Finite Group Databasebeta

← #35213126

Subgroups — #35213126

Order 20864.

270 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 63 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#23782214C211yesmaximal, characteristic
2#23781617C212yesmaximal, complemented
3#23781405C211yesmaximal, characteristic, complemented
4#23781404C212yesmaximal, complemented
5#329×C163C211yesmaximal, characteristic, complemented
6C2×#14579700C411yescharacteristic
7C2×#14579698C411yescharacteristic
8C2×#14579682—22no—
9#19104733C411yescharacteristic
10#19104731C411yescharacteristic
11#19104723C2211yescharacteristic
12#19104708—22no—
13(C22⋊D163)⋊C4—22no—
14C42×D163C2211yescharacteristic
15#19104646C2212yes—
16C8×Dic163—22no—
17#124×C163C2211yescharacteristic
18#97×C163C2212yes—
19#14579700—42no—
20#14579698—42no—
21C2×Dic326C2×C411yescharacteristic
22C2×D652—21no—
23C2×D652C2×C411yescharacteristic
24C2×C22⋊D163—21no—
25C2×C22⋊D163—22no—
26C22×Dic163—21no—
27C2×C4×D163—21no—
28C2×C4×D163C2×C411yescharacteristic
29C2×C4×D163D412yes—
30#14579682—28no—
31C2×C163⋊C8—22no—
32Dic163⋊C4—22no—
33(C2×D163)⋊C4—22no—
34C4×Dic163C2×C411yescharacteristic
35C4×Dic163D412yes—
36C2×Q8×C163C2×C411yescharacteristic
37C2×D4×C163C2×C411yescharacteristic
38C2×C8×C163—22no—
39C22⋊C4×C163—22no—
40C42×C163C2311yescharacteristic
41C22×D163—21no—
42C22×D163C22⋊C411yescharacteristic
43Dic326—41no—
44D652—41no—
45D652—42no—
46C22⋊D163—42no—
47C22⋊D163—44no—
48C2×Dic163—21no—
49C2×Dic163—42no—
50C2×Dic163C22⋊C411yescharacteristic
51C2×Dic163C22⋊C412yes—
52C4×D163—22no—
53C4×D163—24no—
54C4×D163—42no—
55C163⋊C8—24no—
56C23×C163—21no—
57Q8×C163—41no—
58D4×C163—41no—
59C2×C4×C163—21no—
60C2×C4×C163C22×C411yescharacteristic
61C2×C4×C163C2×D412yes—
62C8×C163—24no—
63C22×C163—41no—
64C22×C163—42no—
65C22×C163C2×C22⋊C411yescharacteristic
66C2×D163—21no—
67C2×D163—22no—
68C2×D163—41no—
69C2×D163—42no—
70C4×C163—21no—
71C4×C163—22no—
72C4×C163—41no—
73Dic163—41no—
74Dic163#10012yes—
75Dic163#10414yes—
76C2×C163—81no—
77C2×C163C2×#10011yescharacteristic
78C2×C163C2×#10412yes—
79D163—41no—
80D163—81no—
81C163#152511yescharacteristic, complemented, Sylow
82#1525—1631nomaximal, Sylow
83#532—1631no—
84C4×M16—1632no—
85#329—1631no—
86#329—1632no—
87#329D16311yescharacteristic, complemented
88C2×#41—1631no—
89C2×#41—1631no—
90C2×M16—3262no—
91C4⋊Q8—1631no—
92#127—1631no—
93#124—1631no—
94#124C2×D16311yescharacteristic
95#121—3262no—
96C4×D4—3262no—
97C2×C42—1631no—
98#97—1632no—
99#97C2×D16312yes—
100C4×C8—3262no—
101#41—6522no—
102#41—6522no—
103C2×Q8—1631no—
104C2×Q8C4×D16311yescharacteristic
105C2×D4—3261no—
106C2×D4—3261no—
107C2×D4—3262no—
108C2×D4—1631no—
109C2×D4C4×D16311yescharacteristic
110C22×C4—3261no—
111C22×C4—3261no—
112C22×C4—1631no—
113C22×C4—1632no—
114M16—3268no—
115C2×C8—22no—
116C2×C8—3262no—
117C4⋊C4—3262no—
118C22⋊C4—22no—
119C22⋊C4—3262no—
120C42—1631no—
121C42—1632no—
122C42C22×D16311yescharacteristic
123C23—21no—
124C23—3261no—
125C23—1631no—
126Q8—41no—
127Q8—6521no—
128D4—41no—
129D4—6521no—
130D4—6522no—
131D4—6522no—
132D4—6524no—
133C2×C4—21no—
134C2×C4—3264no—
135C2×C4—3262no—
136C2×C4—3261no—
137C2×C4—6522no—
138C2×C4—6522no—
139C2×C4—1632no—
140C2×C4—1631no—
141C2×C4C2×C4×D16311yescharacteristic
142C2×C4D4×D16312yes—
143C8—24no—
144C8—3264no—
145C22—41no—
146C22—42no—
147C22—3261no—
148C22—3262no—
149C22—6521no—
150C22—6522no—
151C22C22⋊C4×D16311yescharacteristic
152C4—21no—
153C4—22no—
154C4—41no—
155C4—6521no—
156C4—1632no—
157C4—1634no—
158C2—81no—
159C2—13041no—
160C2—6521no—
161C2#100×D16311yescharacteristic
162C2#104×D16312yes—
subgroupG/H
centerC22C22⋊C4×D163
commutatorC2×C4×C163C22×C4
FrattiniC42C22×D163
Fitting#329×C163C2
socleC22×C163C2×C22⋊C4
prime radical#23782214C2
cosocleC42×C163C23
hypercenter#329D163
hypocenterC163#1525
LCSG ▷ C2×C4×C163 ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #23781617C2semidirect
3. #23781405C2semidirect
4. #23781404C2semidirect
5. #329×C163C2semidirect
81. C163#1525semidirect
87. #329D163semidirect

Maximal subgroups

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

subgroupindex
#237822142
#237816172
#237814052
#237814042
#329×C1632
#1525163