Finite Group Databasebeta

← C7×C3⋊(ΓL(1,𝔽8)⋊D25) #36347753

Subgroups — C7×C3⋊(ΓL(1,𝔽8)⋊D25) #36347753

Order 22050.

114 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 114 conjugacy classes of subgroups — too many to draw. Only its 40 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
1C3×C7×ΓL(1,𝔽8)×C25C211yesmaximal, characteristic, complemented, Hall
2C7×D525C311yesmaximal, characteristic, complemented
3C7×ΓL(1,𝔽8)⋊D25—31nomaximal
4C7×C3⋊(C5⋊AGL(1,𝔽7))—51nomaximal
5C7×ΓL(1,𝔽8)×C25—21no—
6C7×ΓL(1,𝔽8)×C25D311yescharacteristic, complemented
7C3×C72×C25C2×C311yescharacteristic, complemented
8C3×C7×D75—71nomaximal
9C3⋊(ΓL(1,𝔽8)⋊D25)C711yesmaximal, characteristic, direct product factor
10C7×D175—31noHall
11C3×C5×C7×ΓL(1,𝔽8)D511yescharacteristic
12C32×C7×C25—71no—
13C3×ΓL(1,𝔽8)×C25C2×C711yescharacteristic, complemented
14C7×D105—51no—
15C7×C5⋊AGL(1,𝔽7)—151no—
16C72×C25C3×D311yescharacteristic, complemented, Hall
17D525C3×C711yescharacteristic, complemented
18C7×D75—71no—
19C3×C7×D25—211no—
20ΓL(1,𝔽8)⋊D25—31no—
21C7×C3⋊AGL(1,𝔽7)—251noHall
22C3×C5×C72C3×D511yescharacteristic
23C5×C7×ΓL(1,𝔽8)—21no—
24C5×C7×ΓL(1,𝔽8)D1511yescharacteristic
25C3×C7×D15—351no—
26C3⋊(C5⋊AGL(1,𝔽7))—51no—
27C3×C7×C25—61no—
28C3×C7×C25—141no—
29C3×C7×C25—141no—
30C3×C7×C25AGL(1,𝔽7)11yescharacteristic, complemented
31C3×C7×C25C2×C3×C711yescharacteristic, complemented
32ΓL(1,𝔽8)×C25—21no—
33ΓL(1,𝔽8)×C25D3×C711yescharacteristic, complemented
34C7×D35—151no—
35C3×D75—71noHall
36C3×C7×ΓL(1,𝔽8)D2511yescharacteristic, complemented, Hall
37D175—31no—
38C7×D25—211no—
39C32×C5×C7—141no—
40C3×C5×ΓL(1,𝔽8)C7×D511yescharacteristic
41C7×D21—251no—
42C7×AGL(1,𝔽7)—751no—
43C5×C72C3×D1511yescharacteristic
44C32×C25—141noHall
45D105—51no—
46C7×D15—351no—
47C3×C7×D5—1051no—
48C5⋊AGL(1,𝔽7)—151no—
49C7×C25—61no—
50C7×C25C3⋊AGL(1,𝔽7)11yescharacteristic, complemented
51C7×C25C3×D3×C711yescharacteristic, complemented
52D75—71no—
53C3×D25—211no—
54C3×C72C3×D2511yescharacteristic, complemented
55C7×ΓL(1,𝔽8)—21no—
56C7×ΓL(1,𝔽8)D7511yescharacteristic, complemented
57C3×D3×C7—1751no—
58C3⋊AGL(1,𝔽7)—251no—
59C3×C5×C7—61no—
60C3×C5×C7—141no—
61C3×C5×C7—141no—
62C3×C5×C7C5⋊AGL(1,𝔽7)11yescharacteristic
63C3×C5×C7C3×C7×D511yescharacteristic
64C5×ΓL(1,𝔽8)—21no—
65C5×ΓL(1,𝔽8)C7×D1511yescharacteristic
66C7×D7—751noHall
67C3×D15—351no—
68C3×C25—141no—
69C3×C25—141no—
70C3×C25C7×AGL(1,𝔽7)11yescharacteristic, complemented
71D35—151no—
72C7×D5—1051no—
73C32×C7—141no—
74C3×ΓL(1,𝔽8)C7×D2511yescharacteristic, complemented
75D25—211noHall
76C72C3×D7511yescharacteristic, complemented, Sylow
77C32×C5—141no—
78C2×C3×C7—5251no—
79D21—251no—
80D3×C7—1751no—
81AGL(1,𝔽7)—751no—
82C5×C7—61no—
83C5×C7C3⋊(C5⋊AGL(1,𝔽7))11yescharacteristic
84C5×C7C3×C7×D1511yescharacteristic
85D15—351no—
86C3×D5—1051no—
87C25C7×C3⋊AGL(1,𝔽7)11yescharacteristic, complemented, Sylow
88C3×C7—61no—
89C3×C7—141no—
90C3×C7—141no—
91C3×C7ΓL(1,𝔽8)⋊D2511yescharacteristic, complemented
92C3×C7C3×C7×D2511yescharacteristic, complemented
93ΓL(1,𝔽8)—21no—
94ΓL(1,𝔽8)C7×D7511yescharacteristic, complemented
95C3×D3—1751noHall
96C3×C5—141no—
97C3×C5—141no—
98C3×C5C7×C5⋊AGL(1,𝔽7)11yescharacteristic
99C2×C7—5251no—
100D7—751no—
101D5—1051no—
102C32—141noSylow
103C7—61no—
104C7C3⋊(ΓL(1,𝔽8)⋊D25)11yescharacteristic, direct product factor
105C7C3×C7×D7511yescharacteristic, complemented
106C2×C3—5251no—
107D3—1751no—
108C5C7×C3⋊(C5⋊AGL(1,𝔽7))11yescharacteristic
109C3—141no—
110C3—141no—
111C3C7×ΓL(1,𝔽8)⋊D2511yescharacteristic, complemented
112C2—5251noSylow
subgroupG/H
center, hypercenterC7C3⋊(ΓL(1,𝔽8)⋊D25)
commutator, cosocle, hypocenterC3×C7×C25C2×C3×C7
FrattiniC5C7×C3⋊(C5⋊AGL(1,𝔽7))
FittingC3×C72×C25C2×C3
socleC3×C5×C72C3×D5
LCSG ▷ C3×C7×C25
UCSC1 ◁ C7

Complements

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

NH ≅ G/Ntype
1. C3×C7×ΓL(1,𝔽8)×C25C2semidirect
2. C7×D525C3semidirect
6. C7×ΓL(1,𝔽8)×C25D3semidirect
7. C3×C72×C25C2×C3semidirect
9. C3⋊(ΓL(1,𝔽8)⋊D25)C7direct (N × H)
13. C3×ΓL(1,𝔽8)×C25C2×C7semidirect
16. C72×C25C3×D3semidirect
17. D525C3×C7semidirect
30. C3×C7×C25AGL(1,𝔽7)semidirect
31. C3×C7×C25C2×C3×C7semidirect
33. ΓL(1,𝔽8)×C25D3×C7semidirect
36. C3×C7×ΓL(1,𝔽8)D25semidirect
50. C7×C25C3⋊AGL(1,𝔽7)semidirect
51. C7×C25C3×D3×C7semidirect
54. C3×C72C3×D25semidirect
56. C7×ΓL(1,𝔽8)D75semidirect
70. C3×C25C7×AGL(1,𝔽7)semidirect
74. C3×ΓL(1,𝔽8)C7×D25semidirect
76. C72C3×D75semidirect
87. C25C7×C3⋊AGL(1,𝔽7)semidirect
91. C3×C7ΓL(1,𝔽8)⋊D25semidirect
92. C3×C7C3×C7×D25semidirect
94. ΓL(1,𝔽8)C7×D75semidirect
104. C7C3⋊(ΓL(1,𝔽8)⋊D25)direct (N × H)
105. C7C3×C7×D75semidirect
111. C3C7×ΓL(1,𝔽8)⋊D25semidirect

Maximal subgroups

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

subgroupindex
C3×C7×ΓL(1,𝔽8)×C252
C7×D5253
C7×ΓL(1,𝔽8)⋊D253
C7×C3⋊(C5⋊AGL(1,𝔽7))5
C3×C7×D757
C3⋊(ΓL(1,𝔽8)⋊D25)7