Finite Group Databasebeta

← C5×ΓL(1,𝔽8)⋊(Dic11⋊C5) #37528329

Subgroups — C5×ΓL(1,𝔽8)⋊(Dic11⋊C5) #37528329

Order 23100.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×ΓL(1,𝔽8)×C11⋊C5C211yesmaximal, characteristic
2C5×C7⋊(Dic11⋊C5)C311yesmaximal, characteristic, complemented, Hall
3C5×ΓL(1,𝔽8)×C11⋊C5C411yescharacteristic, complemented, Hall
4ΓL(1,𝔽8)⋊(Dic11⋊C5)C515yesmaximal, direct product factor
5C5×C11⋊(ΓL(1,𝔽8)⋊C4)C511yesmaximal, characteristic, complemented
6C2×C5×C7×C11⋊C5C2×C311yescharacteristic
7C3×C5×Dic11⋊C5—71nomaximal, Hall
8C2×ΓL(1,𝔽8)×C11⋊C5C2×C515yes—
9C2×C5×C11×ΓL(1,𝔽8)C2×C511yescharacteristic
10C52×ΓL(1,𝔽8)⋊C4—111nomaximal, Hall
11C5×C7×C11⋊C5C3×C411yescharacteristic, complemented, Hall
12C2×C3×C5×C11⋊C5—71no—
13C5×Dic77C3×C511yescharacteristic, complemented
14C7⋊(Dic11⋊C5)C3×C515yescomplemented
15C5×C11×ΓL(1,𝔽8)C4×C511yescharacteristic, complemented
16ΓL(1,𝔽8)×C11⋊C5C4×C515yescomplemented
17C5×Dic11⋊C5—71noHall
18C2×C52×ΓL(1,𝔽8)—111no—
19C11⋊(ΓL(1,𝔽8)⋊C4)C5211yescharacteristic, complemented, Hall
20C3×C5×C11⋊C5—71noHall
21C2×C5×C7×C11C2×C3×C511yescharacteristic
22C2×C7×C11⋊C5C2×C3×C515yes—
23C52×Dic7—111noHall
24C3×C5×Dic11—71no—
25C3×Dic11⋊C5—75no—
26C2×C5×C11⋊C5AGL(1,𝔽7)11yescharacteristic
27C52×ΓL(1,𝔽8)—111noHall
28C2×C11×ΓL(1,𝔽8)C2×C5211yescharacteristic
29C5×ΓL(1,𝔽8)⋊C4—115no—
30C5×ΓL(1,𝔽8)⋊C4—551no—
31C5×C7×C11C3×C4×C511yescharacteristic, complemented
32C7×C11⋊C5C3×C4×C515yescomplemented
33C2×C52×C7—111no—
34C2×C3×C5×C11—71no—
35C2×C3×C11⋊C5—75no—
36Dic77C3×C5211yescharacteristic, complemented, Hall
37C3×C4×C52—771noHall
38C5×C11⋊C5ΓL(1,𝔽8)⋊C411yescharacteristic, complemented, Hall
39C11×ΓL(1,𝔽8)C4×C5211yescharacteristic, complemented, Hall
40C5×Dic11—71no—
41Dic11⋊C5—75no—
42C2×C5×ΓL(1,𝔽8)—115no—
43C2×C5×ΓL(1,𝔽8)AGL(1,𝔽11)11yescharacteristic
44C52×C7—111noHall
45C3×C5×C11—71no—
46C3×C11⋊C5—75no—
47C2×C7×C11C2×C3×C5211yescharacteristic
48C2×C3×C52—771no—
49C5×Dic7—115no—
50C5×Dic7—551no—
51C3×Dic11—71noHall
52C2×C5×C11C5×AGL(1,𝔽7)11yescharacteristic
53C2×C11⋊C5C5×AGL(1,𝔽7)15yes—
54C5×ΓL(1,𝔽8)—115no—
55C5×ΓL(1,𝔽8)Dic11⋊C511yescharacteristic, complemented
56C4×C52—771noHall
57ΓL(1,𝔽8)⋊C4—551noHall
58C7×C11C3×C4×C5211yescharacteristic, complemented, Hall
59C3×C52—771noHall
60C2×C5×C7—115no—
61C2×C5×C7C3×AGL(1,𝔽11)11yescharacteristic
62C2×C3×C11—71no—
63C3×C4×C5—775no—
64C3×C4×C5—3851no—
65C5×C11C5×ΓL(1,𝔽8)⋊C411yescharacteristic, complemented
66C11⋊C5C5×ΓL(1,𝔽8)⋊C415yescomplemented
67C2×C52—111no—
68Dic11—71noHall
69C2×ΓL(1,𝔽8)C5×AGL(1,𝔽11)11yescharacteristic
70C5×C7—115no—
71C5×C7C3×Dic11⋊C511yescharacteristic, complemented
72C3×C11—71noHall
73C2×C3×C5—71no—
74C2×C3×C5—775no—
75Dic7—551noHall
76C52—111noSylow
77C2×C11C52×AGL(1,𝔽7)11yescharacteristic
78ΓL(1,𝔽8)C5×Dic11⋊C511yescharacteristic, complemented, Hall
79C4×C5—775no—
80C4×C5—3851no—
81C3×C5—71no—
82C3×C5—775no—
83C2×C7C3×C5×AGL(1,𝔽11)11yescharacteristic
84C3×C4—3851noHall
85C11C52×ΓL(1,𝔽8)⋊C411yescharacteristic, complemented, Sylow
86C2×C5—115no—
87C2×C5ΓL(1,𝔽8)⋊AGL(1,𝔽11)11yescharacteristic
88C7C3×C5×Dic11⋊C511yescharacteristic, complemented, Sylow
89C2×C3—71no—
90C5—115no—
91C5ΓL(1,𝔽8)⋊(Dic11⋊C5)11yescharacteristic, direct product factor
92C4—3851noSylow
93C3—71noSylow
94C2C5×ΓL(1,𝔽8)⋊AGL(1,𝔽11)11yescharacteristic
subgroupG/H
center, hypercenterC2×C5ΓL(1,𝔽8)⋊AGL(1,𝔽11)
commutator, hypocenterC7×C11C3×C4×C52
FrattiniC2C5×ΓL(1,𝔽8)⋊AGL(1,𝔽11)
Fitting, socleC2×C5×C7×C11C2×C3×C5
prime radicalC2×C5×ΓL(1,𝔽8)×C11⋊C5C2
cosocleC2×C7×C11C2×C3×C52
LCSG ▷ C7×C11
UCSC1 ◁ C2×C5

Complements

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

NH ≅ G/Ntype
2. C5×C7⋊(Dic11⋊C5)C3semidirect
3. C5×ΓL(1,𝔽8)×C11⋊C5C4semidirect
4. ΓL(1,𝔽8)⋊(Dic11⋊C5)C5direct (N × H)
5. C5×C11⋊(ΓL(1,𝔽8)⋊C4)C5semidirect
11. C5×C7×C11⋊C5C3×C4semidirect
13. C5×Dic77C3×C5semidirect
14. C7⋊(Dic11⋊C5)C3×C5semidirect
15. C5×C11×ΓL(1,𝔽8)C4×C5semidirect
16. ΓL(1,𝔽8)×C11⋊C5C4×C5semidirect
19. C11⋊(ΓL(1,𝔽8)⋊C4)C52semidirect
31. C5×C7×C11C3×C4×C5semidirect
32. C7×C11⋊C5C3×C4×C5semidirect
36. Dic77C3×C52semidirect
38. C5×C11⋊C5ΓL(1,𝔽8)⋊C4semidirect
39. C11×ΓL(1,𝔽8)C4×C52semidirect
55. C5×ΓL(1,𝔽8)Dic11⋊C5semidirect
58. C7×C11C3×C4×C52semidirect
65. C5×C11C5×ΓL(1,𝔽8)⋊C4semidirect
66. C11⋊C5C5×ΓL(1,𝔽8)⋊C4semidirect
71. C5×C7C3×Dic11⋊C5semidirect
78. ΓL(1,𝔽8)C5×Dic11⋊C5semidirect
85. C11C52×ΓL(1,𝔽8)⋊C4semidirect
88. C7C3×C5×Dic11⋊C5semidirect
91. C5ΓL(1,𝔽8)⋊(Dic11⋊C5)direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×ΓL(1,𝔽8)×C11⋊C52
C5×C7⋊(Dic11⋊C5)3
ΓL(1,𝔽8)⋊(Dic11⋊C5)5
C5×C11⋊(ΓL(1,𝔽8)⋊C4)5
C3×C5×Dic11⋊C57
C52×ΓL(1,𝔽8)⋊C411