Finite Group Databasebeta

← #35227253

Subgroups — #35227253

Order 20864.

372 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 115 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#23782385C211yesmaximal, characteristic, complemented
2#23782330C211yesmaximal, characteristic, complemented
3#23782023C212yesmaximal, complemented
4#23782017C211yesmaximal, characteristic, complemented
5#23781959C212yesmaximal, complemented
6#23781879C212yesmaximal
7#23781871C211yesmaximal, characteristic, complemented
8#23781792C211yesmaximal, characteristic, complemented
9#23781764C212yesmaximal, complemented
10C4×#14579684C211yesmaximal, characteristic
11#474×C163C211yesmaximal, characteristic, complemented
12C2×Q8×D163—21no—
13C2×#14579688C2212yes—
14C2×#14579684C2211yescharacteristic
15#19104735C2211yescharacteristic
16#19104734C2212yes—
17#19104728—22no—
18#19104722C2212yes—
19#19104714—21no—
20#19104713—22no—
21#19104711—21no—
22C4×Dic326C2211yescharacteristic
23#19104700C2212yes—
24C4×D652C2211yescharacteristic
25#19104697C2212yes—
26#19104694—21no—
27#19104668C2211yescharacteristic
28Dic163⋊C8C2212yes—
29#19104660C2212yes—
30#19104658C2212yes—
31#19104658C2212yes—
32#19104656C2211yescharacteristic
33#19104655C2211yescharacteristic
34#19104645C2212yes—
35C4×C163⋊C8C2211yescharacteristic
36C4⋊Q8×C163C2211yescharacteristic
37C4×D4×C163C2211yescharacteristic
38#107×C163C2212yes—
39C4⋊C8×C163C2211yescharacteristic
40#102×C163C2212yes—
41C2×Dic326—22no—
42C2×Dic326D411yescharacteristic
43C2×Dic326C2312yes—
44Q8×D163—44no—
45C2×D652C2311yescharacteristic
46C2×C4×D163—22no—
47C2×C4×D163—21no—
48#14579688—24no—
49#14579684—22no—
50C2×C163⋊C8C2311yescharacteristic
51C2×C163⋊C8C2311yescharacteristic
52C4⋊Dic163—21no—
53C4⋊Dic163—21no—
54C4⋊Dic163D411yescharacteristic
55C4⋊Dic163C2311yescharacteristic
56Dic163⋊C4—22no—
57Dic163⋊C4—22no—
58Dic163⋊C4D412yes—
59C22⋊Dic163—22no—
60C22⋊Dic163—22no—
61(C2×D163)⋊C4—22no—
62(C2×D163)⋊C4—21no—
63C4×Dic163—22no—
64C4×Dic163C2312yes—
65C2×Q8×C163—21no—
66C2×D4×C163C2311yescharacteristic
67C22×C4×C163—21no—
68C2×C8×C163C2312yes—
69C4⋊C4×C163—21no—
70C4⋊C4×C163C2312yes—
71C4⋊C4×C163C2311yescharacteristic
72C22⋊C4×C163—21no—
73C42×C163C2311yescharacteristic
74C22×D163—21no—
75Dic326—22no—
76Dic326—22no—
77Dic326—22no—
78Dic326—44no—
79D652—21no—
80D652#4112yes—
81C2×Dic163—22no—
82C2×Dic163—21no—
83C2×Dic163C2×D412yes—
84C2×Dic163C2×D412yes—
85C4×D163—44no—
86C4×D163—42no—
87C163⋊C8—21no—
88C163⋊C8—21no—
89C23×C163—21no—
90Q8×C163—42no—
91D4×C163—21no—
92D4×C163—21no—
93C2×C4×C163—21no—
94C2×C4×C163—22no—
95C2×C4×C163—42no—
96C2×C4×C163C2×D411yescharacteristic
97C2×C4×C163C2×D411yescharacteristic
98C2×C4×C163C2411yescharacteristic
99C8×C163—22no—
100C22×C163—41no—
101C22×C163—41no—
102C22×C163—41no—
103C22×C163C22×D411yescharacteristic
104C2×D163—22no—
105C2×D163—41no—
106C2×D163—41no—
107C4×C163—21no—
108C4×C163—41no—
109C4×C163—42no—
110C4×C163AGL(1,ℤ/8)12yes—
111C4×C163C2×#4111yescharacteristic
112C4×C1632−1+411yescharacteristic
113Dic163—22no—
114Dic163—22no—
115Dic163—42no—
116Dic163—41no—
117C2×C163—81no—
118C2×C163#54811yescharacteristic
119C2×C163C2×AGL(1,ℤ/8)11yescharacteristic
120C2×C163#57811yescharacteristic
121D163—42no—
122C163#336711yescharacteristic, complemented, Sylow
123#3367—1631nomaximal, Sylow
124#556—1631no—
125D4×Q8—1631no—
126#489—1632no—
127#487—1631no—
128#477—1632no—
129#474D16311yescharacteristic, complemented
130#463—1632no—
131#461—1631no—
132#446—1631no—
133#440—1632no—
134C4×D8—1631no—
135C22×Q8—3261no—
136C2×SD16—1632no—
137C2×D8—1631no—
138C4⋊Q8—1632no—
139C4⋊Q8—1631no—
140C4⋊Q8C2×D16311yescharacteristic
141#126—3262no—
142#124—1632no—
143C22⋊Q8—3262no—
144C22⋊Q8—3261no—
145C22⋊Q8—3261no—
146C4×Q8—1632no—
147C4×Q8—1631no—
148C4×D4—1632no—
149C4×D4—1631no—
150C4×D4C2×D16311yescharacteristic
151#117—3261no—
152#107—1631no—
153#107C2×D16312yes—
154C4⋊C8—1632no—
155C4⋊C8C2×D16311yescharacteristic
156#103—1632no—
157#103—1632no—
158#103—1632no—
159#102—1631no—
160#102—1631no—
161#102C2×D16312yes—
162#97—1632no—
163C4×C8—1631no—
164C2×Q8—21no—
165C2×Q8—3262no—
166C2×Q8—6524no—
167C2×Q8—1632no—
168C2×Q8—1631no—
169C2×D4—1631no—
170C2×D4C22×D16311yescharacteristic
171C22×C4—21no—
172C22×C4—3262no—
173C22×C4—3261no—
174SD16—3264no—
175D8—3262no—
176C2×C8—1631no—
177C2×C8—1631no—
178C2×C8C22×D16312yes—
179C4⋊C4—21no—
180C4⋊C4—3262no—
181C4⋊C4—3262no—
182C4⋊C4—3261no—
183C4⋊C4—3261no—
184C4⋊C4—1631no—
185C4⋊C4—1631no—
186C4⋊C4—1632no—
187C4⋊C4C22×D16311yescharacteristic
188C4⋊C4C22×D16312yes—
189C22⋊C4—21no—
190C22⋊C4—3262no—
191C22⋊C4—3262no—
192C22⋊C4—3262no—
193C22⋊C4—3261no—
194C42—3262no—
195C42—1632no—
196C42C22×D16311yescharacteristic
197C23—21no—
198C23—3261no—
199Q8—42no—
200Q8—3262no—
201Q8—3262no—
202Q8—3262no—
203Q8—6524no—
204D4—21no—
205D4—21no—
206D4—3261no—
207D4—1632no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—42no—
211C2×C4—3262no—
212C2×C4—3261no—
213C2×C4—6524no—
214C2×C4—6522no—
215C2×C4—1632no—
216C2×C4—1632no—
217C2×C4D4×D16311yescharacteristic
218C2×C4D4×D16311yescharacteristic
219C2×C4C23×D16311yescharacteristic
220C8—22no—
221C8—3261no—
222C8—3261no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—3262no—
227C22—6521no—
228C22—6521no—
229C22C2×D4×D16311yescharacteristic
230C4—21no—
231C4—41no—
232C4—42no—
233C4—3262no—
234C4—3262no—
235C4—6521no—
236C4—6522no—
237C4#1910475912yes—
238C4#41×D16311yescharacteristic
239C4#1910478311yescharacteristic
240C2—81no—
241C2—6522no—
242C2#2378232511yescharacteristic
243C2C2×#1910475911yescharacteristic
244C2#2378248111yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#474×C163C2
socleC22×C163C22×D4
prime radicalC2×#14579684C22
hypercenter#474D163
hypocenterC163#3367
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #23782385C2semidirect
2. #23782330C2semidirect
3. #23782023C2semidirect
4. #23782017C2semidirect
5. #23781959C2semidirect
7. #23781871C2semidirect
8. #23781792C2semidirect
9. #23781764C2semidirect
11. #474×C163C2semidirect
122. C163#3367semidirect
129. #474D163semidirect

Maximal subgroups

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

subgroupindex
#237823852
#237823302
#237820232
#237820172
#237819592
#237818792
#237818712
#237817922
#237817642
C4×#145796842
#474×C1632
#3367163