Finite Group Databasebeta

← C2×C3×C17×#64425 #44472883

Subgroups — C2×C3×C17×#64425 #44472883

Order 29988.

280 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 84 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×#3862C211yesmaximal, characteristic, complemented
2C3×C17×#64425C212yesmaximal, direct product factor
3C2×C3×C17×C7⋊D7C311yesmaximal, characteristic, complemented
4C2×C17×#64425C313yesmaximal, direct product factor
5C3×C17×#3862C2211yescharacteristic, complemented, Hall
6C3×C17×C7⋊D7C2×C312yescomplemented
7C2×C17×#3862C2×C313yescomplemented
8C17×#64425C2×C316yesdirect product factor
9C2×C3×C72×C17C2×C311yescharacteristic, complemented
10C2×C3×C17×AGL(1,𝔽7)—72nomaximal
11C2×C17×C7⋊D7C3211yescharacteristic, complemented, Hall
12C17×#3862C22×C313yescomplemented
13C3×C72×C17C22×C311yescharacteristic, complemented
14C2×C3×C17×ΓL(1,𝔽8)—72no—
15C3×C17×AGL(1,𝔽7)—74no—
16C2×C3×#64425C1711yesmaximal, characteristic, direct product factor, Hall
17C2×C72×C17C2×C3211yescharacteristic, complemented
18C17×C7⋊D7C2×C3212yescomplemented
19C2×C3×D7×C17—72no—
20C2×C3×D7×C17—212no—
21C2×C17×AGL(1,𝔽7)—76no—
22C3×C17×ΓL(1,𝔽8)—72no—
23C2×C3×#3862C2×C1711yescharacteristic, complemented
24C3×#64425C2×C1712yesdirect product factor
25C72×C17C22×C3211yescharacteristic, complemented, Hall
26C2×C3×C7×C17—32no—
27C2×C3×C7×C17AGL(1,𝔽7)12yescomplemented
28C3×D7×C17—74no—
29C3×D7×C17—214no—
30C2×C17×ΓL(1,𝔽8)—76no—
31C17×AGL(1,𝔽7)—712no—
32C22×C32×C17—491noHall
33C2×C3×C7⋊D7C3×C1711yescharacteristic, complemented
34C2×#64425C3×C1713yesdirect product factor
35C2×D7×C17—72no—
36C2×D7×C17—212no—
37C3×#3862C22×C1711yescharacteristic, complemented, Hall
38C3×C7×C17—32no—
39C3×C7×C17C2×AGL(1,𝔽7)12yescomplemented
40C17×ΓL(1,𝔽8)—76no—
41C2×C32×C17—492no—
42C2×C32×C17—491no—
43C2×C3×C72C2×C3×C1711yescharacteristic, complemented
44C3×C7⋊D7C2×C3×C1712yescomplemented
45C2×#3862C2×C3×C1713yescomplemented
46#64425C2×C3×C1716yesdirect product factor
47C2×C3×AGL(1,𝔽7)—72no—
48C2×C7×C17—32no—
49C2×C7×C17C3×AGL(1,𝔽7)12yescomplemented
50D7×C17—74no—
51D7×C17—214no—
52C22×C3×C17—493no—
53C22×C3×C17—491no—
54C2×C7⋊D7C32×C1711yescharacteristic, complemented, Hall
55C32×C17—491noHall
56C3×C72C22×C3×C1711yescharacteristic, complemented
57#3862C22×C3×C1713yescomplemented
58C2×C3×ΓL(1,𝔽8)—72no—
59C3×AGL(1,𝔽7)—74no—
60C7×C17—32no—
61C7×C17C2×C3×AGL(1,𝔽7)12yescomplemented
62C2×C3×C17—496no—
63C2×C3×C17—493no—
64C2×C3×C17—492no—
65C2×C3×C17#6442511yescharacteristic, direct product factor
66C2×C72C2×C32×C1711yescharacteristic, complemented
67C7⋊D7C2×C32×C1712yescomplemented
68C2×C3×D7—72no—
69C2×C3×D7—212no—
70C2×AGL(1,𝔽7)—76no—
71C22×C17—491noHall
72C3×ΓL(1,𝔽8)—72no—
73C3×C17—493no—
74C3×C17C2×#6442511yescharacteristic, direct product factor
75C72C22×C32×C1711yescharacteristic, complemented, Sylow
76C2×C3×C7—32no—
77C2×C3×C7C17×AGL(1,𝔽7)12yescomplemented
78C3×D7—74no—
79C3×D7—214no—
80C2×ΓL(1,𝔽8)—76no—
81AGL(1,𝔽7)—712no—
82C22×C32—491noHall
83C2×C17—492no—
84C2×C17C3×#6442511yescharacteristic, direct product factor
85C2×D7—72no—
86C2×D7—212no—
87C3×C7—32no—
88C3×C7C2×C17×AGL(1,𝔽7)12yescomplemented
89ΓL(1,𝔽8)—76no—
90C2×C32—492no—
91C2×C32—491no—
92C17C2×C3×#6442511yescharacteristic, direct product factor, Sylow
93C2×C7—32no—
94C2×C7C3×C17×AGL(1,𝔽7)12yescomplemented
95D7—74no—
96D7—214no—
97C22×C3—493no—
98C22×C3—491no—
99C32—491noSylow
100C7—32no—
101C7C2×C3×C17×AGL(1,𝔽7)12yescomplemented
102C2×C3—496no—
103C2×C3—492no—
104C2×C3—493no—
105C2×C3C17×#6442511yescharacteristic, direct product factor
106C22—491noSylow
107C3—493no—
108C3C2×C17×#6442511yescharacteristic, direct product factor
109C2—492no—
110C2C3×C17×#6442511yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3×C17#64425
commutator, cosocle, hypocenterC72C22×C32×C17
FrattiniC1G
Fitting, socleC2×C3×C72×C17C2×C3
LCSG ▷ C72
UCSC1 ◁ C2×C3×C17

Complements

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

NH ≅ G/Ntype
1. C2×C3×C17×#3862C2semidirect
2. C3×C17×#64425C2direct (N × H)
3. C2×C3×C17×C7⋊D7C3semidirect
4. C2×C17×#64425C3direct (N × H)
5. C3×C17×#3862C22semidirect
6. C3×C17×C7⋊D7C2×C3semidirect
7. C2×C17×#3862C2×C3semidirect
8. C17×#64425C2×C3direct (N × H)
9. C2×C3×C72×C17C2×C3semidirect
11. C2×C17×C7⋊D7C32semidirect
12. C17×#3862C22×C3semidirect
13. C3×C72×C17C22×C3semidirect
16. C2×C3×#64425C17direct (N × H)
17. C2×C72×C17C2×C32semidirect
18. C17×C7⋊D7C2×C32semidirect
23. C2×C3×#3862C2×C17semidirect
24. C3×#64425C2×C17direct (N × H)
25. C72×C17C22×C32semidirect
27. C2×C3×C7×C17AGL(1,𝔽7)semidirect
33. C2×C3×C7⋊D7C3×C17semidirect
34. C2×#64425C3×C17direct (N × H)
37. C3×#3862C22×C17semidirect
39. C3×C7×C17C2×AGL(1,𝔽7)semidirect
43. C2×C3×C72C2×C3×C17semidirect
44. C3×C7⋊D7C2×C3×C17semidirect
45. C2×#3862C2×C3×C17semidirect
46. #64425C2×C3×C17direct (N × H)
49. C2×C7×C17C3×AGL(1,𝔽7)semidirect
54. C2×C7⋊D7C32×C17semidirect
56. C3×C72C22×C3×C17semidirect
57. #3862C22×C3×C17semidirect
61. C7×C17C2×C3×AGL(1,𝔽7)semidirect
65. C2×C3×C17#64425direct (N × H)
66. C2×C72C2×C32×C17semidirect
67. C7⋊D7C2×C32×C17semidirect
74. C3×C17C2×#64425direct (N × H)
75. C72C22×C32×C17semidirect
77. C2×C3×C7C17×AGL(1,𝔽7)semidirect
84. C2×C17C3×#64425direct (N × H)
88. C3×C7C2×C17×AGL(1,𝔽7)semidirect
92. C17C2×C3×#64425direct (N × H)
94. C2×C7C3×C17×AGL(1,𝔽7)semidirect
101. C7C2×C3×C17×AGL(1,𝔽7)semidirect
105. C2×C3C17×#64425direct (N × H)
108. C3C2×C17×#64425direct (N × H)
110. C2C3×C17×#64425direct (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×#38622
C3×C17×#644252
C2×C3×C17×C7⋊D73
C2×C17×#644253
C2×C3×C17×AGL(1,𝔽7)7
C2×C3×#6442517