Finite Group Databasebeta

← C7×#19129749 #50303885

Subgroups — C7×#19129749 #50303885

Order 38220.

144 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 36 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
1C72×C5⋊(D13⋊C3)C211yesmaximal, characteristic
2C7×C7⋊#63121C311yesmaximal, characteristic, complemented, Hall
3C5×C72×C13⋊C3C411yescharacteristic, complemented, Hall
4C7×#11763979—51nomaximal, Hall
5C72×D65C2×C311yescharacteristic
6#19129749C711yesmaximal, characteristic, direct product factor
7C7×#63121⋊C3—71nomaximal
8C72×D13⋊C3—51no—
9C5×C72×C13C3×C411yescharacteristic, complemented, Hall
10C3×C7×C7⋊AGL(1,𝔽5)—131nomaximal, Hall
11C7×C5⋊(D13⋊C3)—23no—
12C7×C5⋊(D13⋊C3)D711yescharacteristic
13C7×C5⋊(D13⋊C3)C2×C711yescharacteristic
14C7×C7⋊#261—51noHall
15C72×C13⋊C3AGL(1,𝔽5)11yescharacteristic, complemented, Hall
16C7⋊#63121C3×C711yescharacteristic, complemented
17C7×#63121—71no—
18C3×C72×D5—131no—
19C5×C7×C13⋊C3—23no—
20C5×C7×C13⋊C3Dic711yescharacteristic, complemented
21C5×C7×C13⋊C3C4×C711yescharacteristic, complemented
22C72×D13—51no—
23#11763979—51no—
24C7×AGL(1,𝔽13)—351no—
25C7×C7⋊AGL(1,𝔽5)—131noHall
26C7×D65—23no—
27C7×D65C3×D711yescharacteristic
28C7×D65C2×C3×C711yescharacteristic
29#63121⋊C3—71noHall
30C3×C5×C72—131noHall
31C72×C13C3×AGL(1,𝔽5)11yescharacteristic, complemented, Hall
32C3×C7×Dic7—651noHall
33C7×D13⋊C3—51no—
34C7×D13⋊C3—51no—
35C7×D13⋊C3—103no—
36C72×D5—131no—
37C5×C7×C13—23no—
38C5×C7×C13C3×Dic711yescharacteristic, complemented
39C5×C7×C13C3×C4×C711yescharacteristic, complemented
40C3×C7⋊AGL(1,𝔽5)—131no—
41C3×C7×AGL(1,𝔽5)—911no—
42C5⋊(D13⋊C3)C7×D711yescharacteristic
43C7⋊#261—51no—
44C7×#261—351no—
45C2×C3×C72—651no—
46C7×C13⋊C3—23no—
47C7×C13⋊C3C7×AGL(1,𝔽5)11yescharacteristic, complemented
48C7×C13⋊C3C7⋊AGL(1,𝔽5)11yescharacteristic, complemented
49#63121—71noHall
50C5×C72AGL(1,𝔽13)11yescharacteristic, complemented, Hall
51C3×C7×D5—131no—
52C3×C7×D5—131no—
53C3×C7×D5—263no—
54C7×Dic7—651noHall
55C5×C13⋊C3C7×Dic711yescharacteristic, complemented, Hall
56C7×D13—51no—
57C7×D13—51no—
58C7×D13—103no—
59AGL(1,𝔽13)—351noHall
60C3×C72—131noHall
61C7⋊AGL(1,𝔽5)—131no—
62C7×AGL(1,𝔽5)—911no—
63D65C3×C7×D711yescharacteristic
64C3×C5×C7—131no—
65C3×C5×C7—131no—
66C3×C5×C7—263no—
67C2×C72—651no—
68C7×C13—23no—
69C7×C13C3×C7×AGL(1,𝔽5)11yescharacteristic, complemented
70C7×C13C3×C7⋊AGL(1,𝔽5)11yescharacteristic, complemented
71C3×C4×C7—4551no—
72C3×Dic7—651no—
73D13⋊C3—51no—
74C7×D5—131no—
75C7×D5—131no—
76C7×D5—263no—
77C5×C13C3×C7×Dic711yescharacteristic, complemented, Hall
78C3×AGL(1,𝔽5)—911noHall
79#261—351noHall
80C72#63121⋊C311yescharacteristic, complemented, Sylow
81C2×C3×C7—651no—
82C2×C3×C7—651no—
83C2×C3×C7—1303no—
84C13⋊C3C7×C7⋊AGL(1,𝔽5)11yescharacteristic, complemented, Hall
85C5×C7—23no—
86C5×C7C7×AGL(1,𝔽13)11yescharacteristic, complemented
87C5×C7#1176397911yescharacteristic, complemented
88C3×D5—131no—
89C4×C7—4551no—
90Dic7—651no—
91D13—51no—
92C3×C7—131no—
93C3×C7—131no—
94C3×C7—263no—
95AGL(1,𝔽5)—911noHall
96C3×C5—131noHall
97C2×C7—651no—
98C2×C7—651no—
99C2×C7—1303no—
100C13C3×C7×C7⋊AGL(1,𝔽5)11yescharacteristic, complemented, Sylow
101C3×C4—4551noHall
102D5—131no—
103C7—23no—
104C7C7×#63121⋊C311yescharacteristic, complemented
105C7#1912974911yescharacteristic, direct product factor
106C2×C3—651no—
107C5C7×#1176397911yescharacteristic, complemented, Sylow
108C4—4551noSylow
109C3—131noSylow
110C2—651no—
subgroupG/H
center, hypercenterC7#19129749
commutator, hypocenterC5×C7×C13C3×C4×C7
FrattiniC1G
Fitting, socleC5×C72×C13C3×C4
prime radicalC72×C5⋊(D13⋊C3)C2
cosocleC7×D65C2×C3×C7
LCSG ▷ C5×C7×C13
UCSC1 ◁ C7

Complements

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

NH ≅ G/Ntype
2. C7×C7⋊#63121C3semidirect
3. C5×C72×C13⋊C3C4semidirect
6. #19129749C7direct (N × H)
9. C5×C72×C13C3×C4semidirect
15. C72×C13⋊C3AGL(1,𝔽5)semidirect
16. C7⋊#63121C3×C7semidirect
20. C5×C7×C13⋊C3Dic7semidirect
21. C5×C7×C13⋊C3C4×C7semidirect
31. C72×C13C3×AGL(1,𝔽5)semidirect
38. C5×C7×C13C3×Dic7semidirect
39. C5×C7×C13C3×C4×C7semidirect
47. C7×C13⋊C3C7×AGL(1,𝔽5)semidirect
48. C7×C13⋊C3C7⋊AGL(1,𝔽5)semidirect
50. C5×C72AGL(1,𝔽13)semidirect
55. C5×C13⋊C3C7×Dic7semidirect
69. C7×C13C3×C7×AGL(1,𝔽5)semidirect
70. C7×C13C3×C7⋊AGL(1,𝔽5)semidirect
77. C5×C13C3×C7×Dic7semidirect
80. C72#63121⋊C3semidirect
84. C13⋊C3C7×C7⋊AGL(1,𝔽5)semidirect
86. C5×C7C7×AGL(1,𝔽13)semidirect
87. C5×C7#11763979semidirect
100. C13C3×C7×C7⋊AGL(1,𝔽5)semidirect
104. C7C7×#63121⋊C3semidirect
105. C7#19129749direct (N × H)
107. C5C7×#11763979semidirect

Maximal subgroups

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

subgroupindex
C72×C5⋊(D13⋊C3)2
C7×C7⋊#631213
C7×#117639795
#191297497
C7×#63121⋊C37
C3×C7×C7⋊AGL(1,𝔽5)13