Finite Group Databasebeta

← #35214984

Subgroups — #35214984

Order 20864.

358 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 93 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#23782176C211yesmaximal, characteristic
2C2×#19104663C212yesmaximal, complemented
3#23781629C211yesmaximal, characteristic, complemented
4#23781425C212yesmaximal, complemented
5C2×#117×C163C211yesmaximal, characteristic, complemented
6C2×#14579699C2211yescharacteristic
7C2×#14579683C2212yes—
8#19104718—21no—
9#19104714C412yescomplemented
10#19104709C412yescomplemented
11#19104704—21no—
12C4×C22⋊D163—22no—
13#19104694C2211yescharacteristic
14#19104663—24no—
15C22⋊(C163⋊C8)—22no—
16C23×C4×C163C2211yescharacteristic
17#117×C163—24no—
18C2×C4⋊C4×C163—21no—
19C2×C22⋊C4×C163—21no—
20C2×C42×C163C2212yes—
21#14579699—22no—
22#14579699—22no—
23C2×Dic326C2×C411yescharacteristic
24C2×D652C2×C411yescharacteristic
25C2×C22⋊D163—22no—
26C2×C22⋊D163—21no—
27C2×C4×D163—21no—
28#14579683—22no—
29#14579683—22no—
30C2×C163⋊C8—22no—
31C4⋊Dic163C2×C412yes—
32Dic163⋊C4—22no—
33C22⋊Dic163—22no—
34C22⋊Dic163—21no—
35(C2×D163)⋊C4—22no—
36C4×Dic163—21no—
37C24×C163D411yescharacteristic
38C22×C4×C163—22no—
39C22×C4×C163—22no—
40C22×C4×C163—22no—
41C22×C4×C163C2×C412yes—
42C22×C4×C163D411yescharacteristic
43C22×C4×C163D412yes—
44C22×C4×C163C2311yescharacteristic
45C4⋊C4×C163—24no—
46C22⋊C4×C163—24no—
47C42×C163—22no—
48C42×C163D414yescomplemented
49C22×D163—21no—
50Dic326—22no—
51D652—22no—
52C22⋊D163—44no—
53C22⋊D163—41no—
54C22⋊D163—41no—
55C2×Dic163—22no—
56C2×Dic163—21no—
57C4×D163—42no—
58C163⋊C8—42no—
59C23×C163—21no—
60C23×C163—21no—
61C23×C163—42no—
62C23×C163C22⋊C412yes—
63C23×C163C2×D411yescharacteristic
64C2×C4×C163—21no—
65C2×C4×C163—21no—
66C2×C4×C163—24no—
67C2×C4×C163—24no—
68C2×C4×C163—24no—
69C2×C4×C163—44no—
70C2×C4×C163C22⋊C412yes—
71C2×C4×C163C22×C411yescharacteristic
72C2×C4×C163C2×D412yes—
73C2×C4×C163C2×D411yescharacteristic
74C2×C4×C163#4112yes—
75C22×C163—22no—
76C22×C163—22no—
77C22×C163—22no—
78C22×C163—22no—
79C22×C163—42no—
80C22×C163—42no—
81C22×C163C2×C22⋊C411yescharacteristic
82C22×C163#12011yescharacteristic
83C22×C163#12311yescharacteristic
84C2×D163—41no—
85C2×D163—42no—
86C4×C163—22no—
87C4×C163—44no—
88C4×C163C4×D412yes—
89C4×C163#12112yes—
90Dic163—41no—
91Dic163—42no—
92C2×C163—21no—
93C2×C163—21no—
94C2×C163—42no—
95C2×C163#38611yescharacteristic
96C2×C163#42112yes—
97D163—81no—
98C163#189711yescharacteristic, complemented, Sylow
99#1897—1631nomaximal, Sylow
100#525—1631no—
101C2×#117D16311yescharacteristic, complemented
102C2×#104—1632no—
103#407—1631no—
104#339—1632no—
105C2×#41—1631no—
106C23×C4C2×D16311yescharacteristic
107C2×M16—1632no—
108#123—3261no—
109C22⋊Q8—1632no—
110#121—1632no—
111#120—3261no—
112C4×D4—3262no—
113#117—24no—
114#117—1631no—
115C2×C4⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C42C2×D16312yes—
118#104—3264no—
119C22⋊C8—3262no—
120C24C22⋊D16311yescharacteristic
121#41—3262no—
122#41—3262no—
123C2×Q8—1631no—
124C2×D4—3262no—
125C2×D4—3261no—
126C2×D4—1631no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—3261no—
131C22×C4C4×D16312yes—
132C22×C4C22⋊D16311yescharacteristic
133C22×C4D65212yes—
134C22×C4C22×D16311yescharacteristic
135M16—3262no—
136M16—3262no—
137C2×C8—3262no—
138C4⋊C4—24no—
139C4⋊C4—3262no—
140C4⋊C4—1632no—
141C22⋊C4—24no—
142C22⋊C4—3262no—
143C22⋊C4—3261no—
144C22⋊C4—3262no—
145C42—22no—
146C42—3261no—
147C42C22⋊D16314yescomplemented
148C23—21no—
149C23—21no—
150C23—42no—
151C23—3261no—
152C23(C2×D163)⋊C412yes—
153C23C2×C22⋊D16311yescharacteristic
154Q8—3262no—
155D4—3262no—
156D4—6524no—
157D4—6521no—
158D4—6521no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—24no—
162C2×C4—24no—
163C2×C4—24no—
164C2×C4—44no—
165C2×C4—3262no—
166C2×C4—3261no—
167C2×C4—6522no—
168C2×C4(C2×D163)⋊C412yes—
169C2×C4C2×C4×D16311yescharacteristic
170C2×C4C2×C22⋊D16312yes—
171C2×C4C2×D65211yescharacteristic
172C2×C4#1457969912yes—
173C8—6522no—
174C22—22no—
175C22—22no—
176C22—22no—
177C22—22no—
178C22—42no—
179C22—42no—
180C22—6522no—
181C22—6521no—
182C22C2×(C2×D163)⋊C411yescharacteristic
183C22#1910470411yescharacteristic
184C22#1910471811yescharacteristic
185C4—22no—
186C4—44no—
187C4—6522no—
188C4—6521no—
189C4C4×C22⋊D16312yes—
190C4#1910470912yes—
191C2—21no—
192C2—21no—
193C2—42no—
194C2—13041no—
195C2#2378154411yescharacteristic
196C2#2378168312yes—
subgroupG/H
centerC2×C4C2×D652
commutatorC2×C4×C163C22×C4
FrattiniC22×C4C22×D163
FittingC2×#117×C163C2
socleC22×C163C2×C22⋊C4
prime radical#23782176C2
cosocleC22×C4×C163C23
hypercenterC2×#117D163
hypocenterC163#1897
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. C2×#19104663C2semidirect
3. #23781629C2semidirect
4. #23781425C2semidirect
5. C2×#117×C163C2semidirect
9. #19104714C4semidirect
10. #19104709C4semidirect
48. C42×C163D4semidirect
98. C163#1897semidirect
101. C2×#117D163semidirect
147. C42C22⋊D163semidirect

Maximal subgroups

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

subgroupindex
#237821762
C2×#191046632
#237816292
#237814252
C2×#117×C1632
#1897163