Finite Group Databasebeta

← C7×AGL(1,𝔽7)×C37⋊C3 #45657399

Subgroups — C7×AGL(1,𝔽7)×C37⋊C3 #45657399

Order 32634.

100 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 100 conjugacy classes of subgroups — too many to draw. Only its 38 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×ΓL(1,𝔽8)×C37⋊C3C211yesmaximal, characteristic, complemented, Hall
2C7×D7×C37⋊C3C311yesmaximal, characteristic, complemented
3C7×#13095536C311yesmaximal, characteristic, complemented
4C7×#13095535C311yesmaximal, characteristic, complemented
5C7×C37×AGL(1,𝔽7)C311yesmaximal, characteristic, complemented
6C7×#11715912C2×C311yescharacteristic, complemented
7C7×#11715913C2×C311yescharacteristic, complemented
8C7×ΓL(1,𝔽8)×C37C2×C311yescharacteristic, complemented
9C72×C37⋊C3C2×C311yescharacteristic, complemented
10C2×C3×C7×C37⋊C3—71nomaximal
11AGL(1,𝔽7)×C37⋊C3C711yesmaximal, characteristic, direct product factor
12C7×D7×C37C3211yescharacteristic, complemented, Hall
13C3×C7×C37⋊C3—71no—
14ΓL(1,𝔽8)×C37⋊C3C2×C711yescharacteristic, complemented
15C72×C37C2×C3211yescharacteristic, complemented, Hall
16C2×C3×C7×C37—71no—
17C2×C7×C37⋊C3—71no—
18C2×C7×C37⋊C3—71no—
19C2×C7×C37⋊C3—71no—
20#13095536C3×C711yescharacteristic, complemented
21#13095535C3×C711yescharacteristic, complemented
22C37×AGL(1,𝔽7)C3×C711yescharacteristic, complemented
23D7×C37⋊C3C3×C711yescharacteristic, complemented
24C3×C7×AGL(1,𝔽7)—371nomaximal, Hall
25C3×C7×C37—71no—
26#11715913C2×C3×C711yescharacteristic, complemented
27#11715912C2×C3×C711yescharacteristic, complemented
28C7×C37⋊C3—61no—
29C7×C37⋊C3—71no—
30C7×C37⋊C3—71no—
31C7×C37⋊C3AGL(1,𝔽7)11yescharacteristic, direct product factor
32C7×C37⋊C3C2×C3×C711yescharacteristic, complemented
33ΓL(1,𝔽8)×C37C2×C3×C711yescharacteristic, complemented
34C2×C3×C37⋊C3—71noHall
35C2×C7×C37—71no—
36D7×C37C32×C711yescharacteristic, complemented
37C3×C7×ΓL(1,𝔽8)—371noHall
38C3×C37⋊C3—71noHall
39C3×C7×D7—371no—
40C7×AGL(1,𝔽7)—371no—
41C7×AGL(1,𝔽7)—371no—
42C7×AGL(1,𝔽7)C37⋊C311yescharacteristic, direct product factor
43C7×C37—61no—
44C7×C37C3×AGL(1,𝔽7)11yescharacteristic, complemented
45C7×C37C2×C32×C711yescharacteristic, complemented
46C2×C3×C37—71no—
47C2×C37⋊C3—71no—
48C2×C37⋊C3—71no—
49C2×C37⋊C3—71no—
50C3×C72—371no—
51C7×ΓL(1,𝔽8)—371no—
52C7×ΓL(1,𝔽8)—371no—
53C7×ΓL(1,𝔽8)C2×C37⋊C311yescharacteristic, complemented
54C2×C32×C7—2591no—
55C3×AGL(1,𝔽7)—371no—
56C3×C37—71no—
57C37⋊C3—71no—
58C37⋊C3—71no—
59C37⋊C3C7×AGL(1,𝔽7)11yescharacteristic, direct product factor
60C7×D7C3×C37⋊C311yescharacteristic, complemented, Hall
61C2×C37—71noHall
62C32×C7—2591no—
63C3×ΓL(1,𝔽8)—371no—
64C72C2×C3×C37⋊C311yescharacteristic, complemented, Sylow
65C2×C3×C7—2591no—
66C2×C3×C7—2591no—
67C2×C3×C7—2591no—
68C2×C3×C7—71no—
69C3×D7—371no—
70AGL(1,𝔽7)—371no—
71AGL(1,𝔽7)—371no—
72AGL(1,𝔽7)C7×C37⋊C311yescharacteristic, direct product factor
73C37C3×C7×AGL(1,𝔽7)11yescharacteristic, complemented, Sylow
74C3×C7—2591no—
75C3×C7—2591no—
76C3×C7—71no—
77C3×C7—371no—
78C3×C7—371no—
79C3×C7—2221no—
80ΓL(1,𝔽8)—371no—
81ΓL(1,𝔽8)—371no—
82ΓL(1,𝔽8)C2×C7×C37⋊C311yescharacteristic, complemented
83C2×C32—2591noHall
84C2×C7—71no—
85D7C3×C7×C37⋊C311yescharacteristic, complemented
86C32—2591noSylow
87C7—61no—
88C7AGL(1,𝔽7)×C37⋊C311yescharacteristic, direct product factor
89C7C2×C3×C7×C37⋊C311yescharacteristic, complemented
90C2×C3—2591no—
91C2×C3—2591no—
92C2×C3—2591no—
93C2×C3—71no—
94C3—2591no—
95C3—2591no—
96C3—71no—
97C3—371no—
98C2—71noSylow
subgroupG/H
center, hypercenterC7AGL(1,𝔽7)×C37⋊C3
commutator, cosocle, hypocenterC7×C37C2×C32×C7
FrattiniC1G
Fitting, socleC72×C37C2×C32
LCSG ▷ C7×C37
UCSC1 ◁ C7

Complements

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

NH ≅ G/Ntype
1. C7×ΓL(1,𝔽8)×C37⋊C3C2semidirect
2. C7×D7×C37⋊C3C3semidirect
3. C7×#13095536C3semidirect
4. C7×#13095535C3semidirect
5. C7×C37×AGL(1,𝔽7)C3semidirect
6. C7×#11715912C2×C3semidirect
7. C7×#11715913C2×C3semidirect
8. C7×ΓL(1,𝔽8)×C37C2×C3semidirect
9. C72×C37⋊C3C2×C3semidirect
11. AGL(1,𝔽7)×C37⋊C3C7direct (N × H)
12. C7×D7×C37C32semidirect
14. ΓL(1,𝔽8)×C37⋊C3C2×C7semidirect
15. C72×C37C2×C32semidirect
20. #13095536C3×C7semidirect
21. #13095535C3×C7semidirect
22. C37×AGL(1,𝔽7)C3×C7semidirect
23. D7×C37⋊C3C3×C7semidirect
26. #11715913C2×C3×C7semidirect
27. #11715912C2×C3×C7semidirect
31. C7×C37⋊C3AGL(1,𝔽7)direct (N × H)
32. C7×C37⋊C3C2×C3×C7semidirect
33. ΓL(1,𝔽8)×C37C2×C3×C7semidirect
36. D7×C37C32×C7semidirect
42. C7×AGL(1,𝔽7)C37⋊C3direct (N × H)
44. C7×C37C3×AGL(1,𝔽7)semidirect
45. C7×C37C2×C32×C7semidirect
53. C7×ΓL(1,𝔽8)C2×C37⋊C3semidirect
59. C37⋊C3C7×AGL(1,𝔽7)direct (N × H)
60. C7×D7C3×C37⋊C3semidirect
64. C72C2×C3×C37⋊C3semidirect
72. AGL(1,𝔽7)C7×C37⋊C3direct (N × H)
73. C37C3×C7×AGL(1,𝔽7)semidirect
82. ΓL(1,𝔽8)C2×C7×C37⋊C3semidirect
85. D7C3×C7×C37⋊C3semidirect
88. C7AGL(1,𝔽7)×C37⋊C3direct (N × H)
89. C7C2×C3×C7×C37⋊C3semidirect

Maximal subgroups

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

subgroupindex
C7×ΓL(1,𝔽8)×C37⋊C32
C7×D7×C37⋊C33
C7×#130955363
C7×#130955353
C7×C37×AGL(1,𝔽7)3
C2×C3×C7×C37⋊C37
AGL(1,𝔽7)×C37⋊C37
C3×C7×AGL(1,𝔽7)37