Finite Group Databasebeta

← #13078013

Subgroups — #13078013

Order 1408.

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

normal subgroups only

This group has 800 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
1C2×C11×H(5,𝔽2)C211yesmaximal, characteristic, complemented
2#10623335C213yesmaximal, complemented
3#10622857C214yesmaximal, complemented
4#10622856C214yesmaximal, complemented
5C2×#66898C213yesmaximal, complemented
6C11×H(5,𝔽2)—26no—
7C11×H(5,𝔽2)C2214yes—
8C2×C11×#41—23no—
9C22×D4×C11—23no—
10C22×D4×C11C2213yes—
11C22×C22⋊D11C2211yescharacteristic, complemented
12#67006—23no—
13C2×C22⋊Dic11C2213yescomplemented
14#66992—26no—
15#66990C2216yescomplemented
16#66988C2216yescomplemented
17#66982—23no—
18#66898C22112yescomplemented
19C24×C11—21no—
20C24×C11D413yescomplemented
21C24×C11C2311yescharacteristic
22C23×D11C2311yescharacteristic, complemented
23C11×#41—212no—
24C11×#41—212no—
25C2×Q8×C11—21no—
26C2×D4×C11—23no—
27C2×D4×C11—212no—
28C2×D4×C11—26no—
29C2×D4×C11—212no—
30C2×D4×C11—26no—
31C2×D4×C11D416yescomplemented
32C2×D4×C11C2316yes—
33C22×C4×C11—23no—
34C22×C4×C11D413yescomplemented
35C2×C22⋊D11—23no—
36C2×C22⋊D11—23no—
37C2×C22⋊D11—26no—
38C22×Dic11—23no—
39C22×Dic11C2311yescharacteristic, complemented
40C22⋊Dic11—23no—
41C22⋊Dic11—26no—
42C22⋊Dic11C2316yescomplemented
43(C2×D11)⋊C4—26no—
44Dic11⋊C4—26no—
45#3026—111nomaximal, Sylow
46C23×C11—24no—
47C23×C11—23no—
48C23×C11—23no—
49C23×C11—81no—
50C23×C11—83no—
51C23×C11—82no—
52C23×C11C2×D416yes—
53C23×C11C2×D413yes—
54C23×C11C2×D413yes—
55C23×C11C2411yescharacteristic
56C22×D11—23no—
57C22×D11—81no—
58Q8×C11—24no—
59D4×C11—212no—
60D4×C11—212no—
61D4×C11—212no—
62C2×C4×C11—212no—
63C2×C4×C11—23no—
64C2×C4×C11—23no—
65C2×C4×C11C2×D416yes—
66C22⋊D11—41no—
67C22⋊D11—43no—
68C22⋊D11—86no—
69C2×Dic11—23no—
70C2×Dic11—26no—
71C2×Dic11—83no—
72C2×H(5,𝔽2)D1111yescharacteristic, complemented
73#534—113no—
74#458—114no—
75B(4,𝔽2)—114no—
76C2×#99—113no—
77C22×C11—26no—
78C22×C11—41no—
79C22×C11—43no—
80C22×C11—43no—
81C22×C11—46no—
82C22×C11—46no—
83C22×C11—83no—
84C22×C11—84no—
85C22×C11#12011yescharacteristic
86C22×C11#12013yes—
87C22×C11C22×D413yes—
88C2×D11—46no—
89C2×D11—41no—
90C4×C11—26no—
91Dic11—46no—
92Dic11—41no—
93H(5,𝔽2)—26no—
94H(5,𝔽2)C2×D1114yes—
95C2×#41—23no—
96C22×D4—23no—
97C22×D4—111no—
98C22×D4C2×D1113yes—
99#123—116no—
100#123—223no—
101#121—226no—
102#120—116no—
103#120—223no—
104C2×C22⋊C4—113no—
105#99—1112no—
106C2×C11—23no—
107C2×C11—21no—
108C2×C11—23no—
109C2×C11—46no—
110C2×C11C2×#12011yescharacteristic
111D11—81no—
112C24—21no—
113C24—111no—
114C24C22⋊D1113yescomplemented
115C24C22×D1111yescharacteristic
116#41—212no—
117#41—212no—
118C2×Q8—21no—
119C2×D4—23no—
120C2×D4—212no—
121C2×D4—26no—
122C2×D4—212no—
123C2×D4—26no—
124C2×D4—223no—
125C2×D4—223no—
126C2×D4—226no—
127C2×D4C22⋊D1116yescomplemented
128C2×D4C22×D1116yes—
129C22×C4—23no—
130C22×C4—111no—
131C22×C4—223no—
132C22×C4C22⋊D1113yescomplemented
133C4⋊C4—226no—
134C22⋊C4—116no—
135C22⋊C4—223no—
136C22⋊C4—226no—
137C22⋊C4—226no—
138C11#302611yescharacteristic, complemented, Sylow
139C23—24no—
140C23—23no—
141C23—23no—
142C23—81no—
143C23—83no—
144C23—82no—
145C23—223no—
146C23—881no—
147C23C2×C22⋊D1116yes—
148C23C2×C22⋊D1113yes—
149C23C2×C22⋊D1113yes—
150C23C23×D1111yescharacteristic
151Q8—24no—
152D4—212no—
153D4—212no—
154D4—212no—
155D4—441no—
156D4—443no—
157D4—886no—
158C2×C4—212no—
159C2×C4—23no—
160C2×C4—23no—
161C2×C4—226no—
162C2×C4—223no—
163C2×C4—883no—
164C2×C4C2×C22⋊D1116yes—
165C22—26no—
166C22—43no—
167C22—43no—
168C22—41no—
169C22—46no—
170C22—46no—
171C22—83no—
172C22—84no—
173C22—446no—
174C22—441no—
175C22#6700611yescharacteristic
176C22#6700613yes—
177C22C22×C22⋊D1113yes—
178C4—26no—
179C4—446no—
180C4—441no—
181C2—23no—
182C2—21no—
183C2—23no—
184C2—46no—
185C2—881no—
186C2C2×#6700611yescharacteristic
subgroupG/H
centerC2C2×#67006
commutator, cosocleC23×C11C24
FrattiniC23C23×D11
FittingC2×C11×H(5,𝔽2)C2
socleC2×C11C2×#120
hypercenterC2×H(5,𝔽2)D11
hypocenterC11#3026
LCSG ▷ C23×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C2 ◁ C24 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×C11×H(5,𝔽2)C2semidirect
2. #10623335C2semidirect
3. #10622857C2semidirect
4. #10622856C2semidirect
5. C2×#66898C2semidirect
11. C22×C22⋊D11C22semidirect
13. C2×C22⋊Dic11C22semidirect
15. #66990C22semidirect
16. #66988C22semidirect
18. #66898C22semidirect
20. C24×C11D4semidirect
22. C23×D11C23semidirect
31. C2×D4×C11D4semidirect
34. C22×C4×C11D4semidirect
39. C22×Dic11C23semidirect
42. C22⋊Dic11C23semidirect
72. C2×H(5,𝔽2)D11semidirect
114. C24C22⋊D11semidirect
127. C2×D4C22⋊D11semidirect
132. C22×C4C22⋊D11semidirect
138. C11#3026semidirect

Maximal subgroups

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

subgroupindex
C2×C11×H(5,𝔽2)2
#106233352
#106228572
#106228562
C2×#668982
#302611