Finite Group Databasebeta

← ΓL(1,𝔽8)×C3⋊(C7⋊#595) #44472811

Subgroups — ΓL(1,𝔽8)×C3⋊(C7⋊#595) #44472811

Order 29988.

120 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 39 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)×D17C211yesmaximal, characteristic
2C7×C3⋊(C7⋊#595)C311yesmaximal, characteristic, complemented
3ΓL(1,𝔽8)×C7⋊#595—31nomaximal
4C3×C7×C17×ΓL(1,𝔽8)C411yescharacteristic, complemented, Hall
5C3×C72×D17C2×C311yescharacteristic
6C7×ΓL(1,𝔽8)×D17—21no—
7C7×ΓL(1,𝔽8)×D17D311yescharacteristic
8C3×C3⋊(C7⋊#595)—71nomaximal
9ΓL(1,𝔽8)×C3⋊#595—71nomaximal
10C7×C7⋊#595—31noHall
11C7×C17×ΓL(1,𝔽8)—21no—
12C7×C17×ΓL(1,𝔽8)Dic311yescharacteristic, complemented
13C3×C72×C17C3×C411yescharacteristic, complemented
14C32×C7×D17—71no—
15C3×ΓL(1,𝔽8)×D17D711yescharacteristic
16ΓL(1,𝔽8)×Dic21—171nomaximal, Hall
17C72×D17C3×D311yescharacteristic
18C3⋊(C7⋊#595)ΓL(1,𝔽8)11yescharacteristic, direct product factor
19C7×C3⋊#595—71no—
20C3×C7⋊#595—211no—
21ΓL(1,𝔽8)×#595—211no—
22C32×C7×C17—71no—
23C3×C17×ΓL(1,𝔽8)Dic711yescharacteristic, complemented
24C2×C3×C7×ΓL(1,𝔽8)—171no—
25C72×C17C3×Dic311yescharacteristic, complemented, Hall
26C3×C7×D17—61no—
27C3×C7×D17—71no—
28C3×C7×D17—141no—
29C3×C7×D17C2×ΓL(1,𝔽8)11yescharacteristic
30C3×C7×D17C3×D711yescharacteristic
31ΓL(1,𝔽8)×D17—21no—
32ΓL(1,𝔽8)×D17D2111yescharacteristic
33C3×C3⋊#595—491noHall
34C7×Dic21—171no—
35ΓL(1,𝔽8)×Dic7—511no—
36C7⋊#595—31no—
37C7×#595—211no—
38C3×C7×ΓL(1,𝔽8)#59511yescharacteristic, complemented, Hall
39C3×C7×C17—61no—
40C3×C7×C17—71no—
41C3×C7×C17—141no—
42C3×C7×C17C4×ΓL(1,𝔽8)11yescharacteristic, complemented
43C3×C7×C17C3×Dic711yescharacteristic, complemented
44C17×ΓL(1,𝔽8)—21no—
45C17×ΓL(1,𝔽8)Dic2111yescharacteristic, complemented
46C32×D17—71no—
47C2×C3×C72—171no—
48C2×C7×ΓL(1,𝔽8)—171no—
49C2×C7×ΓL(1,𝔽8)—341no—
50C3×Dic21—1191no—
51Dic3×ΓL(1,𝔽8)—1191no—
52C7×D17—61no—
53C7×D17D3×ΓL(1,𝔽8)11yescharacteristic
54C7×D17C3×D2111yescharacteristic
55C3⋊#595—71no—
56C3×#595—1471no—
57C7×Dic7—511noHall
58C32×C17—71noHall
59C3×C72C3×#59511yescharacteristic, complemented
60C7×ΓL(1,𝔽8)—21no—
61C7×ΓL(1,𝔽8)C3⋊#59511yescharacteristic, complemented
62C2×C32×C7—1191no—
63C2×C3×ΓL(1,𝔽8)—171no—
64C7×C17—61no—
65C7×C17Dic3×ΓL(1,𝔽8)11yescharacteristic, complemented
66C7×C17C3×Dic2111yescharacteristic, complemented
67C3×D17—71no—
68C3×D17—141no—
69C3×D17D7×ΓL(1,𝔽8)11yescharacteristic
70C2×C72—171no—
71Dic21—171no—
72C3×Dic7—3571no—
73C7×Dic3—1191no—
74C4×ΓL(1,𝔽8)—3571no—
75#595—211noHall
76C32×C7—71no—
77C3×ΓL(1,𝔽8)C7⋊#59511yescharacteristic, complemented
78C3×C17—71no—
79C3×C17—141no—
80C3×C17ΓL(1,𝔽8)×Dic711yescharacteristic, complemented
81C72C3×C3⋊#59511yescharacteristic, complemented, Sylow
82C2×C3×C7—171no—
83C2×C3×C7—171no—
84C2×C3×C7—1021no—
85C2×C3×C7—1191no—
86C2×C3×C7—2381no—
87C2×ΓL(1,𝔽8)—171no—
88C2×ΓL(1,𝔽8)—341no—
89C3×Dic3—8331noHall
90D17ΓL(1,𝔽8)×D2111yescharacteristic
91C4×C7—3571no—
92Dic7—511no—
93C3×C7—61no—
94C3×C7—71no—
95C3×C7—141no—
96C3×C7ΓL(1,𝔽8)×#59511yescharacteristic, complemented
97C3×C7C3×C7⋊#59511yescharacteristic, complemented
98ΓL(1,𝔽8)—21no—
99ΓL(1,𝔽8)C3⋊(C7⋊#595)11yescharacteristic, direct product factor
100C2×C32—1191no—
101C17ΓL(1,𝔽8)×Dic2111yescharacteristic, complemented, Sylow
102C2×C7—171no—
103C2×C7—171no—
104C2×C7—1021no—
105C3×C4—24991no—
106Dic3—1191no—
107C32—71noSylow
108C7—61no—
109C7ΓL(1,𝔽8)×C3⋊#59511yescharacteristic, complemented
110C7C3×C3⋊(C7⋊#595)11yescharacteristic, complemented
111C2×C3—171no—
112C2×C3—1191no—
113C2×C3—2381no—
114C4—3571noSylow
115C3—71no—
116C3—141no—
117C3ΓL(1,𝔽8)×C7⋊#59511yescharacteristic, complemented
118C2—171no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC3×C72×C17C3×C4
prime radicalC3×C7×ΓL(1,𝔽8)×D17C2
cosocleC3×C72×D17C2×C3
LCSG ▷ C3×C72×C17
UCSC1

Complements

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

NH ≅ G/Ntype
2. C7×C3⋊(C7⋊#595)C3semidirect
4. C3×C7×C17×ΓL(1,𝔽8)C4semidirect
12. C7×C17×ΓL(1,𝔽8)Dic3semidirect
13. C3×C72×C17C3×C4semidirect
18. C3⋊(C7⋊#595)ΓL(1,𝔽8)direct (N × H)
23. C3×C17×ΓL(1,𝔽8)Dic7semidirect
25. C72×C17C3×Dic3semidirect
38. C3×C7×ΓL(1,𝔽8)#595semidirect
42. C3×C7×C17C4×ΓL(1,𝔽8)semidirect
43. C3×C7×C17C3×Dic7semidirect
45. C17×ΓL(1,𝔽8)Dic21semidirect
59. C3×C72C3×#595semidirect
61. C7×ΓL(1,𝔽8)C3⋊#595semidirect
65. C7×C17Dic3×ΓL(1,𝔽8)semidirect
66. C7×C17C3×Dic21semidirect
77. C3×ΓL(1,𝔽8)C7⋊#595semidirect
80. C3×C17ΓL(1,𝔽8)×Dic7semidirect
81. C72C3×C3⋊#595semidirect
96. C3×C7ΓL(1,𝔽8)×#595semidirect
97. C3×C7C3×C7⋊#595semidirect
99. ΓL(1,𝔽8)C3⋊(C7⋊#595)direct (N × H)
101. C17ΓL(1,𝔽8)×Dic21semidirect
109. C7ΓL(1,𝔽8)×C3⋊#595semidirect
110. C7C3×C3⋊(C7⋊#595)semidirect
117. C3ΓL(1,𝔽8)×C7⋊#595semidirect

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)×D172
C7×C3⋊(C7⋊#595)3
ΓL(1,𝔽8)×C7⋊#5953
C3×C3⋊(C7⋊#595)7
ΓL(1,𝔽8)×C3⋊#5957
ΓL(1,𝔽8)×Dic2117