Finite Group Databasebeta

← #35223782

Subgroups — #35223782

Order 20864.

758 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 149 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×#14579700C211yesmaximal, characteristic, complemented
2C2×#19104766C211yesmaximal, characteristic, complemented
3C2×#19104763C211yesmaximal, characteristic, complemented
4#23782177C211yesmaximal, characteristic, complemented
5#23781830C214yesmaximal, complemented
6#23781813C214yesmaximal, complemented
7#23781669C211yesmaximal, characteristic, complemented
8#23781628C211yesmaximal, characteristic, complemented
9#419×C163C211yesmaximal, characteristic, complemented
10C2×#14579700—24no—
11C2×#14579700—22no—
12C2×#14579700—24no—
13C2×#14579700C2214yes—
14C2×#14579699C2211yescharacteristic
15C22×D652—21no—
16C22×D652C2211yescharacteristic
17C22×C4×D163—21no—
18C22×C4×D163C2211yescharacteristic
19#19104766—24no—
20#19104763—24no—
21C2×#14579689C2212yescomplemented
22C2×#14579688C2212yescomplemented
23C2×#14579686C2212yescomplemented
24C2×D1304C2212yescomplemented
25C2×#14579683C2211yescharacteristic
26#19104719—21no—
27#19104715C2212yescomplemented
28#19104710C2212yescomplemented
29#19104705—21no—
30C4×D652—22no—
31#19104655C2214yescomplemented
32(C2×D163)⋊C8C2214yescomplemented
33C22×Q8×C163C2211yescharacteristic
34C2×M16×C163C2211yescharacteristic
35#117×C163C2211yescharacteristic
36#103×C163C2214yescomplemented
37C23×D163—21no—
38C23×D163D411yescharacteristic
39#14579700—216no—
40#14579700—28no—
41#14579700—28no—
42#14579699—22no—
43#14579699—22no—
44C2×Dic326C2311yescharacteristic
45C2×D652—28no—
46C2×D652—24no—
47C2×D652—22no—
48C2×D652—21no—
49C2×D652—22no—
50C2×D652D414yescomplemented
51C2×D652C2311yescharacteristic
52C2×D652C2312yes—
53C2×C22⋊D163—21no—
54C22×Dic163D411yescharacteristic
55C2×C4×D163—28no—
56C2×C4×D163—24no—
57C2×C4×D163—22no—
58C2×C4×D163—22no—
59C2×C4×D163—21no—
60C2×C4×D163D412yes—
61C2×C4×D163C2312yes—
62#14579689—24no—
63#14579688—24no—
64#14579686—24no—
65D1304—24no—
66#14579683—22no—
67C2×C163⋊C8C2312yes—
68C4⋊Dic163—22no—
69(C2×D163)⋊C4—22no—
70(C2×D163)⋊C4—22no—
71C2×Q8×C163—22no—
72C2×Q8×C163—21no—
73C2×Q8×C163D414yescomplemented
74C2×Q8×C163C2312yes—
75C22×C4×C163—21no—
76C22×C4×C163C2311yescharacteristic
77M16×C163—22no—
78C2×C8×C163C2312yes—
79C4⋊C4×C163C2312yes—
80C22⋊C4×C163—21no—
81C42×C163—21no—
82C22×D163—22no—
83C22×D163—24no—
84C22×D163—22no—
85C22×D163—21no—
86C22×D163—42no—
87C22×D163—44no—
88C22×D163C2×D412yes—
89Dic326—21no—
90Dic326—21no—
91D652—216no—
92D652—24no—
93D652—22no—
94D652—21no—
95D652—21no—
96D652—44no—
97D652C2×D414yes—
98C22⋊D163—42no—
99C2×Dic163—22no—
100C2×Dic163—21no—
101C2×Dic163C2×D412yes—
102C4×D163—216no—
103C4×D163—24no—
104C4×D163—22no—
105C4×D163—22no—
106C4×D163—42no—
107C163⋊C8—22no—
108C23×C163C2×D411yescharacteristic
109Q8×C163—24no—
110Q8×C163—22no—
111Q8×C163C2×D414yes—
112C2×C4×C163—24no—
113C2×C4×C163—22no—
114C2×C4×C163—22no—
115C2×C4×C163C2×D411yescharacteristic
116C2×C4×C163C2×D412yes—
117C2×C4×C163C2×D412yes—
118C2×C4×C163C2411yescharacteristic
119C8×C163—22no—
120C22×C163—22no—
121C22×C163#12012yes—
122C22×C163C22×D411yescharacteristic
123C2×D163—22no—
124C2×D163—24no—
125C2×D163—22no—
126C2×D163—22no—
127C2×D163—44no—
128C2×D163—48no—
129C2×D163—44no—
130C2×D163—41no—
131C2×D163—42no—
132C4×C163—24no—
133C4×C163—42no—
134C4×C163#12012yes—
135C4×C163C22×D411yescharacteristic
136C4×C163C22×D411yescharacteristic
137Dic163—22no—
138Dic163—41no—
139C2×C163—22no—
140C2×C163C2×#12011yescharacteristic
141C2×C163#57512yes—
142D163—42no—
143D163—44no—
144D163—81no—
145C163#300511yescharacteristic, complemented, Sylow
146#3005—1631nomaximal, Sylow
147C22×#41—1631no—
148C2×#137—1631no—
149C2×AGL(1,ℤ/8)—1631no—
150#525—1631no—
151#452—1634no—
152#449—1634no—
153#419D16311yescharacteristic, complemented
154#418—1631no—
155#407—1631no—
156C2×#41—3264no—
157C2×#41—3264no—
158C2×#41—3262no—
159C2×#41—1634no—
160C2×#41—1631no—
161C22×Q8C2×D16311yescharacteristic
162C22×D4—3261no—
163C22×D4—1631no—
164C23×C4—3261no—
165C23×C4—1631no—
166#137—3264no—
167AGL(1,ℤ/8)—3264no—
168C2×Q16—1632no—
169C2×SD16—1632no—
170C2×SD16—1632no—
171C2×D8—1632no—
172C2×M16—1631no—
173C2×M16C2×D16311yescharacteristic
174#123—3261no—
175C22⋊Q8—1632no—
176#121—1632no—
177#120—3261no—
178C4×D4—3262no—
179#117C2×D16311yescharacteristic
180#103C2×D16314yescomplemented
181#102—1634no—
182C22⋊C8—1634no—
183C24—3261no—
184C24—1631no—
185#41—32616no—
186#41—3268no—
187#41—3268no—
188#41—3262no—
189#41—3262no—
190C2×Q8—21no—
191C2×Q8—22no—
192C2×Q8—1631no—
193C2×Q8D65214yescomplemented
194C2×Q8C22×D16312yes—
195C2×D4—3268no—
196C2×D4—3264no—
197C2×D4—3262no—
198C2×D4—3261no—
199C2×D4—3261no—
200C2×D4—3262no—
201C2×D4—1634no—
202C2×D4—1631no—
203C2×D4—1632no—
204C22×C4—21no—
205C22×C4—3268no—
206C22×C4—3264no—
207C22×C4—3262no—
208C22×C4—3262no—
209C22×C4—3261no—
210C22×C4—1631no—
211C22×C4—1632no—
212C22×C4—1632no—
213C22×C4C22×D16311yescharacteristic
214Q16—3264no—
215SD16—3264no—
216SD16—3264no—
217D8—3264no—
218M16—22no—
219M16—3262no—
220C2×C8—1632no—
221C2×C8C22×D16312yes—
222C4⋊C4—3262no—
223C4⋊C4C22×D16312yes—
224C22⋊C4—21no—
225C22⋊C4—3262no—
226C22⋊C4—3262no—
227C42—21no—
228C23—3262no—
229C23—3262no—
230C23—3264no—
231C23—3261no—
232C23—6522no—
233C23—6524no—
234C23—1632no—
235C23D4×D16311yescharacteristic
236Q8—22no—
237Q8—24no—
238Q8—3261no—
239Q8—3261no—
240Q8C2×D65214yes—
241D4—32616no—
242D4—3261no—
243D4—3262no—
244D4—3261no—
245D4—3264no—
246D4—6524no—
247D4—6522no—
248D4—1634no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—32616no—
253C2×C4—3262no—
254C2×C4—3262no—
255C2×C4—3262no—
256C2×C4—3264no—
257C2×C4—3261no—
258C2×C4—6522no—
259C2×C4—1632no—
260C2×C4D4×D16312yes—
261C2×C4D4×D16311yescharacteristic
262C2×C4C2×D65212yes—
263C2×C4C23×D16311yescharacteristic
264C8—22no—
265C8—3262no—
266C22—22no—
267C22—3262no—
268C22—3264no—
269C22—3262no—
270C22—3262no—
271C22—6524no—
272C22—6528no—
273C22—6521no—
274C22—6522no—
275C22—6524no—
276C22#1910470512yes—
277C22C2×D4×D16311yescharacteristic
278C4—24no—
279C4—42no—
280C4—3262no—
281C4—6521no—
282C4#1910470512yes—
283C4C2×D4×D16311yescharacteristic
284C4C22×D65211yescharacteristic
285C2—22no—
286C2—13041no—
287C2—6522no—
288C2—6524no—
289C2C2×#1910470511yescharacteristic
290C2#2378246412yes—
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#419×C163C2
socleC22×C163C22×D4
hypercenter#419D163
hypocenterC163#3005
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. C22×#14579700C2semidirect
2. C2×#19104766C2semidirect
3. C2×#19104763C2semidirect
4. #23782177C2semidirect
5. #23781830C2semidirect
6. #23781813C2semidirect
7. #23781669C2semidirect
8. #23781628C2semidirect
9. #419×C163C2semidirect
21. C2×#14579689C22semidirect
22. C2×#14579688C22semidirect
23. C2×#14579686C22semidirect
24. C2×D1304C22semidirect
27. #19104715C22semidirect
28. #19104710C22semidirect
31. #19104655C22semidirect
32. (C2×D163)⋊C8C22semidirect
36. #103×C163C22semidirect
50. C2×D652D4semidirect
73. C2×Q8×C163D4semidirect
145. C163#3005semidirect
153. #419D163semidirect
180. #103C2×D163semidirect
193. C2×Q8D652semidirect

Maximal subgroups

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

subgroupindex
C22×#145797002
C2×#191047662
C2×#191047632
#237821772
#237818302
#237818132
#237816692
#237816282
#419×C1632
#3005163