Finite Group Databasebeta

← C37⋊C3×ΓL(1,𝔽8)⋊D7 #45657405

Subgroups — C37⋊C3×ΓL(1,𝔽8)⋊D7 #45657405

Order 32634.

112 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 30 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
3#24925972C311yesmaximal, characteristic, complemented
4#24925971C311yesmaximal, characteristic, complemented
5C37×ΓL(1,𝔽8)⋊D7C311yesmaximal, 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
11AGL(1,𝔽7)×C37⋊C3—71nomaximal
12C37×C7⋊D7C3211yescharacteristic, complemented, Hall
13C3×C7×C37⋊C3—71no—
14ΓL(1,𝔽8)×C37⋊C3D711yescharacteristic, complemented
15C72×C37C2×C3211yescharacteristic, complemented, Hall
16C3×D7×C37—71no—
17#13095536—71no—
18#13095535—71no—
19C37×AGL(1,𝔽7)—71no—
20D7×C37⋊C3—71no—
21D7×C37⋊C3—71no—
22D7×C37⋊C3—71no—
23D7×C37⋊C3—71no—
24D7×C37⋊C3—212no—
25C3×ΓL(1,𝔽8)⋊D7—371nomaximal, Hall
26C3×C7×C37—71no—
27#11715913C3×D711yescharacteristic, complemented
28#11715912C3×D711yescharacteristic, complemented
29C7×C37⋊C3—32no—
30C7×C37⋊C3—71no—
31C7×C37⋊C3—71no—
32C7×C37⋊C3AGL(1,𝔽7)11yescharacteristic, complemented
33C7×C37⋊C3C3×D711yescharacteristic, complemented
34ΓL(1,𝔽8)×C37C3×D711yescharacteristic, complemented
35C2×C3×C37⋊C3—491noHall
36D7×C37—71no—
37D7×C37—71no—
38D7×C37—212no—
39C3×C7×ΓL(1,𝔽8)—371noHall
40C3×C37⋊C3—71noHall
41C3×C7⋊D7—371no—
42ΓL(1,𝔽8)⋊D7—371no—
43ΓL(1,𝔽8)⋊D7—371no—
44ΓL(1,𝔽8)⋊D7C37⋊C311yescharacteristic, direct product factor
45C7×C37—32no—
46C7×C37C3×AGL(1,𝔽7)11yescharacteristic, complemented
47C7×C37C32×D711yescharacteristic, complemented
48C2×C3×C37—491no—
49C2×C37⋊C3—491no—
50C2×C37⋊C3—491no—
51C2×C37⋊C3—491no—
52C3×C72—371no—
53C7×ΓL(1,𝔽8)—371no—
54C7×ΓL(1,𝔽8)—371no—
55C7×ΓL(1,𝔽8)C2×C37⋊C311yescharacteristic, complemented
56C32×D7—2591no—
57C3×AGL(1,𝔽7)—2591no—
58C3×C37—71no—
59C37⋊C3—71no—
60C37⋊C3—71no—
61C37⋊C3ΓL(1,𝔽8)⋊D711yescharacteristic, direct product factor
62C7⋊D7C3×C37⋊C311yescharacteristic, complemented, Hall
63C2×C37—491noHall
64C32×C7—2591no—
65C3×ΓL(1,𝔽8)—371no—
66C72C2×C3×C37⋊C311yescharacteristic, complemented, Sylow
67C3×D7—2591no—
68C3×D7—2591no—
69C3×D7—2591no—
70C3×D7—2591no—
71C3×D7—71no—
72C3×D7—7772no—
73AGL(1,𝔽7)—2591no—
74AGL(1,𝔽7)—2591no—
75AGL(1,𝔽7)—71no—
76C37C3×ΓL(1,𝔽8)⋊D711yescharacteristic, complemented, Sylow
77C3×C7—2591no—
78C3×C7—2591no—
79C3×C7—71no—
80C3×C7—371no—
81C3×C7—371no—
82C3×C7—1112no—
83ΓL(1,𝔽8)—371no—
84ΓL(1,𝔽8)—371no—
85ΓL(1,𝔽8)D7×C37⋊C311yescharacteristic, complemented
86C2×C32—18131noHall
87D7—71no—
88D7—71no—
89D7—212no—
90C32—2591noSylow
91C7—32no—
92C7AGL(1,𝔽7)×C37⋊C311yescharacteristic, complemented
93C7C3×D7×C37⋊C311yescharacteristic, complemented
94C2×C3—18131no—
95C2×C3—18131no—
96C2×C3—18131no—
97C2×C3—491no—
98C3—2591no—
99C3—2591no—
100C3—71no—
101C3—371no—
102C2—491noSylow
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. C7⋊D7×C37⋊C3C3semidirect
3. #24925972C3semidirect
4. #24925971C3semidirect
5. C37×ΓL(1,𝔽8)⋊D7C3semidirect
6. C7×#11715912C2×C3semidirect
7. C7×#11715913C2×C3semidirect
8. C7×ΓL(1,𝔽8)×C37C2×C3semidirect
9. C72×C37⋊C3C2×C3semidirect
12. C37×C7⋊D7C32semidirect
14. ΓL(1,𝔽8)×C37⋊C3D7semidirect
15. C72×C37C2×C32semidirect
27. #11715913C3×D7semidirect
28. #11715912C3×D7semidirect
32. C7×C37⋊C3AGL(1,𝔽7)semidirect
33. C7×C37⋊C3C3×D7semidirect
34. ΓL(1,𝔽8)×C37C3×D7semidirect
44. ΓL(1,𝔽8)⋊D7C37⋊C3direct (N × H)
46. C7×C37C3×AGL(1,𝔽7)semidirect
47. C7×C37C32×D7semidirect
55. C7×ΓL(1,𝔽8)C2×C37⋊C3semidirect
61. C37⋊C3ΓL(1,𝔽8)⋊D7direct (N × H)
62. C7⋊D7C3×C37⋊C3semidirect
66. C72C2×C3×C37⋊C3semidirect
76. C37C3×ΓL(1,𝔽8)⋊D7semidirect
85. ΓL(1,𝔽8)D7×C37⋊C3semidirect
92. C7AGL(1,𝔽7)×C37⋊C3semidirect
93. 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
C7⋊D7×C37⋊C33
#249259723
#249259713
C37×ΓL(1,𝔽8)⋊D73
C3×D7×C37⋊C37
AGL(1,𝔽7)×C37⋊C37
C3×ΓL(1,𝔽8)⋊D737