Finite Group Databasebeta

← #10621394

Subgroups — #10621394

Order 672.

136 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 41 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
1#66730C211yesmaximal, characteristic, complemented
2D4×D21C211yesmaximal, characteristic, complemented
3C3×C7×D8C211yesmaximal, characteristic, complemented
4#66633C211yesmaximal, characteristic, complemented
5#66632C211yesmaximal, characteristic, complemented
6#66623C211yesmaximal, characteristic, complemented
7#66622C211yesmaximal, characteristic, complemented
8#6474—31nomaximal, Hall
9C22×D21—21no—
10C3×C7×D4C2211yescharacteristic
11C3×C7×D4C2211yescharacteristic, complemented
12#4268—21no—
13#4268—21no—
14C2×Dic21—21no—
15D84C2211yescharacteristic
16C4×D21C2211yescharacteristic
17Dic42C2211yescharacteristic, complemented
18C3×C7×C8C2211yescharacteristic, complemented
19C3⋊(C7⋊C8)C2211yescharacteristic, complemented
20#1159—31no—
21D4×D7—31no—
22C7×D8D311yescharacteristic, complemented
23#1142—31no—
24#1141—31no—
25#1132—31no—
26#1131—31no—
27#911—71nomaximal, Hall
28C22×C3×C7—21no—
29C22×C3×C7—21no—
30C2×D21—21no—
31C2×D21—42no—
32C2×D21D411yescharacteristic
33C3×C4×C7C2311yescharacteristic
34Dic21—21no—
35Dic21D411yescharacteristic
36C22×D7—61no—
37C7×D4C2×D311yescharacteristic
38C7×D4C2×D311yescharacteristic, complemented
39C22⋊D7—61no—
40C22⋊D7—61no—
41C2×Dic7—61no—
42D28—31no—
43C4×D7—31no—
44Dic14—31no—
45C7×C8C2×D311yescharacteristic, complemented
46C7⋊C8—31no—
47#237—71no—
48D3×D4—71no—
49C3×D8D711yescharacteristic, complemented
50#214—71no—
51#213—71no—
52#204—71no—
53ΓL(1,𝔽25)—71no—
54C2×C3×C7—41no—
55C2×C3×C7—41no—
56C2×C3×C7C2×D411yescharacteristic
57D21—21no—
58D21—41no—
59AGL(1,ℤ/8)—211noSylow
60C22×C7—21no—
61C22×C7—21no—
62C2×D7—31no—
63C2×D7—61no—
64C2×D7—122no—
65C4×C7C22×D311yescharacteristic
66Dic7—31no—
67Dic7—61no—
68C22×D3—141no—
69C3×D4C2×D711yescharacteristic
70C3×D4C2×D711yescharacteristic, complemented
71#67—141no—
72#67—141no—
73C2×Dic3—141no—
74D12—71no—
75C4×D3—71no—
76Dic6—71no—
77C3×C8C2×D711yescharacteristic, complemented
78C3⋊C8—71no—
79C3×C7AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
80#41—211no—
81C2×D4—211no—
82SD16—211no—
83SD16—211no—
84D8—211no—
85D8D2111yescharacteristic, complemented
86M16—211no—
87C2×C7—41no—
88C2×C7—41no—
89C2×C7D3×D411yescharacteristic
90D7—61no—
91D7—121no—
92C22×C3—21no—
93C22×C3—21no—
94C2×D3—71no—
95C2×D3—141no—
96C2×D3—282no—
97C3×C4C22×D711yescharacteristic
98Dic3—71no—
99Dic3—141no—
100C23—421no—
101Q8—211no—
102D4—211no—
103D4—421no—
104D4—421no—
105D4C2×D2111yescharacteristic
106D4C2×D2111yescharacteristic, complemented
107C2×C4—211no—
108C2×C4—421no—
109C8—211no—
110C8C2×D2111yescharacteristic, complemented
111C7#91111yescharacteristic, complemented, Sylow
112C2×C3—41no—
113C2×C3—41no—
114C2×C3D4×D711yescharacteristic
115D3—141no—
116D3—281no—
117C22—21no—
118C22—21no—
119C22—211no—
120C22—421no—
121C22—842no—
122C4—211no—
123C4—421no—
124C4C22×D2111yescharacteristic
125C3#647411yescharacteristic, complemented, Sylow
126C2—41no—
127C2—41no—
128C2—421no—
129C2—841no—
130C2D4×D2111yescharacteristic
subgroupG/H
centerC2D4×D21
commutator, cosocleC3×C4×C7C23
FrattiniC4C22×D21
FittingC3×C7×D8C2
socleC2×C3×C7C2×D4
hypercenterD8D21
hypocenterC3×C7AGL(1,ℤ/8)
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C4 ◁ D8

Complements

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

NH ≅ G/Ntype
1. #66730C2semidirect
2. D4×D21C2semidirect
3. C3×C7×D8C2semidirect
4. #66633C2semidirect
5. #66632C2semidirect
6. #66623C2semidirect
7. #66622C2semidirect
11. C3×C7×D4C22semidirect
17. Dic42C22semidirect
18. C3×C7×C8C22semidirect
19. C3⋊(C7⋊C8)C22semidirect
22. C7×D8D3semidirect
38. C7×D4C2×D3semidirect
45. C7×C8C2×D3semidirect
49. C3×D8D7semidirect
70. C3×D4C2×D7semidirect
77. C3×C8C2×D7semidirect
79. C3×C7AGL(1,ℤ/8)semidirect
85. D8D21semidirect
106. D4C2×D21semidirect
110. C8C2×D21semidirect
111. C7#911semidirect
125. C3#6474semidirect

Maximal subgroups

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

subgroupindex
#667302
D4×D212
C3×C7×D82
#666332
#666322
#666232
#666222
#64743
#9117