Finite Group Databasebeta

← #35225187

Subgroups — #35225187

Order 20864.

386 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 117 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#23782271C212yesmaximal, complemented
2#23782042C211yesmaximal, characteristic, complemented
3#23782031C212yesmaximal, complemented
4#23781919C212yesmaximal, complemented
5#23781902C212yesmaximal, complemented
6#23781776C212yesmaximal, complemented
7#23781747C212yesmaximal, complemented
8#23781619C211yesmaximal, characteristic, complemented
9#499×C163C211yesmaximal, characteristic, complemented
10C2×D1304C2212yes—
11C2×D1304C2212yes—
12C2×#14579682C2211yescharacteristic
13C2×C8×D163C2211yescharacteristic
14#19104732C2212yes—
15#19104720—24no—
16#19104710—24no—
17#19104710C2214yes—
18#19104697C2214yes—
19#19104692C2211yescharacteristic
20#19104655C2214yes—
21#19104655C2214yes—
22#19104646C2211yescharacteristic
23#19104645C2211yescharacteristic
24C8×Dic163C2211yescharacteristic
25#125×C163C2212yes—
26#107×C163C2212yes—
27#106×C163C2212yes—
28C4×C8×C163C2211yescharacteristic
29C2×D652—24no—
30C2×D652—24no—
31C2×D652C2314yes—
32C2×C4×D163—24no—
33C2×C4×D163C2311yescharacteristic
34D1304—24no—
35D1304—24no—
36#14579682—22no—
37C2×C163⋊C8C2311yescharacteristic
38C2×C163⋊C8C2311yescharacteristic
39C8×D163—22no—
40Dic163⋊C4D412yes—
41(C2×D163)⋊C4—24no—
42(C2×D163)⋊C4—24no—
43(C2×D163)⋊C4D412yes—
44C4×Dic163C2311yescharacteristic
45C2×C8×C163C2311yescharacteristic
46C2×C8×C163C2311yescharacteristic
47C4⋊C4×C163—24no—
48C4⋊C4×C163C2314yes—
49C42×C163C2311yescharacteristic
50C22×D163—24no—
51C22×D163C2×D411yescharacteristic
52D652—24no—
53D652—24no—
54D652—84no—
55D652—84no—
56C2×Dic163C2×D412yes—
57C2×Dic163C2×D411yescharacteristic
58C4×D163—21no—
59C4×D163—21no—
60C4×D163—84no—
61C163⋊C8—21no—
62C163⋊C8—21no—
63C2×C4×C163—24no—
64C2×C4×C163C2×D412yes—
65C2×C4×C163C2411yescharacteristic
66C8×C163#4112yes—
67C8×C163#4112yes—
68C22×C163C22×D411yescharacteristic
69C2×D163—21no—
70C2×D163—22no—
71C2×D163—44no—
72C2×D163—44no—
73C2×D163—44no—
74C4×C163—22no—
75C4×C163—44no—
76C4×C163C2×#4111yescharacteristic
77C4×C163C2×#4111yescharacteristic
78Dic163—21no—
79Dic163—22no—
80C2×C163#53211yescharacteristic
81C2×C163#57612yes—
82D163—41no—
83D163—84no—
84C163#317611yescharacteristic, complemented, Sylow
85#3176—1631nomaximal, Sylow
86#540—1632no—
87#499D16311yescharacteristic, complemented
88#493—1631no—
89#490—1632no—
90#469—1632no—
91#466—1632no—
92#442—1632no—
93C4×D8—1632no—
94#405—1631no—
95C2×D8—1632no—
96C2×D8—1632no—
97C2×M16—1631no—
98C22×C8—1631no—
99#127—1632no—
100#125C2×D16312yes—
101#123—3264no—
102#121—3264no—
103#121—1634no—
104C4×D4—1634no—
105#117—1631no—
106#107C2×D16312yes—
107#106C2×D16312yes—
108#102—1634no—
109#102—1634no—
110#97—1631no—
111#97—1631no—
112C4×C8—1631no—
113C4×C8C2×D16311yescharacteristic
114C2×D4—3264no—
115C2×D4—3264no—
116C2×D4—1634no—
117C22×C4—3264no—
118C22×C4—1631no—
119D8—3264no—
120D8—3264no—
121M16—3262no—
122C2×C8—3262no—
123C2×C8—1631no—
124C2×C8—1631no—
125C2×C8C22×D16311yescharacteristic
126C2×C8C22×D16311yescharacteristic
127C4⋊C4—24no—
128C4⋊C4—1632no—
129C4⋊C4C22×D16314yes—
130C22⋊C4—3264no—
131C22⋊C4—3264no—
132C22⋊C4—1632no—
133C42—1631no—
134C42C22×D16311yescharacteristic
135C23—3264no—
136C23—1631no—
137D4—13044no—
138D4—13044no—
139D4—3264no—
140D4—3264no—
141C2×C4—24no—
142C2×C4—13044no—
143C2×C4—3261no—
144C2×C4—3261no—
145C2×C4—1632no—
146C2×C4—1631no—
147C2×C4D4×D16312yes—
148C2×C4C23×D16311yescharacteristic
149C8—3261no—
150C8—3261no—
151C8#1457970012yes—
152C8#1457970012yes—
153C22—3262no—
154C22—3261no—
155C22—6524no—
156C22—6524no—
157C22—6524no—
158C22C2×D4×D16311yescharacteristic
159C4—22no—
160C4—44no—
161C4—3262no—
162C4—3261no—
163C4C2×#1457970011yescharacteristic
164C4C2×#1457970011yescharacteristic
165C2—13044no—
166C2—6521no—
167C2#2378221611yescharacteristic
168C2#2378246712yes—
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#499×C163C2
socleC22×C163C22×D4
hypercenter#499D163
hypocenterC163#3176
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #499

Complements

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

NH ≅ G/Ntype
1. #23782271C2semidirect
2. #23782042C2semidirect
3. #23782031C2semidirect
4. #23781919C2semidirect
5. #23781902C2semidirect
6. #23781776C2semidirect
7. #23781747C2semidirect
8. #23781619C2semidirect
9. #499×C163C2semidirect
84. C163#3176semidirect
87. #499D163semidirect

Maximal subgroups

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

subgroupindex
#237822712
#237820422
#237820312
#237819192
#237819022
#237817762
#237817472
#237816192
#499×C1632
#3176163