89C22—31no—
90C22C7×#22611yescharacteristic, complemented
91C4—32no—
92C4—61no—
93C4—121no—
94C4—121no—
95C3—161noSylow
96C2—31no—
97C2—31no—
98C2C7×#102011yescharacteristic
subgroupG/H
center, hypercenterC2×C7#1020
commutator, cosocle, hypocenterC22⋊SL(2,𝔽3)C2×C7
FrattiniC2C7×#1020
FittingC22×C7×Q8D3
socleC23×C7S4
prime radicalC7×C22⋊SL(2,𝔽3)C2
LCSG ▷ C22⋊SL(2,𝔽3)
UCSC1 ◁ C2×C7

Complements

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

NH ≅ G/Ntype
10. #5930C7direct (N × H)
50. C22×C7#226semidirect
86. C7#5930direct (N × H)
90. C22C7×#226semidirect

Maximal subgroups

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

subgroupindex
C7×C22⋊SL(2,𝔽3)2
C7×#4513
C7×SL(2,ℤ/4)4
C7×#2264
#59307