Finite Group Databasebeta

← #35213848

Subgroups — #35213848

Order 20864.

196 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 45 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#23782415C211yesmaximal, characteristic
2#23782074C212yesmaximal, complemented
3#23782060C211yesmaximal, characteristic
4#23781406C212yesmaximal, complemented
5#329×C163C211yesmaximal, characteristic
6C2×#14579689—21no—
7C2×#14579687—21no—
8#19104735C2211yescharacteristic
9#19104733—21no—
10#19104728—22no—
11#19104727C2212yes—
12#19104716—21no—
13#19104712—22no—
14#19104669—22no—
15#19104667—22no—
16#19104646C2211yescharacteristic
17#124×C163C2211yescharacteristic
18#97×C163C2212yes—
19C2×Dic326—21no—
20C2×Dic326—21no—
21C22×Dic163—21no—
22#14579689—42no—
23#14579687—42no—
24C2×C163⋊C8—21no—
25C4⋊Dic163—22no—
26C4⋊Dic163D412yes—
27C4⋊Dic163D412yes—
28Dic163⋊C4—22no—
29Dic163⋊C4—22no—
30Dic163⋊C4—22no—
31C22⋊Dic163—22no—
32C22⋊Dic163—21no—
33C4×Dic163—21no—
34C2×Q8×C163D411yescharacteristic
35C2×D4×C163D411yescharacteristic
36C2×C8×C163—22no—
37C22⋊C4×C163—22no—
38C42×C163C2311yescharacteristic
39Dic326—41no—
40Dic326—41no—
41Dic326—81no—
42C2×Dic163—22no—
43C2×Dic163—22no—
44C2×Dic163—41no—
45C2×Dic163—81no—
46C163⋊C8—22no—
47C23×C163—21no—
48Q8×C163—41no—
49D4×C163—41no—
50C2×C4×C163—21no—
51C2×C4×C163C2×D412yes—
52C2×C4×C163C2×D411yescharacteristic
53C8×C163—24no—
54C22×C163—41no—
55C22×C163—42no—
56C22×C163#12011yescharacteristic
57C4×C163—21no—
58C4×C163—22no—
59C4×C163—41no—
60Dic163—42no—
61Dic163—42no—
62Dic163—81no—
63C2×C163—81no—
64C2×C163#45311yescharacteristic
65C2×C163#45612yes—
66C163#166011yescharacteristic, complemented, Sylow
67#1660—1631nomaximal, Sylow
68#563—1631no—
69#501—1632no—
70#497—1631no—
71#330—1632no—
72#329D16311yescharacteristic
73C2×Q16—3261no—
74C2×SD16—3261no—
75C4⋊Q8—3261no—
76C4⋊Q8—1631no—
77#126—3262no—
78#125—1632no—
79#124C2×D16311yescharacteristic
80#123—3261no—
81C22⋊Q8—3262no—
82#107—3262no—
83#106—3262no—
84#97—1631no—
85#97C2×D16312yes—
86C2×Q8—3261no—
87C2×Q8—3261no—
88C2×Q8D65211yescharacteristic
89C2×D4D65211yescharacteristic
90C22×C4—3261no—
91Q16—6522no—
92SD16—6522no—
93C2×C8—22no—
94C2×C8—3261no—
95C4⋊C4—3262no—
96C4⋊C4—3262no—
97C4⋊C4—3262no—
98C4⋊C4—3262no—
99C4⋊C4—1632no—
100C4⋊C4—1632no—
101C22⋊C4—22no—
102C22⋊C4—3262no—
103C22⋊C4—3261no—
104C42—3261no—
105C42C22×D16311yescharacteristic
106C23—21no—
107Q8—41no—
108Q8—13041no—
109Q8—6521no—
110Q8—6521no—
111D4—41no—
112C2×C4—21no—
113C2×C4—13041no—
114C2×C4—3262no—
115C2×C4—3262no—
116C2×C4—6521no—
117C2×C4D4×D16312yes—
118C2×C4C2×D65211yescharacteristic
119C8—24no—
120C8—3262no—
121C22—41no—
122C22—42no—
123C22#1910470511yescharacteristic
124C4—21no—
125C4—22no—
126C4—41no—
127C4—13041no—
128C4—6522no—
129C4—6522no—
130C2—81no—
131C2#2378183511yescharacteristic
132C2#2378184912yes—
subgroupG/H
centerC22#19104705
commutator, cosocleC42×C163C23
FrattiniC42C22×D163
Fitting#329×C163C2
socleC22×C163#120
prime radicalC2×D4×C163D4
hypercenter#329D163
hypocenterC163#1660
LCSG ▷ C42×C163 ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #23782074C2semidirect
4. #23781406C2semidirect
66. C163#1660semidirect

Maximal subgroups

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

subgroupindex
#237824152
#237820742
#237820602
#237814062
#329×C1632
#1660163