Finite Group Databasebeta

← C3⋊#20301295 #36348010

Subgroups — C3⋊#20301295 #36348010

Order 22050.

168 conjugacy classes of subgroups; only the 22 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 22 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×C7⋊(C52⋊C3)C211yesmaximal, characteristic, complemented, Hall
2C3⋊(C5⋊(C5⋊(C7⋊D7)))C311yesmaximal, characteristic, complemented
3#20301295—31nomaximal
4C7×C7⋊(C52⋊C3)—21no—
5C7×C7⋊(C52⋊C3)D311yescharacteristic, complemented
6C3×C52×C72C2×C311yescharacteristic, complemented
7C3⋊#11760384—71nomaximal
8C3⋊#11760386—71nomaximal
9C5⋊(C5⋊(C7⋊D7))—31noHall
10C3×C7⋊(C52⋊C3)D711yescharacteristic, complemented
11C3×C7×C52⋊C3—141no—
12C3⋊(C5⋊(C7⋊D7))—152no—
13C52×C72C3×D311yescharacteristic, complemented, Hall
14C3⋊(C5⋊D35)—71no—
15C3⋊(C5⋊D35)—71no—
16C3⋊(C5⋊D35)—212no—
17#11760386—211no—
18#11760384—211no—
19C3⋊(ΓL(1,𝔽8)⋊D7)—251nomaximal, Hall
20C3×C5×C72—32no—
21C3×C52×C7—32no—
22C3×C52×C7AGL(1,𝔽7)11yescharacteristic, complemented
23C3×C52×C7C3×D711yescharacteristic, complemented
24C7⋊(C52⋊C3)—21no—
25C7⋊(C52⋊C3)D2111yescharacteristic, complemented
26C7×C52⋊C3—71no—
27C7×C52⋊C3—141no—
28C5⋊(C7⋊D7)—452no—
29C3⋊((C5⋊D5)⋊C3)—491noHall
30C3×C7×ΓL(1,𝔽8)—501noHall
31C5⋊D35—211no—
32C5⋊D35—211no—
33C5⋊D35—632no—
34C3⋊(C7⋊D7)—251no—
35ΓL(1,𝔽8)⋊D7—751no—
36C5×C72—32no—
37C3×C52⋊C3—141noHall
38D105—10512no—
39D105—1052no—
40D105—1052no—
41C52×C7—32no—
42C52×C7C3⋊AGL(1,𝔽7)11yescharacteristic, complemented
43C52×C7C3×D2111yescharacteristic, complemented
44C3⋊(C5⋊D5)—491no—
45(C5⋊D5)⋊C3—1471no—
46C3×C72(C5⋊D5)⋊C311yescharacteristic, complemented
47C7×ΓL(1,𝔽8)—501no—
48C7×ΓL(1,𝔽8)—501no—
49C3×D21—1751no—
50C3⋊AGL(1,𝔽7)—1751no—
51C3×C5×C7—312no—
52C3×C5×C7—32no—
53C3×C5×C7—32no—
54C7⋊D7—751noHall
55C3×C52ΓL(1,𝔽8)⋊D711yescharacteristic, complemented
56C52⋊C3—141no—
57C52⋊C3—141no—
58D35—3152no—
59D35—3152no—
60D35—31512no—
61C32×C7—3501no—
62C3×ΓL(1,𝔽8)—501no—
63C5⋊D5—1471noHall
64C72C3⋊((C5⋊D5)⋊C3)11yescharacteristic, complemented, Sylow
65D21—5252no—
66D21—1751no—
67D21—1751no—
68C3×D7—5251no—
69AGL(1,𝔽7)—5251no—
70C5×C7—312no—
71C5×C7—32no—
72C5×C7—32no—
73D15—7352no—
74C52C3⋊(ΓL(1,𝔽8)⋊D7)11yescharacteristic, complemented, Sylow
75C3×C7—32no—
76C3×C7—3501no—
77C3×C7—3501no—
78C3×C7#1176038411yescharacteristic, complemented
79C3×C7#1176038611yescharacteristic, complemented
80ΓL(1,𝔽8)—501no—
81ΓL(1,𝔽8)—501no—
82C3×D3—12251noHall
83C3×C5—32no—
84D7—5251no—
85D7—5251no—
86D7—15752no—
87D5—22052no—
88C32—3501noSylow
89C7—32no—
90C7C3⋊#1176038611yescharacteristic, complemented
91C7C3⋊#1176038411yescharacteristic, complemented
92C2×C3—36751no—
93D3—12251no—
94C5—32no—
95C3—3501no—
96C3—3501no—
97C3#2030129511yescharacteristic, complemented
98C2—36751noSylow
subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC3×C52×C72C2×C3
LCSG ▷ C3×C52×C72
UCSC1

Complements

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

NH ≅ G/Ntype
1. C3×C7×C7⋊(C52⋊C3)C2semidirect
2. C3⋊(C5⋊(C5⋊(C7⋊D7)))C3semidirect
5. C7×C7⋊(C52⋊C3)D3semidirect
6. C3×C52×C72C2×C3semidirect
10. C3×C7⋊(C52⋊C3)D7semidirect
13. C52×C72C3×D3semidirect
22. C3×C52×C7AGL(1,𝔽7)semidirect
23. C3×C52×C7C3×D7semidirect
25. C7⋊(C52⋊C3)D21semidirect
42. C52×C7C3⋊AGL(1,𝔽7)semidirect
43. C52×C7C3×D21semidirect
46. C3×C72(C5⋊D5)⋊C3semidirect
55. C3×C52ΓL(1,𝔽8)⋊D7semidirect
64. C72C3⋊((C5⋊D5)⋊C3)semidirect
74. C52C3⋊(ΓL(1,𝔽8)⋊D7)semidirect
78. C3×C7#11760384semidirect
79. C3×C7#11760386semidirect
90. C7C3⋊#11760386semidirect
91. C7C3⋊#11760384semidirect
97. C3#20301295semidirect

Maximal subgroups

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

subgroupindex
C3×C7×C7⋊(C52⋊C3)2
C3⋊(C5⋊(C5⋊(C7⋊D7)))3
#203012953
C3⋊#117603847
C3⋊#117603867
C3⋊(ΓL(1,𝔽8)⋊D7)25