Finite Group Databasebeta

← #35215882

Subgroups — #35215882

Order 20864.

246 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 75 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#23782296C211yesmaximal, characteristic
2#23781707C212yesmaximal
3#23781534C211yesmaximal, characteristic
4#23781426C212yesmaximal
5#343×C163C211yesmaximal, characteristic
6#19104734—22no—
7#19104726—22no—
8#19104714—22no—
9#19104694C2212yes—
10C2×C4⋊Dic163C2211yescharacteristic
11C2×Dic163⋊C4—22no—
12C2×C4×Dic163C2211yescharacteristic
13#19104669—22no—
14#19104667—22no—
15C2×M16×C163C2212yes—
16C2×C4⋊C4×C163C2211yescharacteristic
17C2×Dic326—22no—
18C22×Dic163—21no—
19C22×Dic163—21no—
20C4⋊Dic163—22no—
21C4⋊Dic163—22no—
22C4⋊Dic163D412yes—
23C4⋊Dic163D412yes—
24Dic163⋊C4—24no—
25Dic163⋊C4—44no—
26C22⋊Dic163—22no—
27C4×Dic163—22no—
28C4×Dic163—21no—
29C4×Dic163Q812yescomplemented
30C22×C4×C163D412yes—
31C22×C4×C163C2311yescharacteristic
32M16×C163—22no—
33M16×C163Q814yescomplemented
34C2×C8×C163—22no—
35C4⋊C4×C163—22no—
36Dic326—82no—
37C2×Dic163—22no—
38C2×Dic163—22no—
39C2×Dic163—22no—
40C2×Dic163—22no—
41C2×Dic163—22no—
42C2×Dic163—42no—
43C2×Dic163—41no—
44C23×C163C2×D411yescharacteristic
45C2×C4×C163—22no—
46C2×C4×C163—42no—
47C2×C4×C163C2×D412yes—
48C2×C4×C163C2×Q811yescharacteristic
49C2×C4×C163C2×Q812yes—
50C2×C4×C163#4111yescharacteristic
51C8×C163—24no—
52C22×C163—21no—
53C22×C163—21no—
54C22×C163#12011yescharacteristic
55C22×C163C22⋊Q811yescharacteristic
56C22×C163C4⋊Q811yescharacteristic
57C4×C163—42no—
58C4×C163C22⋊Q812yes—
59C4×C163C4⋊Q812yes—
60Dic163—24no—
61Dic163—42no—
62Dic163—42no—
63Dic163—81no—
64C2×C163—21no—
65C2×C163—21no—
66C2×C163#39511yescharacteristic
67C2×C163#45311yescharacteristic
68C2×C163#45611yescharacteristic
69C163#206011yescharacteristic, complemented, Sylow
70#2060—1631nomaximal, Sylow
71#544—1631no—
72#428—1632no—
73#384—1631no—
74#343D16311yescharacteristic
75#339—1632no—
76C2×M16C2×D16312yes—
77C4⋊Q8—3262no—
78#125—3262no—
79C22⋊Q8—3262no—
80#117—1632no—
81C2×C4⋊C4—3262no—
82C2×C4⋊C4—1631no—
83C2×C4⋊C4C2×D16311yescharacteristic
84C2×C42—1631no—
85#107—3262no—
86#106—3262no—
87C2×Q8—3262no—
88C22×C4—3261no—
89C22×C4—3261no—
90C22×C4D65212yes—
91C22×C4C22×D16311yescharacteristic
92M16—22no—
93M16Dic32614yescomplemented
94C2×C8—22no—
95C4⋊C4—22no—
96C4⋊C4—3262no—
97C4⋊C4—3264no—
98C4⋊C4—3262no—
99C4⋊C4—6524no—
100C4⋊C4—1632no—
101C4⋊C4—1632no—
102C22⋊C4—3262no—
103C42—3262no—
104C42—3261no—
105C42—1632no—
106C23C2×D65211yescharacteristic
107Q8—13042no—
108C2×C4—22no—
109C2×C4—42no—
110C2×C4—3262no—
111C2×C4—3262no—
112C2×C4—3262no—
113C2×C4—3262no—
114C2×C4—3262no—
115C2×C4—6522no—
116C2×C4—6521no—
117C2×C4D4×D16312yes—
118C2×C4Q8×D16311yescharacteristic
119C2×C4C2×Dic32612yes—
120C2×C4#1457969811yescharacteristic
121C8—24no—
122C22—21no—
123C22—21no—
124C22#1910470511yescharacteristic
125C22#1910471511yescharacteristic
126C22#1910473511yescharacteristic
127C4—42no—
128C4—13041no—
129C4—3264no—
130C4—6522no—
131C4—6522no—
132C4#1910471212yes—
133C4#1910473412yes—
134C2—21no—
135C2—21no—
136C2#2378158611yescharacteristic
137C2#2378183511yescharacteristic
138C2#2378185011yescharacteristic
subgroupG/H
centerC22#19104705
commutator, cosocleC22×C4×C163C23
FrattiniC22×C4C22×D163
Fitting#343×C163C2
socleC22×C163#120
prime radicalC23×C163C2×D4
hypercenter#343D163
hypocenterC163#2060
LCSG ▷ C22×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
29. C4×Dic163Q8semidirect
33. M16×C163Q8semidirect
69. C163#2060semidirect
93. M16Dic326semidirect

Maximal subgroups

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

subgroupindex
#237822962
#237817072
#237815342
#237814262
#343×C1632
#2060163