Finite Group Databasebeta

← #35222885

Subgroups — #35222885

Order 20864.

644 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 189 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×#19104733C211yesmaximal, characteristic, complemented
2C2×#19104715C211yesmaximal, characteristic
3C2×#19104711C211yesmaximal, characteristic
4C2×C42×D163C211yesmaximal, characteristic
5#23781571C211yesmaximal, characteristic, complemented
6#23781562C214yesmaximal, complemented
7#23781534C211yesmaximal, characteristic, complemented
8#23781524C214yesmaximal
9#391×C163C211yesmaximal, characteristic, complemented
10C22×Dic326C2211yescharacteristic
11C22×C4×D163C2211yescharacteristic
12C22×C4×D163C2212yes—
13#19104733—24no—
14#19104715—24no—
15#19104711—24no—
16C2×C4⋊Dic163C2211yescharacteristic
17C2×C4⋊Dic163C2212yes—
18C2×Dic163⋊C4C2212yes—
19C2×Dic163⋊C4C2214yes—
20C2×Dic163⋊C4C2214yes—
21C2×(C2×D163)⋊C4C2212yes—
22C2×(C2×D163)⋊C4C2212yes—
23C2×C4×Dic163C2212yes—
24C2×C4×Dic163C2211yescharacteristic
25C42×D163—24no—
26C163⋊#95C2214yes—
27C22×Q8×C163C2211yescharacteristic
28C2×C4⋊C4×C163C2211yescharacteristic
29C2×C42×C163C2211yescharacteristic
30#95×C163C2214yes—
31C23×D163C2311yescharacteristic
32C2×Dic326—22no—
33C2×Dic326—22no—
34C2×Dic326—22no—
35C22×Dic163C2312yes—
36C22×Dic163C2312yes—
37C22×Dic163C2311yescharacteristic
38C22×Dic163C2312yes—
39C2×C4×D163—24no—
40C2×C4×D163—24no—
41C2×C4×D163—22no—
42C2×C4×D163—22no—
43C2×C4×D163D414yes—
44C2×C4×D163Q818yescomplemented
45C4⋊Dic163—24no—
46C4⋊Dic163—22no—
47Dic163⋊C4—24no—
48Dic163⋊C4—28no—
49Dic163⋊C4—28no—
50(C2×D163)⋊C4—24no—
51(C2×D163)⋊C4—24no—
52C4×Dic163—24no—
53C4×Dic163D414yes—
54C2×Q8×C163—22no—
55C2×Q8×C163—22no—
56C2×Q8×C163—22no—
57C22×C4×C163C2312yes—
58C22×C4×C163C2312yes—
59C22×C4×C163C2312yes—
60C22×C4×C163C2311yescharacteristic
61C4⋊C4×C163—22no—
62C42×C163—22no—
63C22×D163—21no—
64C22×D163—22no—
65C22×D163—21no—
66C22×D163C2×D412yes—
67C22×D163C2×Q814yes—
68Dic326—44no—
69C2×Dic163—21no—
70C2×Dic163—21no—
71C2×Dic163—22no—
72C2×Dic163—22no—
73C2×Dic163—22no—
74C2×Dic163—24no—
75C2×Dic163—22no—
76C2×Dic163C2×D412yes—
77C2×Dic163C2×D414yes—
78C2×Dic163C2×Q814yes—
79C2×Dic163#4114yes—
80C4×D163—28no—
81C4×D163—216no—
82C23×C163C2411yescharacteristic
83Q8×C163—44no—
84C2×C4×C163—22no—
85C2×C4×C163—22no—
86C2×C4×C163—22no—
87C2×C4×C163—22no—
88C2×C4×C163—22no—
89C2×C4×C163—22no—
90C2×C4×C163—24no—
91C2×C4×C163C2×D412yes—
92C2×C4×C163C2×D412yes—
93C2×C4×C163C2×Q814yes—
94C2×C4×C163#4112yes—
95C22×C163C22×D411yescharacteristic
96C22×C163C22×D411yescharacteristic
97C22×C163C22×Q812yes—
98C22×C163C2×#4112yes—
99C22×C163C2×#4111yescharacteristic
100C2×D163—22no—
101C2×D163—24no—
102C2×D163—22no—
103C2×D163—24no—
104C2×D163C4⋊Q814yescomplemented
105C4×C163—24no—
106C4×C163—42no—
107C4×C163—42no—
108C4×C163#12114yes—
109Dic163—22no—
110Dic163—42no—
111Dic163—42no—
112Dic163C22⋊Q818yescomplemented
113C2×C163C2×#12111yescharacteristic
114C2×C163C2×C22⋊Q812yes—
115C2×C163C2×C4⋊Q811yescharacteristic
116C2×C163#53211yescharacteristic
117C2×C163#53312yes—
118D163—24no—
119C163#285711yescharacteristic, complemented, Sylow
120#2857—1631nomaximal, Sylow
121C2×C4⋊Q8—1631no—
122C2×C22⋊Q8—1631no—
123C2×C22⋊Q8—1631no—
124C22×C42—1631no—
125#391—1631no—
126#391D16311yescharacteristic, complemented
127#389—1634no—
128#384—1631no—
129#380—1634no—
130C22×Q8—1631no—
131C22×Q8C2×D16311yescharacteristic
132C23×C4—1631no—
133C23×C4—1632no—
134C4⋊Q8—3264no—
135C22⋊Q8—3264no—
136C22⋊Q8—3264no—
137C2×C4⋊C4—1631no—
138C2×C4⋊C4—1632no—
139C2×C4⋊C4—1632no—
140C2×C4⋊C4—1634no—
141C2×C4⋊C4—1634no—
142C2×C4⋊C4C2×D16311yescharacteristic
143C2×C22⋊C4—1632no—
144C2×C22⋊C4—1632no—
145C2×C42—3264no—
146C2×C42—1632no—
147C2×C42—1631no—
148C2×C42C2×D16311yescharacteristic
149#95—1634no—
150#95C2×D16314yes—
151C24—1631no—
152C2×Q8—22no—
153C2×Q8—22no—
154C2×Q8—22no—
155C2×Q8—3262no—
156C2×Q8—3262no—
157C2×Q8—3262no—
158C22×C4—3264no—
159C22×C4—3264no—
160C22×C4—3262no—
161C22×C4—3262no—
162C22×C4—1632no—
163C22×C4—1632no—
164C22×C4—1631no—
165C22×C4—1634no—
166C22×C4—1638no—
167C22×C4—1632no—
168C22×C4C22×D16312yes—
169C22×C4C22×D16312yes—
170C22×C4C22×D16312yes—
171C22×C4C22×D16311yescharacteristic
172C4⋊C4—22no—
173C4⋊C4—3264no—
174C4⋊C4—3264no—
175C4⋊C4—3268no—
176C4⋊C4—3268no—
177C4⋊C4—3262no—
178C22⋊C4—3264no—
179C22⋊C4—3264no—
180C42—22no—
181C42—3264no—
182C42—1634no—
183C23—3261no—
184C23—3262no—
185C23—3261no—
186C23—1632no—
187C23—1634no—
188C23C23×D16311yescharacteristic
189Q8—44no—
190Q8—6524no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—24no—
197C2×C4—22no—
198C2×C4—32616no—
199C2×C4—3261no—
200C2×C4—3261no—
201C2×C4—3268no—
202C2×C4—3262no—
203C2×C4—3262no—
204C2×C4—3262no—
205C2×C4—3264no—
206C2×C4—3262no—
207C2×C4—1632no—
208C2×C4—1634no—
209C2×C4—1634no—
210C2×C4—1634no—
211C2×C4D4×D16312yes—
212C2×C4D4×D16312yes—
213C2×C4Q8×D16314yes—
214C2×C4#1457969912yes—
215C22—3262no—
216C22—3264no—
217C22—3262no—
218C22—3264no—
219C22—1634no—
220C22C2×D4×D16311yescharacteristic
221C22C2×D4×D16311yescharacteristic
222C22C2×Q8×D16312yes—
223C22#41×D16312yes—
224C22C2×#1457969911yescharacteristic
225C4—24no—
226C4—42no—
227C4—42no—
228C4—3262no—
229C4—6522no—
230C4—6522no—
231C4—1638no—
232C4#1910470814yes—
233C2—3264no—
234C2C2×#1910470811yescharacteristic
235C2C22⋊Q8×D16312yes—
236C2C4⋊Q8×D16311yescharacteristic
237C2#2378221411yescharacteristic
238C2#2378221912yes—
subgroupG/H
center, FrattiniC23C23×D163
commutator, socle, cosocleC23×C163C24
Fitting#391×C163C2
prime radicalC23×D163C23
hypercenter#391D163
hypocenterC163#2857
LCSG ▷ C23×C163 ▷ C163
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#19104733C2semidirect
5. #23781571C2semidirect
6. #23781562C2semidirect
7. #23781534C2semidirect
9. #391×C163C2semidirect
44. C2×C4×D163Q8semidirect
104. C2×D163C4⋊Q8semidirect
112. Dic163C22⋊Q8semidirect
119. C163#2857semidirect
126. #391D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047332
C2×#191047152
C2×#191047112
C2×C42×D1632
#237815712
#237815622
#237815342
#237815242
#391×C1632
#2857163