Finite Group Databasebeta

← #13057199

Subgroups — #13057199

Order 1344.

144 conjugacy classes of subgroups; only the 23 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 23 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
1C7×2−1+4⋊C3C211yesmaximal, characteristic, complemented
2#10621504C211yesmaximal, characteristic, complemented
3C7⋊GU(2,𝔽3)C211yesmaximal, characteristic, complemented
4#90017—31nomaximal, Hall
5C7×#41⋊C3C2211yescharacteristic
6C2×C7×SL(2,𝔽3)—21no—
7C7⋊GL(2,𝔽3)—21no—
8#66632—41nomaximal
9C7×2−1+4D311yescharacteristic, complemented
10#6512—31no—
11#6512—31no—
12#6510—31no—
13#6411—31no—
14#6411—31no—
15#6410—31no—
16#5431—71nomaximal, Hall
17C3×C7×D4—41no—
18D84—41no—
19C7×SL(2,𝔽3)D411yescharacteristic
20C3⋊(C7⋊C8)—41no—
21C7×#41—31no—
22C7×#41—61no—
23C7×#41C2×D311yescharacteristic
24C2×C7×Q8—21no—
25C2×C7×Q8—31no—
26C2×D28—31no—
27#1143—61no—
28#1143—61no—
29#1141—61no—
30#1141—61no—
31(C2×D7)⋊C4—62no—
32C4×Dic7—31no—
33#1136—31no—
34#1136—31no—
352−1+4⋊C3D711yescharacteristic, complemented
36#986—71no—
37GU(2,𝔽3)—71no—
38C22×C3×C7—81no—
39C2×D21—81no—
40C3×C4×C7—41no—
41#454—211noSylow
42C22×D7—61no—
43C7×Q8—31no—
44C7×Q8—61no—
45C7×Q8#6711yescharacteristic
46C7×D4—31no—
47C7×D4—61no—
48C7×D4S411yescharacteristic
49C2×C4×C7—31no—
50C2×C4×C7—32no—
51C2×C4×C7—61no—
52C2×Dic7—61no—
53D28—61no—
54D28—61no—
55C7⋊C8—61no—
56C7⋊C8—61no—
57#41⋊C3C2×D711yescharacteristic
58C2×SL(2,𝔽3)—21no—
59GL(2,𝔽3)—141no—
60#213—281no—
61C2×C3×C7—41no—
62C2×C3×C7—161no—
63D21—161no—
642−1+4D2111yescharacteristic, complemented
65AGL(1,ℤ/8)—211no—
66AGL(1,ℤ/8)—211no—
67#124—211no—
68#104—211no—
69#104—211no—
70#101—211no—
71C22×C7—21no—
72C22×C7—31no—
73C2×D7—122no—
74C2×D7—121no—
75C4×C7—31no—
76C4×C7—61no—
77C4×C7C2×S411yescharacteristic
78Dic7—121no—
79C3×D4—41no—
80D12—281no—
81SL(2,𝔽3)C22⋊D711yescharacteristic
82C3⋊C8—281no—
83C3×C7—41noHall
84#41—31no—
85#41—61no—
86#41C2×D2111yescharacteristic
87C2×Q8—21no—
88C2×Q8—31no—
89C2×D4—211no—
90SD16—421no—
91SD16—421no—
92D8—421no—
93D8—421no—
94M16—211no—
95M16—211no—
96C22⋊C4—422no—
97C42—211no—
98C2×C7—41no—
99C2×C7—61no—
100C2×C7GL(2,ℤ/4)11yescharacteristic
101D7—241no—
102C22×C3—81no—
103C2×D3—561no—
104C3×C4—41no—
105C23—421no—
106Q8—31no—
107Q8—61no—
108Q8#426811yescharacteristic
109D4—31no—
110D4—61no—
111D4—421no—
112D4—421no—
113D4C7⋊S411yescharacteristic
114C2×C4—31no—
115C2×C4—32no—
116C2×C4—61no—
117C2×C4—421no—
118C8—421no—
119C8—421no—
120C7#543111yescharacteristic, complemented, Sylow
121C2×C3—41no—
122C2×C3—161no—
123D3—1121no—
124C22—21no—
125C22—31no—
126C22—842no—
127C22—841no—
128C4—31no—
129C4—61no—
130C4—841no—
131C4C2×C7⋊S411yescharacteristic
132C3—41noSylow
133C2—41no—
134C2—61no—
135C2—1681no—
136C2#1062150611yescharacteristic
subgroupG/H
centerC2#10621506
commutator, cosocleC7×#41⋊C3C22
FrattiniC4C2×C7⋊S4
FittingC7×2−1+4D3
socleC2×C7GL(2,ℤ/4)
hypercenterD4C7⋊S4
hypocenterC7×SL(2,𝔽3)D4
LCSG ▷ C7×#41⋊C3 ▷ C7×SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C4 ◁ D4

Complements

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

NH ≅ G/Ntype
1. C7×2−1+4⋊C3C2semidirect
2. #10621504C2semidirect
3. C7⋊GU(2,𝔽3)C2semidirect
9. C7×2−1+4D3semidirect
35. 2−1+4⋊C3D7semidirect
64. 2−1+4D21semidirect
120. C7#5431semidirect

Maximal subgroups

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

subgroupindex
C7×2−1+4⋊C32
#106215042
C7⋊GU(2,𝔽3)2
#900173
#666324
#54317