Finite Group Databasebeta

← #13057155

Subgroups — #13057155

Order 1344.

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

normal subgroups only

This group has 134 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
1C2×C7⋊#226C211yesmaximal, characteristic
2C4×C7×SL(2,𝔽3)C211yesmaximal, characteristic
3C7⋊#859C211yesmaximal, characteristic
4#89802—31nomaximal, Hall
5C2×C7×SL(2,𝔽3)C2211yescharacteristic
6C7⋊#226C412yescomplemented
7#66629—41nomaximal
8C4×C7×Q8D311yescharacteristic
9C2×#1144—31no—
10C4×Dic14—31no—
11#6409—31no—
12#6384—31no—
13#6382—31no—
14#6377—31no—
15#5388—71nomaximal, Hall
16C2×C3×C4×C7—41no—
17C2×Dic21—41no—
18C2×Dic21—41no—
19C7×SL(2,𝔽3)C2×C411yescharacteristic
20C2×C7×Q8C2×D311yescharacteristic
21C2×Dic14—31no—
22C7×C4⋊C4—31no—
23C42×C7—31no—
24#1144—32no—
25#1144—32no—
26C4⋊Dic7—31no—
27Dic7⋊C4—61no—
28C4×Dic7—61no—
29C2×C7⋊C8—31no—
30C2×C7⋊C8—31no—
31C2×#226—71no—
32C4×SL(2,𝔽3)D711yescharacteristic
33#859—71no—
34C22×C3×C7—41no—
35C3×C4×C7—81no—
36Dic21—42no—
37Dic21—81no—
38#441—211noSylow
39C7×Q8—31no—
40C7×Q8C4×D311yescharacteristic
41C2×C4×C7—31no—
42C2×C4×C7—31no—
43C2×C4×C7S411yescharacteristic
44C2×Dic7—61no—
45C2×Dic7—61no—
46Dic14—32no—
47Dic14—61no—
48C7⋊C8—32no—
49C7⋊C8—61no—
50C2×SL(2,𝔽3)C2×D711yescharacteristic
51#226—72no—
52Dic3⋊C4—281no—
53C2×C3×C7—41no—
54C2×C3×C7—41no—
55C2×C3×C7—41no—
56C2×Q16—211no—
57C4×Q8—211no—
58C4×Q8D2111yescharacteristic
59#106—211no—
60#103—211no—
61#103—211no—
62#97—211no—
63C22×C7C2×S411yescharacteristic
64C4×C7—21no—
65C4×C7—31no—
66C4×C7—31no—
67C4×C7—61no—
68Dic7—62no—
69Dic7—121no—
70C2×C3×C4—41no—
71C2×Dic3—281no—
72C2×Dic3—281no—
73SL(2,𝔽3)C4×D711yescharacteristic
74C3×C7—41noHall
75C2×Q8—211no—
76C2×Q8C2×D2111yescharacteristic
77Q16—212no—
78Q16—212no—
79C2×C8—211no—
80C2×C8—211no—
81C4⋊C4—31no—
82C4⋊C4—211no—
83C4⋊C4—421no—
84C42—31no—
85C42—421no—
86C2×C7C4×S411yescharacteristic
87C2×C7#98311yescharacteristic
88C2×C7#98411yescharacteristic
89C22×C3—41no—
90C3×C4—81no—
91Dic3—282no—
92Dic3—561no—
93Q8—31no—
94Q8—212no—
95Q8—421no—
96Q8C4×D2111yescharacteristic
97C2×C4—31no—
98C2×C4—31no—
99C2×C4—421no—
100C2×C4—421no—
101C2×C4C7⋊S411yescharacteristic
102C8—212no—
103C8—421no—
104C7#538811yescharacteristic, complemented, Sylow
105C2×C3—41no—
106C2×C3—41no—
107C2×C3—41no—
108C22C2×C7⋊S411yescharacteristic
109C4—21no—
110C4—31no—
111C4—31no—
112C4—61no—
113C4—422no—
114C4—841no—
115C3—41noSylow
116C2C4×C7⋊S411yescharacteristic
117C2#1062150111yescharacteristic
118C2#1062150211yescharacteristic
subgroupG/H
center, FrattiniC22C2×C7⋊S4
commutator, hypocenterC7×SL(2,𝔽3)C2×C4
FittingC4×C7×Q8D3
socleC22×C7C2×S4
prime radical, cosocleC2×C7×SL(2,𝔽3)C22
hypercenterC2×C4C7⋊S4
LCSG ▷ C7×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
6. C7⋊#226C4semidirect
104. C7#5388semidirect

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
C7⋊#8592
#898023
#666294
#53887