Finite Group Databasebeta

← #35217801

Subgroups — #35217801

Order 20864.

560 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 147 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
1C2×(C22⋊D163)⋊C4C211yesmaximal, characteristic
2C2×#19104693C211yesmaximal, characteristic, complemented
3#23781603C211yesmaximal, characteristic, complemented
4#23781597C211yesmaximal, characteristic, complemented
5#23781591C212yesmaximal, complemented
6#23781590C211yesmaximal, characteristic, complemented
7#23781581C212yesmaximal
8C4⋊C4×Dic163C212yesmaximal, complemented
9C22⋊C4×Dic163C212yesmaximal, complemented
10#23781512C211yesmaximal, characteristic, complemented
11#399×C163C211yesmaximal, characteristic, complemented
12C22×C22⋊D163C2211yescharacteristic
13C23×Dic163C2211yescharacteristic
14C22×C4×D163C2211yescharacteristic
15(C22⋊D163)⋊C4—24no—
16#19104693—24no—
17C2×C4⋊Dic163C2212yes—
18C2×C4⋊Dic163C2211yescharacteristic
19C2×Dic163⋊C4C2211yescharacteristic
20C2×C22⋊Dic163C2212yes—
21C2×(C2×D163)⋊C4C2212yes—
22C2×(C2×D163)⋊C4C2212yes—
23C2×(C2×D163)⋊C4C2211yescharacteristic
24C2×C4×Dic163C2212yes—
25C2×C4×Dic163C2212yes—
26C2×C4×Dic163C2211yescharacteristic
27C163⋊#95C2211yescharacteristic
28C163⋊#95C2212yes—
29C163⋊#95C2212yes—
30C163⋊#95C2212yes—
31C163⋊#95C2212yes—
32C2×C4⋊C4×C163C2212yes—
33C2×C4⋊C4×C163C2211yescharacteristic
34C2×C22⋊C4×C163C2212yes—
35C2×C22⋊C4×C163C2211yescharacteristic
36#95×C163C2211yescharacteristic
37C23×D163C2311yescharacteristic
38C2×C22⋊D163—22no—
39C2×C22⋊D163—22no—
40C2×C22⋊D163—22no—
41C22×Dic163—22no—
42C22×Dic163—24no—
43C22×Dic163C2311yescharacteristic
44C22×Dic163C2312yes—
45C22×Dic163C2312yes—
46C22×Dic163C2311yescharacteristic
47C22×Dic163C2311yescharacteristic
48C2×C4×D163—24no—
49C2×C4×D163D414yes—
50C4⋊Dic163—24no—
51C4⋊Dic163—22no—
52Dic163⋊C4—22no—
53C22⋊Dic163—24no—
54(C2×D163)⋊C4—24no—
55(C2×D163)⋊C4—22no—
56(C2×D163)⋊C4—24no—
57C4×Dic163—24no—
58C4×Dic163—22no—
59C4×Dic163—24no—
60C24×C163C2311yescharacteristic
61C22×C4×C163C2311yescharacteristic
62C22×C4×C163C2312yes—
63C22×C4×C163C2311yescharacteristic
64C22×C4×C163C2312yes—
65C4⋊C4×C163—22no—
66C4⋊C4×C163—24no—
67C22⋊C4×C163—22no—
68C22⋊C4×C163—24no—
69C22×D163—22no—
70C22×D163—22no—
71C22×D163—21no—
72C22×D163—21no—
73C22×D163C2×D412yes—
74C22⋊D163—28no—
75C2×Dic163—21no—
76C2×Dic163—21no—
77C2×Dic163—22no—
78C2×Dic163—21no—
79C2×Dic163—22no—
80C2×Dic163—22no—
81C2×Dic163—22no—
82C2×Dic163—22no—
83C2×Dic163—22no—
84C2×Dic163—22no—
85C2×Dic163—44no—
86C2×Dic163C2×D412yes—
87C2×Dic163#4112yes—
88C2×Dic163#4114yes—
89C4×D163—28no—
90C23×C163—21no—
91C23×C163—22no—
92C23×C163—22no—
93C23×C163—21no—
94C23×C163#4112yes—
95C23×C163C2411yescharacteristic
96C2×C4×C163—22no—
97C2×C4×C163—22no—
98C2×C4×C163—22no—
99C2×C4×C163—21no—
100C2×C4×C163—22no—
101C2×C4×C163—22no—
102C2×C4×C163—22no—
103C2×C4×C163—22no—
104C2×C4×C163C2×D412yes—
105C2×C4×C163#4114yes—
106C22×C163—22no—
107C22×C163—24no—
108C22×C163—41no—
109C22×C163—42no—
110C22×C163—41no—
111C22×C163C22×D411yescharacteristic
112C22×C163C2×#4112yes—
113C22×C163C2×#4111yescharacteristic
114C22×C163C2×#4112yes—
115C22×C163C2×#4111yescharacteristic
116C2×D163—22no—
117C2×D163—22no—
118C2×D163—22no—
119C2×D163—42no—
120C2×D163—42no—
121C4×C163—22no—
122C4×C163—24no—
123C4×C163—41no—
124C4×C163—42no—
125Dic163—22no—
126Dic163—41no—
127Dic163—42no—
128Dic163—42no—
129Dic163#12314yes—
130C2×C163—42no—
131C2×C163C2×#12311yescharacteristic
132C2×C163#52511yescharacteristic
133C2×C163#52912yes—
134C2×C163#52912yes—
135C2×C163#53211yescharacteristic
136D163—42no—
137C163#239511yescharacteristic, complemented, Sylow
138#2395—1631nomaximal, Sylow
139C2×C4×D4—1631no—
140C2×#117—1631no—
141#400—1631no—
142#399D16311yescharacteristic, complemented
143#398—1631no—
144#397—1632no—
145#397—1631no—
146#394—1632no—
147C4×C4⋊C4—1632no—
148C4×C22⋊C4—1632no—
149C4×C22⋊C4—1631no—
150C22×D4—1631no—
151C23×C4—1631no—
152C23×C4—1631no—
153C4×D4—3264no—
154#117—3264no—
155C2×C4⋊C4—1631no—
156C2×C4⋊C4—1631no—
157C2×C4⋊C4—1632no—
158C2×C4⋊C4C2×D16311yescharacteristic
159C2×C4⋊C4C2×D16312yes—
160C2×C22⋊C4—1632no—
161C2×C22⋊C4—1632no—
162C2×C22⋊C4—1631no—
163C2×C22⋊C4—1632no—
164C2×C22⋊C4C2×D16311yescharacteristic
165C2×C22⋊C4C2×D16312yes—
166C2×C42—1632no—
167C2×C42—1631no—
168C2×C42—1632no—
169#95—1632no—
170#95—1631no—
171#95—1632no—
172#95—1632no—
173#95—1632no—
174#95C2×D16311yescharacteristic
175C24—1631no—
176C24C22×D16311yescharacteristic
177C2×D4—3262no—
178C2×D4—3262no—
179C2×D4—3262no—
180C22×C4—3264no—
181C22×C4—3262no—
182C22×C4—3264no—
183C22×C4—1632no—
184C22×C4—1631no—
185C22×C4—1632no—
186C22×C4—1631no—
187C22×C4—1634no—
188C22×C4—1631no—
189C22×C4C22×D16311yescharacteristic
190C22×C4C22×D16312yes—
191C22×C4C22×D16311yescharacteristic
192C22×C4C22×D16312yes—
193C4⋊C4—22no—
194C4⋊C4—24no—
195C4⋊C4—3264no—
196C4⋊C4—3262no—
197C4⋊C4—3262no—
198C22⋊C4—22no—
199C22⋊C4—24no—
200C22⋊C4—3264no—
201C22⋊C4—3262no—
202C22⋊C4—3264no—
203C22⋊C4—3264no—
204C42—3264no—
205C42—3262no—
206C42—3264no—
207C23—21no—
208C23—22no—
209C23—21no—
210C23—22no—
211C23—3262no—
212C23—3261no—
213C23—3262no—
214C23—3261no—
215C23—1632no—
216C23#1457970012yes—
217C23C23×D16311yescharacteristic
218D4—3268no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—3268no—
228C2×C4—3261no—
229C2×C4—3261no—
230C2×C4—3262no—
231C2×C4—3261no—
232C2×C4—3262no—
233C2×C4—3262no—
234C2×C4—3262no—
235C2×C4—3262no—
236C2×C4—3262no—
237C2×C4—3262no—
238C2×C4—6524no—
239C2×C4—1632no—
240C2×C4—1632no—
241C2×C4—1634no—
242C2×C4D4×D16312yes—
243C2×C4#1457969814yes—
244C22—22no—
245C22—24no—
246C22—41no—
247C22—42no—
248C22—41no—
249C22—3262no—
250C22—3262no—
251C22—3262no—
252C22—6522no—
253C22—6522no—
254C22C2×D4×D16311yescharacteristic
255C22#41×D16312yes—
256C22#41×D16311yescharacteristic
257C22C2×#1457969812yes—
258C22C2×#1457970011yescharacteristic
259C4—22no—
260C4—24no—
261C4—41no—
262C4—42no—
263C4—3262no—
264C4—6521no—
265C4—6522no—
266C4—6522no—
267C4—1634no—
268C2—42no—
269C2—6522no—
270C2#123×D16311yescharacteristic
271C2#2378217811yescharacteristic
272C2#2378219212yes—
273C2#2378220112yes—
274C2#2378221311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D163
commutator, socle, cosocleC23×C163C24
Fitting#399×C163C2
prime radicalC22×C22⋊D163C22
hypercenter#399D163
hypocenterC163#2395
LCSG ▷ C23×C163 ▷ C163
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
2. C2×#19104693C2semidirect
3. #23781603C2semidirect
4. #23781597C2semidirect
5. #23781591C2semidirect
6. #23781590C2semidirect
8. C4⋊C4×Dic163C2semidirect
9. C22⋊C4×Dic163C2semidirect
10. #23781512C2semidirect
11. #399×C163C2semidirect
137. C163#2395semidirect
142. #399D163semidirect

Maximal subgroups

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

subgroupindex
C2×(C22⋊D163)⋊C42
C2×#191046932
#237816032
#237815972
#237815912
#237815902
#237815812
C4⋊C4×Dic1632
C22⋊C4×Dic1632
#237815122
#399×C1632
#2395163