Finite Group Databasebeta

← #13056704

Subgroups — #13056704

Order 1344.

220 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 67 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
1C3×C7×#125C211yesmaximal, characteristic, complemented
2#10621357C211yesmaximal, characteristic
3#10620648C214yesmaximal, complemented
4#10620642C211yesmaximal, characteristic, complemented
5#89839—31nomaximal, Hall
6C2×D84—22no—
7C2×D84C2212yes—
8C3×C7×C4⋊C4—22no—
9C3×C7×C4⋊C4C2212yes—
10C3×C42×C7C2211yescharacteristic
11C2×C3⋊(C7⋊C8)C2212yes—
12C7×#125D311yescharacteristic, complemented
13#6437—31no—
14#6383—34no—
15#6377—31no—
16#5067—71nomaximal, Hall
17C22×D21—22no—
18C2×C3×C4×C7—22no—
19C2×C3×C4×C7D412yes—
20C2×C3×C4×C7C2311yescharacteristic
21D84—24no—
22D84—44no—
23C3⋊(C7⋊C8)—22no—
24C2×D28—32no—
25C2×D28—62no—
26C7×C4⋊C4—22no—
27C7×C4⋊C4C2×D312yes—
28C42×C7C2×D311yescharacteristic
29C2×C7⋊C8—32no—
30C3×#125D711yescharacteristic, complemented
31#874—71no—
32#809—74no—
33#803—71no—
34C22×C3×C7C2×D411yescharacteristic
35C2×D21—42no—
36C2×D21—44no—
37C3×C4×C7—22no—
38C3×C4×C7—42no—
39C3×C4×C7#4112yes—
40#490—211noSylow
41C22×D7—62no—
42C2×C4×C7—22no—
43C2×C4×C7#6712yes—
44C2×C4×C7C22×D311yescharacteristic
45D28—64no—
46D28—124no—
47C7⋊C8—62no—
48C2×D12—72no—
49C2×D12—142no—
50C3×C4⋊C4—22no—
51C3×C4⋊C4C2×D712yes—
52C3×C42C2×D711yescharacteristic
53C2×C3⋊C8—72no—
54C2×C3×C7#12411yescharacteristic
55C2×C3×C7AGL(1,ℤ/8)12yes—
56D21—82no—
57#127—211no—
58#125D2111yescharacteristic, complemented
59#102—214no—
60#97—211no—
61C22×C7C2×#6711yescharacteristic
62C2×D7—122no—
63C2×D7—124no—
64C4×C7—22no—
65C4×C7—42no—
66C4×C7#23912yes—
67C22×D3—142no—
68C2×C3×C4—22no—
69C2×C3×C4C22⋊D712yes—
70C2×C3×C4C22×D711yescharacteristic
71D12—144no—
72D12—284no—
73C3⋊C8—142no—
74C3×C7#49011yescharacteristic, complemented, Hall
75C2×D4—212no—
76C2×D4—422no—
77C2×C8—212no—
78C4⋊C4—22no—
79C4⋊C4C2×D2112yes—
80C42C2×D2111yescharacteristic
81C2×C7#94711yescharacteristic
82C2×C7#94912yes—
83D7—242no—
84C22×C3C2×C22⋊D711yescharacteristic
85C2×D3—282no—
86C2×D3—284no—
87C3×C4—22no—
88C3×C4—42no—
89C3×C4#116112yes—
90C23—422no—
91D4—424no—
92D4—844no—
93C2×C4—22no—
94C2×C4#426812yes—
95C2×C4C22×D2111yescharacteristic
96C8—422no—
97C7#506711yescharacteristic, complemented, Sylow
98C2×C3#651011yescharacteristic
99C2×C3#651212yes—
100D3—562no—
101C22—842no—
102C22—844no—
103C22C2×#426811yescharacteristic
104C4—22no—
105C4—42no—
106C4#6673212yes—
107C3#8983911yescharacteristic, complemented, Sylow
108C2—1682no—
109C2#1062143011yescharacteristic
110C2#1062143212yes—
subgroupG/H
centerC22C2×#4268
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
FittingC3×C7×#125C2
socleC22×C3×C7C2×D4
prime radical#10621357C2
hypercenter#125D21
hypocenterC3×C7#490
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C42 ◁ #125

Complements

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

NH ≅ G/Ntype
1. C3×C7×#125C2semidirect
3. #10620648C2semidirect
4. #10620642C2semidirect
12. C7×#125D3semidirect
30. C3×#125D7semidirect
58. #125D21semidirect
74. C3×C7#490semidirect
97. C7#5067semidirect
107. C3#89839semidirect

Maximal subgroups

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

subgroupindex
C3×C7×#1252
#106213572
#106206482
#106206422
#898393
#50677