Finite Group Databasebeta

← #72259151

Subgroups — #72259151

Order 7920.

100 conjugacy classes of subgroups; only the 21 normal subgroups are shown.

normal subgroups only

This group has 100 conjugacy classes of subgroups — too many to draw. Only its 21 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
1#72259113C211yesmaximal, characteristic, complemented
2C11⋊C5×GO+(4,𝔽2)C211yesmaximal, characteristic
3#72228776C211yesmaximal, characteristic, complemented
4D32×C11⋊C5—21no—
5(C11⋊C5)⋊#157—21no—
6#157×C11⋊C5C2211yescharacteristic
7#13097490C511yesmaximal, characteristic, complemented, Hall
8C3⋊D3×C11⋊C5D411yescharacteristic
9C3×D3×C11⋊C5—41no—
10#4324⋊C5—91nomaximal, Hall
11C11⋊PSU(3,𝔽2)C2×C511yescharacteristic, complemented
12C11×GO+(4,𝔽2)C2×C511yescharacteristic
13C11⋊AGL(1,𝔽9)C2×C511yescharacteristic, complemented
14C5×AΓL(1,𝔽9)—111nomaximal, Hall
15C2×D3×C11⋊C5—121no—
16C32×C11⋊C5SD1611yescharacteristic, complemented, Hall
17D4×C11⋊C5—91no—
18(C11⋊C5)⋊Q8—91no—
19(C11⋊C5)⋊C8—91no—
20D32×C11—21no—
21C11⋊#157—21no—
22C11×#157C22×C511yescharacteristic
23C5×PSU(3,𝔽2)—111no—
24C5×GO+(4,𝔽2)—221no—
25C5×AGL(1,𝔽9)—111no—
26C2×C3×C11⋊C5—121no—
27D3×C11⋊C5—41no—
28D3×C11⋊C5—121no—
29C22×C11⋊C5—181no—
30C4×C11⋊C5—91no—
31Dic11⋊C5—181no—
32C11×C3⋊D3C5×D411yescharacteristic
33C3×D3×C11—41no—
34C5×D32—221no—
35C5×#157—221no—
36C5×#157—221no—
37#4324—91noHall
38C3×C11⋊C5—41no—
39AΓL(1,𝔽9)—111noHall
40C2×D3×C11—121no—
41C2×C11⋊C5—91no—
42C2×C11⋊C5—121no—
43C32×C11C5×SD1611yescharacteristic, complemented, Hall
44C5×C3⋊D3—221no—
45C3×C5×D3—441no—
46D4×C11—91no—
47Dic22—91no—
48C11⋊C8—91no—
49C5×SD16—991noHall
50PSU(3,𝔽2)—111no—
51GO+(4,𝔽2)AGL(1,𝔽11)11yescharacteristic
52AGL(1,𝔽9)—111no—
53C2×C3×C11—121no—
54D3×C11—41no—
55D3×C11—121no—
56C2×C5×D3—1321no—
57C11⋊C5AΓL(1,𝔽9)11yescharacteristic, complemented, Hall
58C32×C5—221noHall
59C22×C11—181no—
60C4×C11—91no—
61Dic11—181no—
62C5×Q8—991no—
63C5×D4—1981no—
64C5×C8—991no—
65D32—21no—
66#157—221no—
67#157C2×AGL(1,𝔽11)11yescharacteristic
68C3×C11—41no—
69C2×C3×C5—1321no—
70C5×D3—441no—
71C5×D3—1321no—
72C2×C11—91no—
73C2×C11—121no—
74C22×C5—1981no—
75C4×C5—1981no—
76C4×C5—1981no—
77C3⋊D3C22⋊AGL(1,𝔽11)11yescharacteristic
78C3×D3—41no—
79SD16—991noSylow
80C3×C5—441no—
81C2×D3—121no—
82C11C5×AΓL(1,𝔽9)11yescharacteristic, complemented, Sylow
83C2×C5—1321no—
84C2×C5—1981no—
85C32#4324⋊C511yescharacteristic, complemented, Sylow
86Q8—991no—
87D4—91no—
88C8—991no—
89C2×C3—121no—
90D3—41no—
91D3—121no—
92C5—221noSylow
93C22—181no—
94C4—91no—
95C4—1981no—
96C3—41no—
97C2—91no—
98C2—121no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, cosocleC11×#157C22×C5
Fitting, socle, hypocenterC32×C11C5×SD16
prime radicalC11⋊C5×GO+(4,𝔽2)C2
LCSG ▷ C11×#157 ▷ C11×C3⋊D3 ▷ C32×C11
UCSC1

Complements

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

NH ≅ G/Ntype
1. #72259113C2semidirect
3. #72228776C2semidirect
7. #13097490C5semidirect
11. C11⋊PSU(3,𝔽2)C2×C5semidirect
13. C11⋊AGL(1,𝔽9)C2×C5semidirect
16. C32×C11⋊C5SD16semidirect
43. C32×C11C5×SD16semidirect
57. C11⋊C5AΓL(1,𝔽9)semidirect
82. C11C5×AΓL(1,𝔽9)semidirect
85. C32#4324⋊C5semidirect

Maximal subgroups

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

subgroupindex
#722591132
C11⋊C5×GO+(4,𝔽2)2
#722287762
#130974905
#4324⋊C59
C5×AΓL(1,𝔽9)11