Finite Group Databasebeta

← #13057163

Subgroups — #13057163

Order 1344.

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

normal subgroups only

This group has 146 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
1C2×C7⋊GL(2,𝔽3)C211yesmaximal, characteristic
2C2×C7⋊#226C211yesmaximal, characteristic, complemented
3C4×C7×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#89800—31nomaximal, Hall
5C2×C7×SL(2,𝔽3)C2211yescharacteristic
6C7⋊GL(2,𝔽3)—21no—
7C7⋊#226—21no—
8#66631—41nomaximal
9C4×C7×Q8D311yescharacteristic, complemented
10C2×#1144—31no—
11C2×#1143—31no—
12#6438—31no—
13#6384—31no—
14#6383—31no—
15C4⋊(C7⋊C8)—31no—
16#5395—71nomaximal, Hall
17C22×D21—41no—
18C2×C3×C4×C7—41no—
19C2×Dic21—41no—
20C7×SL(2,𝔽3)D411yescharacteristic
21C2×C7×Q8C2×D311yescharacteristic
22C2×D28—31no—
23C2×Dic14—31no—
24C7×C4⋊C4—31no—
25C42×C7—31no—
26#1144—61no—
27#1144—61no—
28#1143—61no—
29#1143—61no—
30(C2×D7)⋊C4—61no—
31(C2×D7)⋊C4—61no—
32C2×C7⋊C8—31no—
33C2×C7⋊C8—31no—
34C2×GL(2,𝔽3)—71no—
35C2×#226—71no—
36C4×SL(2,𝔽3)D711yescharacteristic, complemented
37C22×C3×C7—41no—
38C2×D21—42no—
39C2×D21—81no—
40C2×D21—81no—
41C3×C4×C7—81no—
42Dic21—81no—
43#464—211noSylow
44C22×D7—61no—
45C7×Q8—31no—
46C7×Q8D1211yescharacteristic
47C2×C4×C7—31no—
48C2×C4×C7—31no—
49C2×C4×C7S411yescharacteristic
50C2×Dic7—61no—
51D28—61no—
52D28—61no—
53Dic14—61no—
54Dic14—61no—
55C7⋊C8—61no—
56C7⋊C8—61no—
57C2×SL(2,𝔽3)C2×D711yescharacteristic
58GL(2,𝔽3)—141no—
59#226—141no—
60(C2×D3)⋊C4—281no—
61C2×C3×C7—41no—
62C2×C3×C7—41no—
63C2×C3×C7—41no—
64D21—82no—
65C2×Q16—211no—
66C2×SD16—211no—
67#124—211no—
68C4×Q8D2111yescharacteristic, complemented
69C4⋊C8—211no—
70#103—211no—
71#102—211no—
72C22×C7C2×S411yescharacteristic
73C2×D7—121no—
74C2×D7—121no—
75C2×D7—121no—
76C4×C7—21no—
77C4×C7—31no—
78C4×C7—31no—
79C4×C7—61no—
80Dic7—121no—
81C22×D3—281no—
82C2×C3×C4—41no—
83C2×Dic3—281no—
84SL(2,𝔽3)D2811yescharacteristic
85C3×C7—41noHall
86C2×Q8—211no—
87C2×Q8C2×D2111yescharacteristic
88C2×D4—211no—
89Q16—421no—
90Q16—421no—
91SD16—421no—
92SD16—421no—
93C2×C8—211no—
94C2×C8—211no—
95C4⋊C4—31no—
96C22⋊C4—421no—
97C22⋊C4—421no—
98C42—31no—
99C2×C7C4⋊S411yescharacteristic
100C2×C7#98311yescharacteristic
101C2×C7#98511yescharacteristic
102D7—241no—
103C22×C3—41no—
104C2×D3—282no—
105C2×D3—561no—
106C2×D3—561no—
107C3×C4—81no—
108Dic3—561no—
109C23—421no—
110Q8—31no—
111Q8—421no—
112Q8—421no—
113Q8D8411yescharacteristic
114D4—421no—
115D4—421no—
116C2×C4—31no—
117C2×C4—31no—
118C2×C4—421no—
119C2×C4C7⋊S411yescharacteristic
120C8—421no—
121C8—421no—
122C7#539511yescharacteristic, complemented, Sylow
123C2×C3—41no—
124C2×C3—41no—
125C2×C3—41no—
126D3—562no—
127C22—841no—
128C22—841no—
129C22—841no—
130C22C2×C7⋊S411yescharacteristic
131C4—21no—
132C4—31no—
133C4—31no—
134C4—61no—
135C4—841no—
136C3—41noSylow
137C2—1681no—
138C2C4⋊(C7⋊S4)11yescharacteristic
139C2#1062150111yescharacteristic
140C2#1062150311yescharacteristic
subgroupG/H
center, FrattiniC22C2×C7⋊S4
commutator, cosocleC2×C7×SL(2,𝔽3)C22
FittingC4×C7×Q8D3
socleC22×C7C2×S4
prime radicalC2×C7⋊GL(2,𝔽3)C2
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
2. C2×C7⋊#226C2semidirect
3. C4×C7×SL(2,𝔽3)C2semidirect
9. C4×C7×Q8D3semidirect
36. C4×SL(2,𝔽3)D7semidirect
68. C4×Q8D21semidirect
122. C7#5395semidirect

Maximal subgroups

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

subgroupindex
C2×C7⋊GL(2,𝔽3)2
C2×C7⋊#2262
C4×C7×SL(2,𝔽3)2
#898003
#666314
#53957