Finite Group Databasebeta

← D7×ΓL(1,𝔽8)×C37⋊C3 #45657447

Subgroups — D7×ΓL(1,𝔽8)×C37⋊C3 #45657447

Order 32634.

100 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 100 conjugacy classes of subgroups — too many to draw. Only its 33 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
2D7×#11715912C311yesmaximal, characteristic, complemented
3D7×#11715913C311yesmaximal, characteristic, complemented
4D7×ΓL(1,𝔽8)×C37C311yesmaximal, characteristic, complemented
5C7×D7×C37⋊C3C311yesmaximal, characteristic, complemented
6C7×#11715912C2×C311yescharacteristic, complemented
7C7×#11715913C2×C311yescharacteristic, complemented
8C7×ΓL(1,𝔽8)×C37C2×C311yescharacteristic, complemented
9C72×C37⋊C3C2×C311yescharacteristic, complemented
10C3×D7×C37⋊C3—71nomaximal
11C2×ΓL(1,𝔽8)×C37⋊C3—71nomaximal
12C7×D7×C37C3211yescharacteristic, complemented, Hall
13C3×C7×C37⋊C3—71no—
14ΓL(1,𝔽8)×C37⋊C3D711yescharacteristic, direct product factor
15C72×C37C2×C3211yescharacteristic, complemented, Hall
16C3×D7×C37—71no—
17C2×#11715912—71no—
18C2×#11715913—71no—
19C2×ΓL(1,𝔽8)×C37—71no—
20C2×C7×C37⋊C3—71no—
21D7×C37⋊C3—71no—
22D7×C37⋊C3—71no—
23D7×C37⋊C3ΓL(1,𝔽8)11yescharacteristic, direct product factor
24C3×D7×ΓL(1,𝔽8)—371nomaximal, Hall
25C3×C7×C37—71no—
26#11715913C3×D711yescharacteristic, complemented
27#11715912C3×D711yescharacteristic, complemented
28C7×C37⋊C3—61no—
29C7×C37⋊C3—71no—
30C7×C37⋊C3—71no—
31C7×C37⋊C3C2×ΓL(1,𝔽8)11yescharacteristic, complemented
32C7×C37⋊C3C3×D711yescharacteristic, complemented
33ΓL(1,𝔽8)×C37C3×D711yescharacteristic, complemented
34C2×C3×C37⋊C3—491noHall
35C2×C7×C37—71no—
36D7×C37C3×ΓL(1,𝔽8)11yescharacteristic, complemented
37C3×C7×ΓL(1,𝔽8)—371noHall
38C3×C37⋊C3—71noHall
39C3×C7×D7—371no—
40D7×ΓL(1,𝔽8)—371no—
41D7×ΓL(1,𝔽8)—371no—
42D7×ΓL(1,𝔽8)C37⋊C311yescharacteristic, direct product factor
43C7×C37—61no—
44C7×C37C2×C3×ΓL(1,𝔽8)11yescharacteristic, complemented
45C7×C37C32×D711yescharacteristic, complemented
46C2×C3×C37—491no—
47C2×C37⋊C3—71no—
48C2×C37⋊C3—491no—
49C2×C37⋊C3—491no—
50C3×C72—371no—
51C7×ΓL(1,𝔽8)—371no—
52C7×ΓL(1,𝔽8)—371no—
53C7×ΓL(1,𝔽8)C2×C37⋊C311yescharacteristic, complemented
54C32×D7—2591no—
55C2×C3×ΓL(1,𝔽8)—2591no—
56C3×C37—71no—
57C37⋊C3—71no—
58C37⋊C3—71no—
59C37⋊C3D7×ΓL(1,𝔽8)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—
66C3×D7—2591no—
67C3×D7—2591no—
68C3×D7—71no—
69C3×D7—371no—
70C2×ΓL(1,𝔽8)—2591no—
71C2×ΓL(1,𝔽8)—2591no—
72C2×ΓL(1,𝔽8)—71no—
73C37C3×D7×ΓL(1,𝔽8)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)D7×C37⋊C311yescharacteristic, direct product factor
83C2×C32—18131noHall
84C2×C7—71no—
85D7ΓL(1,𝔽8)×C37⋊C311yescharacteristic, direct product factor
86C32—2591noSylow
87C7—61no—
88C7C2×ΓL(1,𝔽8)×C37⋊C311yescharacteristic, complemented
89C7C3×D7×C37⋊C311yescharacteristic, complemented
90C2×C3—2591no—
91C2×C3—18131no—
92C2×C3—18131no—
93C2×C3—491no—
94C3—2591no—
95C3—2591no—
96C3—71no—
97C3—371no—
98C2—71noSylow
subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC72×C37C2×C32
LCSG ▷ C72×C37
UCSC1

Complements

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

NH ≅ G/Ntype
1. C7×ΓL(1,𝔽8)×C37⋊C3C2semidirect
2. D7×#11715912C3semidirect
3. D7×#11715913C3semidirect
4. D7×ΓL(1,𝔽8)×C37C3semidirect
5. C7×D7×C37⋊C3C3semidirect
6. C7×#11715912C2×C3semidirect
7. C7×#11715913C2×C3semidirect
8. C7×ΓL(1,𝔽8)×C37C2×C3semidirect
9. C72×C37⋊C3C2×C3semidirect
12. C7×D7×C37C32semidirect
14. ΓL(1,𝔽8)×C37⋊C3D7direct (N × H)
15. C72×C37C2×C32semidirect
23. D7×C37⋊C3ΓL(1,𝔽8)direct (N × H)
26. #11715913C3×D7semidirect
27. #11715912C3×D7semidirect
31. C7×C37⋊C3C2×ΓL(1,𝔽8)semidirect
32. C7×C37⋊C3C3×D7semidirect
33. ΓL(1,𝔽8)×C37C3×D7semidirect
36. D7×C37C3×ΓL(1,𝔽8)semidirect
42. D7×ΓL(1,𝔽8)C37⋊C3direct (N × H)
44. C7×C37C2×C3×ΓL(1,𝔽8)semidirect
45. C7×C37C32×D7semidirect
53. C7×ΓL(1,𝔽8)C2×C37⋊C3semidirect
59. C37⋊C3D7×ΓL(1,𝔽8)direct (N × H)
60. C7×D7C3×C37⋊C3semidirect
64. C72C2×C3×C37⋊C3semidirect
73. C37C3×D7×ΓL(1,𝔽8)semidirect
82. ΓL(1,𝔽8)D7×C37⋊C3direct (N × H)
85. D7ΓL(1,𝔽8)×C37⋊C3direct (N × H)
88. C7C2×ΓL(1,𝔽8)×C37⋊C3semidirect
89. C7C3×D7×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
D7×#117159123
D7×#117159133
D7×ΓL(1,𝔽8)×C373
C7×D7×C37⋊C33
C3×D7×C37⋊C37
C2×ΓL(1,𝔽8)×C37⋊C37
C3×D7×ΓL(1,𝔽8)37