Finite Group Databasebeta

← C3×#23756143 #44472647

Subgroups — C3×#23756143 #44472647

Order 29988.

360 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 102 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
1C2×C3×C17×#3861C211yesmaximal, characteristic
2C3×C7⋊Dic119C311yesmaximal, characteristic, complemented
3#23756143C313yesmaximal, direct product factor
4C3×C17×#3861C411yescharacteristic, complemented, Hall
5C2×C17×#3861C2×C313yes—
6C2×C3×C72×C17C2×C311yescharacteristic
7C3×ΓL(1,𝔽8)⋊Dic17—72nomaximal
8C3×ΓL(1,𝔽8)⋊Dic17—75nomaximal
9C3×ΓL(1,𝔽8)⋊Dic17—71nomaximal
10C7⋊Dic119C3211yescharacteristic, complemented, Hall
11C17×#3861C3×C413yescomplemented
12C3×C72×C17C3×C411yescharacteristic, complemented
13C2×C3×C17×ΓL(1,𝔽8)—72no—
14C2×C3×C17×ΓL(1,𝔽8)—75no—
15C2×C3×C17×ΓL(1,𝔽8)—71no—
16C3×#10596874—171nomaximal, Hall
17C2×C72×C17C2×C3211yescharacteristic
18C3×Dic119—72no—
19C3×Dic119—215no—
20C3×Dic119—211no—
21ΓL(1,𝔽8)⋊Dic17—76no—
22ΓL(1,𝔽8)⋊Dic17—715no—
23ΓL(1,𝔽8)⋊Dic17—73no—
24C3×C17×ΓL(1,𝔽8)—72no—
25C3×C17×ΓL(1,𝔽8)—75no—
26C3×C17×ΓL(1,𝔽8)—71no—
27C2×C3×#3861D1711yescharacteristic
28C72×C17C32×C411yescharacteristic, complemented, Hall
29C2×C3×C7×C17AGL(1,𝔽7)12yes—
30C2×C3×C7×C17AGL(1,𝔽7)15yes—
31C2×C3×C7×C17AGL(1,𝔽7)11yescharacteristic
32C2×C17×ΓL(1,𝔽8)—76no—
33C2×C17×ΓL(1,𝔽8)—715no—
34C2×C17×ΓL(1,𝔽8)—73no—
35C32×Dic17—491noHall
36C3×C7⋊Dic7—171no—
37#10596874—173no—
38Dic119—72no—
39Dic119—215no—
40Dic119—211no—
41C3×#3861Dic1711yescharacteristic, complemented, Hall
42C3×C7×C17ΓL(1,𝔽8)⋊C412yescomplemented
43C3×C7×C17ΓL(1,𝔽8)⋊C415yescomplemented
44C3×C7×C17ΓL(1,𝔽8)⋊C411yescharacteristic, complemented
45C17×ΓL(1,𝔽8)—76no—
46C17×ΓL(1,𝔽8)—715no—
47C17×ΓL(1,𝔽8)—73no—
48C2×C32×C17—491no—
49C2×C3×C72C3×D1711yescharacteristic
50C2×#3861C3×D1713yes—
51C3×ΓL(1,𝔽8)⋊C4—1192no—
52C3×ΓL(1,𝔽8)⋊C4—1195no—
53C3×ΓL(1,𝔽8)⋊C4—1191no—
54C2×C7×C17C3×AGL(1,𝔽7)12yes—
55C2×C7×C17C3×AGL(1,𝔽7)15yes—
56C2×C7×C17C3×AGL(1,𝔽7)11yescharacteristic
57C3×Dic17—493no—
58C3×Dic17—1471no—
59C7⋊Dic7—171noHall
60C32×C17—491noHall
61C3×C72C3×Dic1711yescharacteristic, complemented
62#3861C3×Dic1713yescomplemented
63C2×C3×ΓL(1,𝔽8)—72no—
64C2×C3×ΓL(1,𝔽8)—75no—
65C2×C3×ΓL(1,𝔽8)—71no—
66C7×C17C3×ΓL(1,𝔽8)⋊C412yescomplemented
67C7×C17C3×ΓL(1,𝔽8)⋊C415yescomplemented
68C7×C17C3×ΓL(1,𝔽8)⋊C411yescharacteristic, complemented
69C2×C3×C17—493no—
70C2×C3×C17#6442411yescharacteristic
71C2×C72C32×D1711yescharacteristic
72C3×Dic7—3572no—
73C3×Dic7—3575no—
74C3×Dic7—1191no—
75ΓL(1,𝔽8)⋊C4—1196no—
76ΓL(1,𝔽8)⋊C4—11915no—
77ΓL(1,𝔽8)⋊C4—1193no—
78Dic17—1471noHall
79C3×ΓL(1,𝔽8)—72no—
80C3×ΓL(1,𝔽8)—75no—
81C3×ΓL(1,𝔽8)—71no—
82C3×C17—493no—
83C3×C17#1059687411yescharacteristic, complemented
84C72C32×Dic1711yescharacteristic, complemented, Sylow
85C2×C3×C7C17⋊AGL(1,𝔽7)12yes—
86C2×C3×C7C17⋊AGL(1,𝔽7)15yes—
87C2×C3×C7C17⋊AGL(1,𝔽7)11yescharacteristic
88C2×ΓL(1,𝔽8)—76no—
89C2×ΓL(1,𝔽8)—715no—
90C2×ΓL(1,𝔽8)—73no—
91C32×C4—8331noHall
92C2×C17C3×#6442411yescharacteristic
93Dic7—3572no—
94Dic7—3575no—
95Dic7—1191no—
96C3×C7ΓL(1,𝔽8)⋊Dic1712yescomplemented
97C3×C7ΓL(1,𝔽8)⋊Dic1715yescomplemented
98C3×C7ΓL(1,𝔽8)⋊Dic1711yescharacteristic, complemented
99ΓL(1,𝔽8)—76no—
100ΓL(1,𝔽8)—715no—
101ΓL(1,𝔽8)—73no—
102C2×C32—491no—
103C17C3×#1059687411yescharacteristic, complemented, Sylow
104C2×C7C3×C17⋊AGL(1,𝔽7)12yes—
105C2×C7C3×C17⋊AGL(1,𝔽7)15yes—
106C2×C7C3×C17⋊AGL(1,𝔽7)11yescharacteristic
107C3×C4—8333no—
108C3×C4—24991no—
109C32—491noSylow
110C7C3×ΓL(1,𝔽8)⋊Dic1712yescomplemented
111C7C3×ΓL(1,𝔽8)⋊Dic1715yescomplemented
112C7C3×ΓL(1,𝔽8)⋊Dic1711yescharacteristic, complemented
113C2×C3—493no—
114C2×C3C17⋊#6442411yescharacteristic
115C4—24991noSylow
116C3—493no—
117C3#2375614311yescharacteristic, direct product factor
118C2C3×C17⋊#6442411yescharacteristic
subgroupG/H
center, hypercenterC2×C3C17⋊#64424
commutator, hypocenterC72×C17C32×C4
FrattiniC2C3×C17⋊#64424
Fitting, socleC2×C3×C72×C17C2×C3
prime radicalC2×C3×C17×#3861C2
cosocleC2×C72×C17C2×C32
LCSG ▷ C72×C17
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
2. C3×C7⋊Dic119C3semidirect
3. #23756143C3direct (N × H)
4. C3×C17×#3861C4semidirect
10. C7⋊Dic119C32semidirect
11. C17×#3861C3×C4semidirect
12. C3×C72×C17C3×C4semidirect
28. C72×C17C32×C4semidirect
41. C3×#3861Dic17semidirect
42. C3×C7×C17ΓL(1,𝔽8)⋊C4semidirect
43. C3×C7×C17ΓL(1,𝔽8)⋊C4semidirect
44. C3×C7×C17ΓL(1,𝔽8)⋊C4semidirect
61. C3×C72C3×Dic17semidirect
62. #3861C3×Dic17semidirect
66. C7×C17C3×ΓL(1,𝔽8)⋊C4semidirect
67. C7×C17C3×ΓL(1,𝔽8)⋊C4semidirect
68. C7×C17C3×ΓL(1,𝔽8)⋊C4semidirect
83. C3×C17#10596874semidirect
84. C72C32×Dic17semidirect
96. C3×C7ΓL(1,𝔽8)⋊Dic17semidirect
97. C3×C7ΓL(1,𝔽8)⋊Dic17semidirect
98. C3×C7ΓL(1,𝔽8)⋊Dic17semidirect
103. C17C3×#10596874semidirect
110. C7C3×ΓL(1,𝔽8)⋊Dic17semidirect
111. C7C3×ΓL(1,𝔽8)⋊Dic17semidirect
112. C7C3×ΓL(1,𝔽8)⋊Dic17semidirect
117. C3#23756143direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C17×#38612
C3×C7⋊Dic1193
#237561433
C3×ΓL(1,𝔽8)⋊Dic177
C3×#1059687417