Finite Group Databasebeta

← #13078022

Subgroups — #13078022

Order 1408.

716 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 161 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
1C11×#585C211yesmaximal, characteristic
2#10623356C213yesmaximal, complemented
3#10622856C218yesmaximal, complemented
4#10622853C213yesmaximal, complemented
5C11×2−1+4—23no—
6C11×H(5,𝔽2)—23no—
7C11×H(5,𝔽2)C2214yes—
8C2×C11×#41—26no—
9C2×C11×#41C2213yes—
10C22×Dic22C2211yescharacteristic, complemented
11#66997—23no—
12#66989—23no—
13#66988C22112yescomplemented
14#66979C2213yescomplemented
15#66977—26no—
16#66898C22112yescomplemented
17C11×#41—24no—
18C11×#41—212no—
19C11×#41—26no—
20C11×#41—212no—
21C11×#41—26no—
22C2×Q8×C11—26no—
23C2×Q8×C11D413yes—
24C2×D4×C11—26no—
25C2×D4×C11—212no—
26C2×D4×C11D413yes—
27C2×D4×C11C2316yes—
28C22×C4×C11—24no—
29C22×C4×C11D416yes—
30C22×C4×C11C2311yescharacteristic
31C22×Dic11C2312yes—
32C2×Dic22—23no—
33C2×Dic22—23no—
34C22⋊Dic11—212no—
35C22⋊Dic11C2316yes—
36C4⋊Dic11—23no—
37Dic11⋊C4—212no—
38C4×Dic11—23no—
39#3028—111nomaximal, Sylow
40C23×C11—24no—
41C23×C11C2×D416yes—
42C23×C11C2411yescharacteristic
43Q8×C11—24no—
44Q8×C11—26no—
45D4×C11—212no—
46D4×C11—212no—
47D4×C11—26no—
48C2×C4×C11—212no—
49C2×C4×C11—26no—
50C2×C4×C11—26no—
51C2×C4×C11C2×D416yes—
52C2×C4×C11C2×D413yes—
53C2×C4×C11C2×D413yes—
54C2×Dic11—26no—
55C2×Dic11—26no—
56Dic22—41no—
57Dic22—43no—
58#585D1111yescharacteristic
59#536—113no—
60#458—118no—
61#410—113no—
62C22×C11—26no—
63C22×C11—41no—
64C22×C11—46no—
65C22×C11—84no—
66C22×C11C22×D413yes—
67C4×C11—26no—
68C4×C11#12011yescharacteristic
69C4×C11#12013yes—
70Dic11—46no—
71Dic11—42no—
722−1+4—23no—
73H(5,𝔽2)—23no—
74H(5,𝔽2)C2×D1114yes—
75C2×#41—26no—
76C2×#41C2×D1113yes—
77C22×Q8—111no—
78C4⋊Q8—223no—
79#124—223no—
80#123—1112no—
81C22⋊Q8—226no—
82#117—113no—
83#99—1112no—
84C2×C11—23no—
85C2×C11—46no—
86C2×C11C2×#12011yescharacteristic
87#41—24no—
88#41—212no—
89#41—26no—
90#41—212no—
91#41—26no—
92C2×Q8—26no—
93C2×Q8—223no—
94C2×Q8—223no—
95C2×Q8C22⋊D1113yes—
96C2×D4—26no—
97C2×D4—212no—
98C2×D4C22⋊D1113yes—
99C2×D4C22×D1116yes—
100C22×C4—24no—
101C22×C4—112no—
102C22×C4C22⋊D1116yes—
103C22×C4C22×D1111yescharacteristic
104C4⋊C4—223no—
105C4⋊C4—2212no—
106C22⋊C4—116no—
107C22⋊C4—2212no—
108C42—223no—
109C11#302811yescharacteristic, complemented, Sylow
110C23—24no—
111C23C2×C22⋊D1116yes—
112C23C23×D1111yescharacteristic
113Q8—24no—
114Q8—26no—
115Q8—441no—
116Q8—443no—
117D4—212no—
118D4—212no—
119D4—26no—
120C2×C4—212no—
121C2×C4—26no—
122C2×C4—26no—
123C2×C4—226no—
124C2×C4—226no—
125C2×C4C2×C22⋊D1116yes—
126C2×C4C2×C22⋊D1113yes—
127C2×C4C2×C22⋊D1113yes—
128C22—26no—
129C22—41no—
130C22—46no—
131C22—84no—
132C22C22×C22⋊D1113yes—
133C4—26no—
134C4—446no—
135C4—442no—
136C4#6700611yescharacteristic
137C4#6700613yes—
138C2—23no—
139C2—46no—
140C2C2×#6700611yescharacteristic
subgroupG/H
centerC2C2×#67006
commutator, cosocleC23×C11C24
FrattiniC23C23×D11
Fitting, prime radicalC11×#585C2
socleC2×C11C2×#120
hypercenter#585D11
hypocenterC11#3028
LCSG ▷ C23×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C2 ◁ C22×C4 ◁ #585

Complements

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

NH ≅ G/Ntype
2. #10623356C2semidirect
3. #10622856C2semidirect
4. #10622853C2semidirect
10. C22×Dic22C22semidirect
13. #66988C22semidirect
14. #66979C22semidirect
16. #66898C22semidirect
109. C11#3028semidirect

Maximal subgroups

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

subgroupindex
C11×#5852
#106233562
#106228562
#106228532
#302811