Finite Group Databasebeta

← #66988⋊C5 #13182535

Subgroups — #66988⋊C5 #13182535

Order 1760.

156 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 55 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×D4×C11⋊C5C211yesmaximal, characteristic
2C22×Dic11⋊C5C211yesmaximal, characteristic, complemented
3C22⋊(Dic11⋊C5)C211yesmaximal, characteristic, complemented
4C22⋊(Dic11⋊C5)C212yesmaximal, complemented
5(Dic11⋊C4)⋊C5C212yesmaximal, complemented
6C23×C11⋊C5C2211yescharacteristic
7C23×C11⋊C5C2211yescharacteristic
8D4×C11⋊C5—22no—
9C2×C4×C11⋊C5C2211yescharacteristic
10C2×Dic11⋊C5—22no—
11C2×Dic11⋊C5C2212yes—
12C2×Dic11⋊C5C2212yes—
13#66988C511yesmaximal, characteristic, complemented, Hall
14C22×C11⋊C5—22no—
15C22×C11⋊C5—21no—
16C22×C11⋊C5—22no—
17C22×C11⋊C5D412yes—
18C22×C11⋊C5C2311yescharacteristic
19C4×C11⋊C5—21no—
20Dic11⋊C5—22no—
21Dic11⋊C5—22no—
22C2×D4×C11C2×C511yescharacteristic
23C22×Dic11C2×C511yescharacteristic, complemented
24C22⋊Dic11C2×C512yescomplemented
25C22⋊Dic11C2×C511yescharacteristic, complemented
26Dic11⋊C4C2×C512yescomplemented
27C5×#123—111nomaximal, Hall
28C2×C11⋊C5—22no—
29C2×C11⋊C5—41no—
30C2×C11⋊C5C2×D411yescharacteristic
31C2×C11⋊C5#4112yes—
32C23×C11C22×C511yescharacteristic
33C23×C11C22×C511yescharacteristic
34D4×C11—22no—
35C2×C4×C11C22×C511yescharacteristic
36C2×Dic11—22no—
37C2×Dic11C22×C512yes—
38C2×Dic11C22×C512yes—
39C2×C5×D4—111no—
40C22×C4×C5—111no—
41C5×C4⋊C4—112no—
42C5×C22⋊C4—111no—
43C5×C22⋊C4—112no—
44C11⋊C5#12311yescharacteristic, complemented, Hall
45C22×C11—22no—
46C22×C11—21no—
47C22×C11—22no—
48C22×C11C5×D412yes—
49C22×C11C23×C511yescharacteristic
50C4×C11—21no—
51Dic11—22no—
52Dic11—22no—
53C23×C5—111no—
54C23×C5—111no—
55C5×D4—222no—
56C2×C4×C5—112no—
57C2×C4×C5—112no—
58C2×C4×C5—111no—
59C2×C4×C5—222no—
60#123—111noSylow
61C2×C11—22no—
62C2×C11—41no—
63C2×C11C2×C5×D411yescharacteristic
64C2×C11C5×#4112yes—
65C22×C5—112no—
66C22×C5—111no—
67C22×C5—222no—
68C22×C5—221no—
69C22×C5—222no—
70C4×C5—221no—
71C4×C5—222no—
72C4×C5—222no—
73C2×D4AGL(1,𝔽11)11yescharacteristic
74C22×C4—111no—
75C4⋊C4—112no—
76C22⋊C4—111no—
77C22⋊C4—112no—
78C11C5×#12311yescharacteristic, complemented, Sylow
79C2×C5—112no—
80C2×C5—111no—
81C2×C5—222no—
82C2×C5—441no—
83C23C2×AGL(1,𝔽11)11yescharacteristic
84C23C2×AGL(1,𝔽11)11yescharacteristic
85D4—22no—
86C2×C4—112no—
87C2×C4—112no—
88C2×C4—222no—
89C2×C4C2×AGL(1,𝔽11)11yescharacteristic
90C5—111noSylow
91C22—22no—
92C22—22no—
93C22—21no—
94C22C22⋊AGL(1,𝔽11)12yes—
95C22C22×AGL(1,𝔽11)11yescharacteristic
96C4—21no—
97C4—222no—
98C4—222no—
99C2—22no—
100C2—41no—
101C2#4341⋊C512yes—
102C2C2×C22⋊AGL(1,𝔽11)11yescharacteristic
subgroupG/H
center, FrattiniC22C22×AGL(1,𝔽11)
commutator, socle, cosocleC22×C11C23×C5
FittingC2×D4×C11C2×C5
prime radicalC2×D4×C11⋊C5C2
hypercenterC2×D4AGL(1,𝔽11)
hypocenterC11C5×#123
LCSG ▷ C22×C11 ▷ C11
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C22×Dic11⋊C5C2semidirect
3. C22⋊(Dic11⋊C5)C2semidirect
4. C22⋊(Dic11⋊C5)C2semidirect
5. (Dic11⋊C4)⋊C5C2semidirect
13. #66988C5semidirect
23. C22×Dic11C2×C5semidirect
24. C22⋊Dic11C2×C5semidirect
25. C22⋊Dic11C2×C5semidirect
26. Dic11⋊C4C2×C5semidirect
44. C11⋊C5#123semidirect
78. C11C5×#123semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C11⋊C52
C22×Dic11⋊C52
C22⋊(Dic11⋊C5)2
(Dic11⋊C4)⋊C52
#669885
C5×#12311