Finite Group Databasebeta

← #35218179

Subgroups — #35218179

Order 20864.

800 conjugacy classes of subgroups; only the 193 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 193 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
1C22×Q8×D163C211yesmaximal, characteristic
2C2×#19104735C211yesmaximal, characteristic, complemented
3C2×#19104733C211yesmaximal, characteristic, complemented
4C2×#19104715C211yesmaximal, characteristic
5#23781586C214yesmaximal, complemented
6#23781570C211yesmaximal, characteristic, complemented
7#23781539C214yesmaximal
8C4×(C2×D163)⋊C4C211yesmaximal, characteristic
9#391×C163C211yesmaximal, characteristic, complemented
10C22×Dic326C2211yescharacteristic
11C22×Dic326C2212yes—
12C22×Dic326C2211yescharacteristic
13C2×Q8×D163—24no—
14C2×Q8×D163—24no—
15C2×Q8×D163—24no—
16C22×C4×D163C2212yes—
17C22×C4×D163C2211yescharacteristic
18#19104735—24no—
19#19104733—24no—
20#19104715—24no—
21C2×C4⋊Dic163C2212yes—
22C2×C4⋊Dic163C2214yes—
23C2×Dic163⋊C4C2214yes—
24C2×(C2×D163)⋊C4C2212yes—
25C2×(C2×D163)⋊C4C2212yes—
26C2×C4×Dic163C2211yescharacteristic
27C163⋊#95C2212yes—
28C163⋊#95C2214yes—
29C22×Q8×C163C2211yescharacteristic
30C2×C4⋊C4×C163C2211yescharacteristic
31C2×C42×C163C2211yescharacteristic
32#95×C163C2214yes—
33C23×D163C2311yescharacteristic
34C2×Dic326—24no—
35C2×Dic326—24no—
36C2×Dic326—22no—
37C2×Dic326—24no—
38C2×Dic326—22no—
39C2×Dic326—22no—
40C2×Dic326—22no—
41C2×Dic326—22no—
42C2×Dic326D414yes—
43Q8×D163—232no—
44C22×Dic163C2312yes—
45C22×Dic163C2312yes—
46C22×Dic163C2311yescharacteristic
47C22×Dic163C2312yes—
48C2×C4×D163—24no—
49C2×C4×D163—24no—
50C2×C4×D163—22no—
51C2×C4×D163—24no—
52C2×C4×D163—22no—
53C2×C4×D163D414yes—
54C4⋊Dic163—24no—
55C4⋊Dic163—28no—
56Dic163⋊C4—28no—
57(C2×D163)⋊C4—24no—
58(C2×D163)⋊C4Q818yescomplemented
59C4×Dic163—22no—
60C2×Q8×C163—22no—
61C2×Q8×C163—22no—
62C2×Q8×C163D414yes—
63C22×C4×C163C2312yes—
64C22×C4×C163C2312yes—
65C22×C4×C163C2312yes—
66C22×C4×C163C2311yescharacteristic
67C4⋊C4×C163—22no—
68C42×C163—22no—
69C22×D163—21no—
70C22×D163—21no—
71C22×D163—22no—
72C22×D163C2×D412yes—
73C22×D163C2×Q814yes—
74Dic326—216no—
75Dic326—28no—
76Dic326—44no—
77C2×Dic163—22no—
78C2×Dic163—22no—
79C2×Dic163—21no—
80C2×Dic163—22no—
81C2×Dic163—21no—
82C2×Dic163—22no—
83C2×Dic163—22no—
84C2×Dic163—22no—
85C2×Dic163—22no—
86C2×Dic163C2×D414yes—
87C2×Dic163C2×D412yes—
88C2×Dic163C2×Q814yescomplemented
89C4×D163—216no—
90C4×D163—28no—
91C23×C163C2411yescharacteristic
92Q8×C163—28no—
93C2×C4×C163—22no—
94C2×C4×C163—22no—
95C2×C4×C163—22no—
96C2×C4×C163—22no—
97C2×C4×C163—22no—
98C2×C4×C163—24no—
99C2×C4×C163C2×D412yes—
100C2×C4×C163C2×D412yes—
101C2×C4×C163C2×D414yes—
102C2×C4×C163C2×D412yes—
103C2×C4×C163C2×Q814yescomplemented
104C22×C163C22×D411yescharacteristic
105C22×C163C22×D411yescharacteristic
106C22×C163C22×D411yescharacteristic
107C22×C163C22×Q812yes—
108C22×C1632−1+412yes—
109C2×D163—22no—
110C2×D163—22no—
111C2×D163—24no—
112C2×D163—24no—
113C2×D163C4⋊Q814yescomplemented
114C4×C163—24no—
115C4×C163—24no—
116C4×C163—42no—
117C4×C163#12014yes—
118Dic163—24no—
119Dic163—22no—
120Dic163—24no—
121Dic163—42no—
122C2×C163C2×#12011yescharacteristic
123C2×C163C2×C4⋊Q811yescharacteristic
124C2×C163#53611yescharacteristic
125C2×C163D4×Q812yes—
126C2×C163D4×Q812yes—
127D163—24no—
128C163#243411yescharacteristic, complemented, Sylow
129#2434—1631nomaximal, Sylow
130C23×Q8—1631no—
131C2×C4⋊Q8—1631no—
132C2×C4⋊Q8—1631no—
133C2×C22⋊Q8—1631no—
134#395—1634no—
135#391—1631no—
136#391D16311yescharacteristic, complemented
137#385—1634no—
138C4×C22⋊C4—1631no—
139C22×Q8—3264no—
140C22×Q8—3264no—
141C22×Q8—3264no—
142C22×Q8—1631no—
143C22×Q8—1632no—
144C22×Q8—1631no—
145C22×Q8C2×D16311yescharacteristic
146C23×C4—1632no—
147C23×C4—1631no—
148C4⋊Q8—3264no—
149C4⋊Q8—3264no—
150C22⋊Q8—3264no—
151C2×C4⋊C4—1632no—
152C2×C4⋊C4—1634no—
153C2×C4⋊C4—1634no—
154C2×C4⋊C4C2×D16311yescharacteristic
155C2×C22⋊C4—1632no—
156C2×C22⋊C4—1632no—
157C2×C42—1631no—
158C2×C42C2×D16311yescharacteristic
159#95—1632no—
160#95—1634no—
161#95C2×D16314yes—
162C24—1631no—
163C2×Q8—22no—
164C2×Q8—22no—
165C2×Q8—3264no—
166C2×Q8—3264no—
167C2×Q8—32632no—
168C2×Q8—3262no—
169C2×Q8—3262no—
170C2×Q8—3262no—
171C2×Q8—3264no—
172C2×Q8—3262no—
173C2×Q8—3262no—
174C2×Q8—1634no—
175C2×Q8D65214yes—
176C22×C4—3264no—
177C22×C4—3264no—
178C22×C4—3262no—
179C22×C4—3264no—
180C22×C4—3262no—
181C22×C4—1632no—
182C22×C4—1632no—
183C22×C4—1632no—
184C22×C4—1631no—
185C22×C4—1634no—
186C22×C4C22×D16312yes—
187C22×C4C22×D16312yes—
188C22×C4C22×D16312yes—
189C22×C4C22×D16311yescharacteristic
190C4⋊C4—22no—
191C4⋊C4—3264no—
192C4⋊C4—3268no—
193C4⋊C4—3268no—
194C22⋊C4—3264no—
195C22⋊C4—1638no—
196C42—22no—
197C42—3262no—
198C23—3261no—
199C23—3261no—
200C23—3262no—
201C23—1632no—
202C23—1634no—
203C23C23×D16311yescharacteristic
204Q8—28no—
205Q8—32616no—
206Q8—3268no—
207Q8—6524no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—24no—
214C2×C4—32616no—
215C2×C4—3262no—
216C2×C4—3261no—
217C2×C4—3262no—
218C2×C4—3262no—
219C2×C4—3261no—
220C2×C4—3262no—
221C2×C4—3268no—
222C2×C4—3262no—
223C2×C4—3262no—
224C2×C4—3262no—
225C2×C4—1634no—
226C2×C4—1632no—
227C2×C4—1634no—
228C2×C4D4×D16312yes—
229C2×C4D4×D16312yes—
230C2×C4C2×D65214yes—
231C2×C4C2×D65212yes—
232C2×C4Q8×D16314yescomplemented
233C22—3262no—
234C22—3262no—
235C22—3264no—
236C22—3264no—
237C22—1634no—
238C22C2×D4×D16311yescharacteristic
239C22C2×D4×D16311yescharacteristic
240C22C22×D65211yescharacteristic
241C22C2×Q8×D16312yes—
242C22#1910478212yes—
243C4—24no—
244C4—24no—
245C4—42no—
246C4—3264no—
247C4—3262no—
248C4—3264no—
249C4—6522no—
250C4#1910470514yes—
251C2—3264no—
252C2C2×#1910470511yescharacteristic
253C2C4⋊Q8×D16311yescharacteristic
254C2#2378223511yescharacteristic
255C2#2378232812yes—
256C2Q8×D65212yes—
subgroupG/H
center, FrattiniC23C23×D163
commutator, socle, cosocleC23×C163C24
Fitting#391×C163C2
prime radicalC23×D163C23
hypercenter#391D163
hypocenterC163#2434
LCSG ▷ C23×C163 ▷ C163
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
2. C2×#19104735C2semidirect
3. C2×#19104733C2semidirect
5. #23781586C2semidirect
6. #23781570C2semidirect
9. #391×C163C2semidirect
58. (C2×D163)⋊C4Q8semidirect
88. C2×Dic163C2×Q8semidirect
103. C2×C4×C163C2×Q8semidirect
113. C2×D163C4⋊Q8semidirect
128. C163#2434semidirect
136. #391D163semidirect
232. C2×C4Q8×D163semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D1632
C2×#191047352
C2×#191047332
C2×#191047152
#237815862
#237815702
#237815392
C4×(C2×D163)⋊C42
#391×C1632
#2434163