Finite Group Databasebeta

← #35215890

Subgroups — #35215890

Order 20864.

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

normal subgroups only

This group has 314 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#23782291C211yesmaximal, characteristic
2#23781540C212yesmaximal
3#23781537C211yesmaximal, characteristic
4#23781431C212yesmaximal
5#342×C163C211yesmaximal, characteristic
6C23×Dic163C2211yescharacteristic
7#19104714—21no—
8#19104714—21no—
9#19104712—22no—
10#19104712—22no—
11C2×C4⋊Dic163—22no—
12C2×Dic163⋊C4—21no—
13C2×C22⋊Dic163C2212yes—
14C2×C22⋊Dic163C2211yescharacteristic
15C163⋊#95—22no—
16C163⋊#95—21no—
17C2×C22⋊C4×C163C2211yescharacteristic
18C2×C22⋊C4×C163C2212yes—
19C2×Dic326—21no—
20C2×Dic326—21no—
21C22×Dic163—21no—
22C22×Dic163—22no—
23C22×Dic163—21no—
24C22×Dic163—42no—
25C22×Dic163D412yes—
26C22×Dic163D412yes—
27C4⋊Dic163—21no—
28C4⋊Dic163—21no—
29C4⋊Dic163—44no—
30Dic163⋊C4—22no—
31Dic163⋊C4—22no—
32Dic163⋊C4—42no—
33C22⋊Dic163—22no—
34C22⋊Dic163—22no—
35C22⋊Dic163—21no—
36C22⋊Dic163Q812yescomplemented
37C24×C163C2311yescharacteristic
38C22×C4×C163—22no—
39C22×C4×C163D411yescharacteristic
40C22×C4×C163D411yescharacteristic
41C22⋊C4×C163—22no—
42C22⋊C4×C163—22no—
43C22⋊C4×C163Q814yescomplemented
44Dic326—81no—
45Dic326—81no—
46C2×Dic163—22no—
47C2×Dic163—24no—
48C2×Dic163—22no—
49C2×Dic163—22no—
50C2×Dic163—22no—
51C2×Dic163—41no—
52C2×Dic163—41no—
53C2×Dic163—41no—
54C2×Dic163—42no—
55C2×Dic163—41no—
56C2×Dic163—42no—
57C2×Dic163—42no—
58C23×C163—41no—
59C23×C163—41no—
60C23×C163C2×D412yes—
61C23×C163C2×D411yescharacteristic
62C23×C163C2×Q811yescharacteristic
63C23×C163C2×Q812yes—
64C23×C163#4111yescharacteristic
65C2×C4×C163—21no—
66C2×C4×C163—21no—
67C2×C4×C163—24no—
68C2×C4×C163—42no—
69C2×C4×C163—42no—
70C2×C4×C163—41no—
71C2×C4×C163—41no—
72C22×C163—22no—
73C22×C163—21no—
74C22×C163—22no—
75C22×C163—21no—
76C22×C163—41no—
77C22×C163—42no—
78C22×C163—41no—
79C22×C163#12011yescharacteristic
80C22×C163C22⋊Q812yes—
81C22×C163C22⋊Q811yescharacteristic
82C22×C163C4⋊Q812yes—
83C22×C163C4⋊Q811yescharacteristic
84C4×C163—44no—
85C4×C163—41no—
86C4×C163—41no—
87Dic163—42no—
88Dic163—42no—
89Dic163—42no—
90Dic163—42no—
91C2×C163—21no—
92C2×C163—22no—
93C2×C163—22no—
94C2×C163—21no—
95C2×C163#39511yescharacteristic
96C2×C163B(4,𝔽2)11yescharacteristic
97C2×C163#45811yescharacteristic
98C163#206211yescharacteristic, complemented, Sylow
99#2062—1631nomaximal, Sylow
100#543—1631no—
101#385—1631no—
102#385—1632no—
103#342—1632no—
104#342D16311yescharacteristic
105C23×C4—1631no—
106C22⋊Q8—3262no—
107C22⋊Q8—3262no—
108C22⋊Q8—3261no—
109C22⋊Q8—3261no—
110C2×C4⋊C4—3262no—
111C2×C4⋊C4—3261no—
112C2×C22⋊C4—1632no—
113C2×C22⋊C4—1631no—
114C2×C22⋊C4C2×D16311yescharacteristic
115C2×C22⋊C4C2×D16312yes—
116#95—3262no—
117#95—3261no—
118C24C22×D16311yescharacteristic
119C2×Q8—3261no—
120C2×Q8—3261no—
121C22×C4—22no—
122C22×C4—3262no—
123C22×C4—3261no—
124C22×C4—3261no—
125C22×C4—6522no—
126C22×C4—1632no—
127C22×C4—1632no—
128C22×C4D65211yescharacteristic
129C22×C4D65211yescharacteristic
130C4⋊C4—3262no—
131C4⋊C4—3262no—
132C4⋊C4—3261no—
133C4⋊C4—3261no—
134C4⋊C4—6522no—
135C4⋊C4—6524no—
136C22⋊C4—22no—
137C22⋊C4—22no—
138C22⋊C4—3262no—
139C22⋊C4—3262no—
140C22⋊C4—3261no—
141C22⋊C4—1632no—
142C22⋊C4Dic32614yescomplemented
143C23—41no—
144C23—41no—
145C23D4×D16312yes—
146C23C2×D65211yescharacteristic
147C23Q8×D16311yescharacteristic
148C23C2×Dic32612yes—
149C23#1457969811yescharacteristic
150Q8—13041no—
151Q8—13041no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—24no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—3264no—
160C2×C4—3262no—
161C2×C4—3262no—
162C2×C4—3262no—
163C2×C4—3262no—
164C2×C4—6521no—
165C2×C4—6522no—
166C2×C4—6521no—
167C2×C4—6521no—
168C2×C4—6521no—
169C2×C4—6522no—
170C2×C4—6522no—
171C22—22no—
172C22—21no—
173C22—22no—
174C22—21no—
175C22—41no—
176C22—42no—
177C22—41no—
178C22#1910470511yescharacteristic
179C22#1910471212yes—
180C22#1910471511yescharacteristic
181C22#1910473412yes—
182C22#1910473511yescharacteristic
183C4—44no—
184C4—41no—
185C4—41no—
186C4—6522no—
187C4—6522no—
188C4—6522no—
189C4—6522no—
190C2—22no—
191C2—21no—
192C2—22no—
193C2—21no—
194C2#2378158611yescharacteristic
195C2#2378185311yescharacteristic
196C2#2378185611yescharacteristic
subgroupG/H
centerC22#19104705
commutator, prime radical, cosocleC24×C163C23
FrattiniC24C22×D163
Fitting#342×C163C2
socleC22×C163#120
hypercenter#342D163
hypocenterC163#2062
LCSG ▷ C24×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
36. C22⋊Dic163Q8semidirect
43. C22⋊C4×C163Q8semidirect
98. C163#2062semidirect
142. C22⋊C4Dic326semidirect

Maximal subgroups

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

subgroupindex
#237822912
#237815402
#237815372
#237814312
#342×C1632
#2062163