Finite Group Databasebeta

← #35225136

Subgroups — #35225136

Order 20864.

364 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 125 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#23782199C211yesmaximal, characteristic
2#23782071C211yesmaximal, characteristic, complemented
3#23782057C211yesmaximal, characteristic, complemented
4#23782033C211yesmaximal, characteristic, complemented
5#23782027C211yesmaximal, characteristic, complemented
6#23781921C212yesmaximal, complemented
7#23781913C212yesmaximal
8C4×#14579689C212yesmaximal, complemented
9C4×#14579686C212yesmaximal, complemented
10C4×#14579683C211yesmaximal, characteristic, complemented
11#533×C163C211yesmaximal, characteristic, complemented
12C2×#14579689C2212yes—
13C2×#14579686C2212yes—
14C2×#14579683C2212yes—
15#19104729—21no—
16#19104727C2211yescharacteristic
17#19104723C2211yescharacteristic
18#19104718—21no—
19#19104714C2211yescharacteristic
20#19104709C2211yescharacteristic
21C4×Dic326C2211yescharacteristic
22C4×D652C2211yescharacteristic
23C4×C22⋊D163—21no—
24#19104692—21no—
25#19104668C2212yes—
26#19104666C2212yes—
27#19104659C2212yes—
28#19104659C2212yes—
29#19104658C2212yes—
30#19104655C2212yes—
31#19104646C2211yescharacteristic
32#19104646C2211yescharacteristic
33C4×C163⋊C8C2212yes—
34C4⋊Q8×C163C2211yescharacteristic
35C4⋊Q8×C163C2211yescharacteristic
36#125×C163—21no—
37C22⋊Q8×C163—21no—
38C22⋊Q8×C163C2212yes—
39C4×Q8×C163—21no—
40C4×Q8×C163C2212yes—
41#117×C163—21no—
42C2×C42×C163C2211yescharacteristic
43C2×Dic326C2311yescharacteristic
44C2×D652C2311yescharacteristic
45C2×C22⋊D163—21no—
46C2×C4×D163—21no—
47#14579689—24no—
48#14579686—24no—
49#14579683—24no—
50C2×C163⋊C8C2312yes—
51C2×C163⋊C8C2312yes—
52C4⋊Dic163C2311yescharacteristic
53C4⋊Dic163C2311yescharacteristic
54Dic163⋊C4—21no—
55Dic163⋊C4—21no—
56Dic163⋊C4—21no—
57Dic163⋊C4—21no—
58C22⋊Dic163—21no—
59(C2×D163)⋊C4—21no—
60(C2×D163)⋊C4—21no—
61(C2×D163)⋊C4—21no—
62(C2×D163)⋊C4—21no—
63C4×Dic163—21no—
64C2×Q8×C163—21no—
65C2×Q8×C163C2312yes—
66C22×C4×C163D411yescharacteristic
67C22×C4×C163D411yescharacteristic
68C22×C4×C163C2311yescharacteristic
69C4⋊C4×C163—21no—
70C4⋊C4×C163—21no—
71C4⋊C4×C163—21no—
72C4⋊C4×C163—22no—
73C4⋊C4×C163—22no—
74C4⋊C4×C163C2312yes—
75C22⋊C4×C163—22no—
76C42×C163—22no—
77C42×C163D411yescharacteristic
78C42×C163D411yescharacteristic
79C42×C163C2311yescharacteristic
80C42×C163C2311yescharacteristic
81C22×D163—21no—
82Dic326—22no—
83D652—22no—
84C22⋊D163—42no—
85C2×Dic163—21no—
86C2×Dic163—21no—
87C2×Dic163—21no—
88C4×D163—42no—
89C163⋊C8—22no—
90C163⋊C8#4114yes—
91C23×C163C2×D411yescharacteristic
92Q8×C163—22no—
93Q8×C163—22no—
94Q8×C163—42no—
95C2×C4×C163—22no—
96C2×C4×C163—22no—
97C2×C4×C163—22no—
98C2×C4×C163—22no—
99C2×C4×C163—41no—
100C2×C4×C163C2×D411yescharacteristic
101C2×C4×C163C2×D411yescharacteristic
102C2×C4×C163C2×D411yescharacteristic
103C2×C4×C163C2×D411yescharacteristic
104C2×C4×C163C2×D411yescharacteristic
105C2×C4×C163C2411yescharacteristic
106C22×C163—21no—
107C22×C163—21no—
108C22×C163—21no—
109C22×C163C22×D411yescharacteristic
110C2×D163—41no—
111C2×D163—41no—
112C2×D163—41no—
113C4×C163—21no—
114C4×C163—21no—
115C4×C163—21no—
116C4×C163—21no—
117C4×C163—42no—
118C4×C163—42no—
119C4×C163#13712yes—
120C4×C163C2×#4112yes—
121Dic163—41no—
122Dic163—41no—
123Dic163—41no—
124C2×C163—41no—
125C2×C163#53211yescharacteristic
126C2×C163C2×#13711yescharacteristic
127C2×C163#57511yescharacteristic
128D163—81no—
129C163#317211yescharacteristic, complemented, Sylow
130#3172—1631nomaximal, Sylow
131#533D16311yescharacteristic, complemented
132#529—1631no—
133#501—1631no—
134#497—1631no—
135#491—1631no—
136#489—1631no—
137#470—1632no—
138#468—1632no—
139C4×Q16—1632no—
140C4×SD16—1632no—
141C4×M16—1631no—
142C2×Q16—1632no—
143C2×SD16—1632no—
144C2×M16—1632no—
145C4⋊Q8C2×D16311yescharacteristic
146C4⋊Q8C2×D16311yescharacteristic
147#126—3261no—
148#125—21no—
149#125—1631no—
150#124—1631no—
151#123—3261no—
152C22⋊Q8—21no—
153C22⋊Q8—1631no—
154C22⋊Q8C2×D16312yes—
155#121—1631no—
156C4×Q8—21no—
157C4×Q8—1631no—
158C4×Q8C2×D16312yes—
159C4×D4—3261no—
160C4×D4—1631no—
161#117—21no—
162#117—3261no—
163C2×C42C2×D16311yescharacteristic
164#107—1632no—
165#106—1632no—
166#103—1632no—
167#103—1632no—
168#103—1632no—
169#102—1632no—
170#97—1631no—
171#97—1631no—
172C4×C8—1632no—
173C2×Q8—21no—
174C2×Q8—1631no—
175C2×Q8C22×D16312yes—
176C2×D4—3261no—
177C2×D4—1631no—
178C22×C4—3261no—
179C22×C4C22⋊D16311yescharacteristic
180C22×C4C22⋊D16311yescharacteristic
181C22×C4C22×D16311yescharacteristic
182Q16—3264no—
183SD16—3264no—
184M16—3264no—
185C2×C8—1632no—
186C2×C8—1632no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—3261no—
193C4⋊C4—3261no—
194C4⋊C4—3261no—
195C4⋊C4—3261no—
196C4⋊C4—1631no—
197C4⋊C4—1631no—
198C4⋊C4C22×D16312yes—
199C22⋊C4—22no—
200C22⋊C4—3261no—
201C22⋊C4—3261no—
202C22⋊C4—3261no—
203C22⋊C4—3261no—
204C22⋊C4—3261no—
205C42—22no—
206C42—3261no—
207C42C22⋊D16311yescharacteristic
208C42C22⋊D16311yescharacteristic
209C42C22×D16311yescharacteristic
210C42C22×D16311yescharacteristic
211C23—3261no—
212C23C2×C22⋊D16311yescharacteristic
213Q8—22no—
214Q8—22no—
215Q8—42no—
216Q8—3262no—
217D4—3262no—
218D4—6522no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—41no—
224C2×C4—3261no—
225C2×C4—3261no—
226C2×C4—3261no—
227C2×C4—6522no—
228C2×C4C2×C22⋊D16311yescharacteristic
229C2×C4C2×C22⋊D16311yescharacteristic
230C2×C4C2×C22⋊D16311yescharacteristic
231C2×C4C2×C22⋊D16311yescharacteristic
232C2×C4C2×C22⋊D16311yescharacteristic
233C2×C4C23×D16311yescharacteristic
234C8—3262no—
235C8—1634no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—6521no—
240C22—6521no—
241C22—6521no—
242C22C22×C22⋊D16311yescharacteristic
243C4—21no—
244C4—21no—
245C4—21no—
246C4—21no—
247C4—42no—
248C4—42no—
249C4—6521no—
250C4—6521no—
251C4—6521no—
252C4#1910476612yes—
253C4#41×D16312yes—
254C2—41no—
255C2—13041no—
256C2#2378221111yescharacteristic
257C2C2×#1910476611yescharacteristic
258C2#2378246311yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#533×C163C2
socleC22×C163C22×D4
prime radical#19104709C22
hypercenter#533D163
hypocenterC163#3172
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
2. #23782071C2semidirect
3. #23782057C2semidirect
4. #23782033C2semidirect
5. #23782027C2semidirect
6. #23781921C2semidirect
8. C4×#14579689C2semidirect
9. C4×#14579686C2semidirect
10. C4×#14579683C2semidirect
11. #533×C163C2semidirect
129. C163#3172semidirect
131. #533D163semidirect

Maximal subgroups

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

subgroupindex
#237821992
#237820712
#237820572
#237820332
#237820272
#237819212
#237819132
C4×#145796892
C4×#145796862
C4×#145796832
#533×C1632
#3172163