Finite Group Databasebeta

← #92695⋊C3 #13094733

Subgroups — #92695⋊C3 #13094733

Order 1512.

188 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 37 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×C3×C3⋊AGL(1,𝔽7)C212yesmaximal, complemented
2ΓL(1,𝔽8)×C3⋊Dic3C211yesmaximal, characteristic, complemented
3#92695C311yesmaximal, characteristic, complemented
4#4246⋊C3—32nomaximal
5C2×C32×ΓL(1,𝔽8)C2211yescharacteristic
6C3×C3⋊AGL(1,𝔽7)—22no—
7C2×C3×D21C2×C312yescomplemented
8C2×C3⋊AGL(1,𝔽7)—22no—
9C2×C3⋊AGL(1,𝔽7)D312yescomplemented
10C2×C3×AGL(1,𝔽7)—32no—
11C7×C3⋊Dic3C2×C311yescharacteristic, complemented
12Dic3×ΓL(1,𝔽8)—32no—
13Dic3×ΓL(1,𝔽8)—61no—
14C3×#625—71nomaximal, Hall
15C32×ΓL(1,𝔽8)D411yescharacteristic, complemented, Hall
16#4246—32no—
17C4⋊AGL(1,𝔽7)—91no—
18C2×C32×C7C22×C311yescharacteristic
19C3×D21—22no—
20C2×C3×ΓL(1,𝔽8)—22no—
21C2×C3×ΓL(1,𝔽8)—21no—
22C2×C3×ΓL(1,𝔽8)—41no—
23C2×C3×ΓL(1,𝔽8)C2×D312yes—
24C3⋊AGL(1,𝔽7)—24no—
25C3⋊AGL(1,𝔽7)—22no—
26C3×AGL(1,𝔽7)—62no—
27C2×C32×D3—72no—
28C3×C3⋊Dic3—71no—
29C2×D21C3×D312yescomplemented
30C2×C3×D7—32no—
31C2×AGL(1,𝔽7)—32no—
32C2×AGL(1,𝔽7)—62no—
33C7×Dic3—32no—
34C7×Dic3—61no—
35C4×ΓL(1,𝔽8)—91no—
36C3×#67—212no—
37#625—71no—
38C32×C7C3×D411yescharacteristic, complemented
39C3×ΓL(1,𝔽8)—22no—
40C3×ΓL(1,𝔽8)—21no—
41C3×ΓL(1,𝔽8)—41no—
42C3×ΓL(1,𝔽8)#6712yescomplemented
43D28—91noHall
44C2×C33—71no—
45C32×D3—142no—
46C2×C3×C7—21no—
47C2×C3×C7C2×C3×D312yes—
48D21—22no—
49C3×D7—62no—
50C2×ΓL(1,𝔽8)—22no—
51C2×ΓL(1,𝔽8)—41no—
52C2×ΓL(1,𝔽8)D3211yescharacteristic
53AGL(1,𝔽7)—64no—
54AGL(1,𝔽7)—62no—
55C22×C32—212no—
56C2×C3×D3—72no—
57C2×C3×D3—72no—
58C2×C3×D3—142no—
59C3⋊Dic3AGL(1,𝔽7)11yescharacteristic, complemented
60C3×Dic3—212no—
61C3×Dic3—421no—
62C2×D7—32no—
63C4×C7—91no—
64C33—71noSylow
65C3×D4—631no—
66#67—212no—
67C3×C7—21no—
68C3×C7C3×#6712yescomplemented
69ΓL(1,𝔽8)—22no—
70ΓL(1,𝔽8)—41no—
71ΓL(1,𝔽8)#62511yescharacteristic, complemented
72C2×C32—72no—
73C2×C32—141no—
74C2×C32—142no—
75C2×C32—281no—
76C2×C32—422no—
77C2×C32C2×AGL(1,𝔽7)11yescharacteristic
78C3×D3—142no—
79C3×D3—144no—
80C3×D3—142no—
81C2×C7C3×D3211yescharacteristic
82D7—62no—
83C22×C3—212no—
84C22×C3—212no—
85C22×C3—422no—
86C2×D3—72no—
87C3×C4—631no—
88Dic3—32no—
89Dic3—61no—
90C32—72no—
91C32—141no—
92C32—142no—
93C32—281no—
94C32C4⋊AGL(1,𝔽7)11yescharacteristic, complemented
95D4—631noSylow
96C7C3×#62511yescharacteristic, complemented, Sylow
97C2×C3—21no—
98C2×C3—71no—
99C2×C3—142no—
100C2×C3—281no—
101C2×C3—422no—
102C2×C3—424no—
103C2×C3—422no—
104C2×C3D3×AGL(1,𝔽7)12yes—
105D3—142no—
106C22—212no—
107C4—91no—
108C3—21no—
109C3—71no—
110C3—142no—
111C3—281no—
112C3#4246⋊C312yescomplemented
113C2—422no—
114C2(C3⋊AGL(1,𝔽7))⋊D311yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2(C3⋊AGL(1,𝔽7))⋊D3
commutator, Fitting, socle, cosocleC2×C32×C7C22×C3
hypocenterC32×C7C3×D4
LCSG ▷ C2×C32×C7 ▷ C32×C7
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C2×C3×C3⋊AGL(1,𝔽7)C2semidirect
2. ΓL(1,𝔽8)×C3⋊Dic3C2semidirect
3. #92695C3semidirect
7. C2×C3×D21C2×C3semidirect
9. C2×C3⋊AGL(1,𝔽7)D3semidirect
11. C7×C3⋊Dic3C2×C3semidirect
15. C32×ΓL(1,𝔽8)D4semidirect
29. C2×D21C3×D3semidirect
38. C32×C7C3×D4semidirect
42. C3×ΓL(1,𝔽8)#67semidirect
59. C3⋊Dic3AGL(1,𝔽7)semidirect
68. C3×C7C3×#67semidirect
71. ΓL(1,𝔽8)#625semidirect
94. C32C4⋊AGL(1,𝔽7)semidirect
96. C7C3×#625semidirect
112. C3#4246⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C3⋊AGL(1,𝔽7)2
ΓL(1,𝔽8)×C3⋊Dic32
#926953
#4246⋊C33
C3×#6257