Finite Group Databasebeta

← #35224478

Subgroups — #35224478

Order 20864.

462 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 127 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×#19104710C211yesmaximal, characteristic
2#117×D163C211yesmaximal, characteristic, complemented
3#23781992C212yesmaximal, complemented
4#23781991C212yesmaximal, complemented
5#23781985C212yesmaximal, complemented
6#23781980C212yesmaximal, complemented
7#23781705C211yesmaximal, characteristic, complemented
8#23781671C211yesmaximal, characteristic, complemented
9#23781669C211yesmaximal, characteristic, complemented
10#23781628C211yesmaximal, characteristic, complemented
11#428×C163C211yesmaximal, characteristic, complemented
12C22×D652—21no—
13C22×D652C2211yescharacteristic
14C22×C4×D163C2211yescharacteristic
15C2×#14579683C2211yescharacteristic
16#19104710—21no—
17#19104710—21no—
18#19104710C2212yes—
19#19104710C2212yes—
20#19104694C2211yescharacteristic
21#19104693C2211yescharacteristic, complemented
22#19104693C2211yescharacteristic, complemented
23#19104692—21no—
24#19104691—21no—
25C4⋊C4×D163C2211yescharacteristic, complemented
26C4⋊C4×D163C2211yescharacteristic, complemented
27C2×(C2×D163)⋊C4—21no—
28C22⋊C4×D163—21no—
29C42×D163—21no—
30#19104668C2212yescomplemented
31#19104666C2212yescomplemented
32#19104657C2212yescomplemented
33#19104657C2212yescomplemented
34#19104655C2212yescomplemented
35#19104655C2212yescomplemented
36(C2×D163)⋊C8C2212yescomplemented
37(C2×D163)⋊C8C2212yescomplemented
38C2×M16×C163C2211yescharacteristic
39#117×C163C2211yescharacteristic
40C2×C4⋊C4×C163C2211yescharacteristic
41#107×C163C2212yescomplemented
42#106×C163C2212yescomplemented
43C23×D163—21no—
44C23×D163D411yescharacteristic
45C2×D652—22no—
46C2×D652—22no—
47C2×D652—21no—
48C2×D652—22no—
49C2×D652—22no—
50C2×D652—44no—
51C2×D652C2312yes—
52C22×Dic163D411yescharacteristic
53C2×C4×D163—21no—
54C2×C4×D163—21no—
55C2×C4×D163—22no—
56C2×C4×D163—22no—
57C2×C4×D163D411yescharacteristic
58C2×C4×D163D411yescharacteristic
59C2×C4×D163C2311yescharacteristic
60C2×C4×D163C2311yescharacteristic
61#14579683—22no—
62C2×C163⋊C8C2312yes—
63C4⋊Dic163C2311yescharacteristic
64C4⋊Dic163C2311yescharacteristic
65Dic163⋊C4—21no—
66Dic163⋊C4—21no—
67C22⋊Dic163—21no—
68(C2×D163)⋊C4—21no—
69(C2×D163)⋊C4—21no—
70(C2×D163)⋊C4—22no—
71(C2×D163)⋊C4—22no—
72C4×Dic163—21no—
73C4×Dic163—21no—
74C4×Dic163—21no—
75C22×C4×C163—21no—
76C22×C4×C163C2311yescharacteristic
77M16×C163—22no—
78C2×C8×C163C2312yes—
79C4⋊C4×C163—21no—
80C4⋊C4×C163C2311yescharacteristic
81C4⋊C4×C163C2312yes—
82C4⋊C4×C163C2311yescharacteristic
83C22⋊C4×C163—21no—
84C42×C163—21no—
85C22×D163—22no—
86C22×D163—22no—
87C22×D163—22no—
88C22×D163—41no—
89C22×D163—42no—
90C22×D163—42no—
91C22×D163—42no—
92C22×D163C2×D411yescharacteristic
93C22×D163C2×D411yescharacteristic
94D652—22no—
95D652—22no—
96D652—44no—
97D652—44no—
98D652—42no—
99C2×Dic163—22no—
100C2×Dic163—21no—
101C2×Dic163—21no—
102C2×Dic163C2×D411yescharacteristic
103C2×Dic163C2×D411yescharacteristic
104C4×D163—22no—
105C4×D163—22no—
106C4×D163—21no—
107C4×D163—21no—
108C4×D163—21no—
109C4×D163—21no—
110C4×D163—42no—
111C4×D163—42no—
112C163⋊C8—22no—
113C23×C163C2×D411yescharacteristic
114C2×C4×C163—21no—
115C2×C4×C163—21no—
116C2×C4×C163—22no—
117C2×C4×C163—42no—
118C2×C4×C163C2×D411yescharacteristic
119C2×C4×C163#4112yes—
120C2×C4×C163#4112yes—
121C2×C4×C163C2411yescharacteristic
122C8×C163—22no—
123C22×C163—21no—
124C22×C163—21no—
125C22×C163#12312yes—
126C22×C163C22×D411yescharacteristic
127C2×D163—21no—
128C2×D163—21no—
129C2×D163—21no—
130C2×D163—21no—
131C2×D163—21no—
132C2×D163—44no—
133C2×D163—42no—
134C2×D163—42no—
135C2×D163—42no—
136C2×D163—84no—
137C2×D163AGL(1,ℤ/8)12yes—
138C4×C163—42no—
139C4×C163—41no—
140C4×C163—41no—
141C4×C163#12312yes—
142C4×C163C2×#4111yescharacteristic
143C4×C163C2×#4111yescharacteristic
144Dic163—21no—
145Dic163—21no—
146Dic163—41no—
147Dic163—41no—
148C2×C163—22no—
149C2×C163C2×#12311yescharacteristic
150C2×C163C2×AGL(1,ℤ/8)11yescharacteristic
151C2×C163#57511yescharacteristic
152D163—22no—
153D163—41no—
154D163—82no—
155C163#309411yescharacteristic, complemented, Sylow
156#3094—1631nomaximal, Sylow
157C2×#121—1631no—
158C2×#117—1631no—
159#482—1632no—
160#482—1632no—
161#481—1632no—
162#480—1632no—
163#428—1631no—
164#428D16311yescharacteristic, complemented
165#418—1631no—
166#418—1631no—
167#407—1631no—
168C22×D4—3261no—
169C22×D4—1631no—
170C23×C4—1631no—
171C2×M16—1631no—
172C2×M16C2×D16311yescharacteristic
173#121—3261no—
174#121—3261no—
175#121—1632no—
176#121—1632no—
177#117—3261no—
178#117—3261no—
179#117—1631no—
180#117—1631no—
181#117—1631no—
182#117C2×D16311yescharacteristic
183C2×C4⋊C4—1631no—
184C2×C4⋊C4—1631no—
185C2×C4⋊C4C2×D16311yescharacteristic
186C2×C22⋊C4—3261no—
187C2×C22⋊C4—3261no—
188C2×C42—3261no—
189#107—1632no—
190#107C2×D16312yescomplemented
191#106—1632no—
192#106C2×D16312yescomplemented
193#102—1632no—
194#102—1632no—
195#102—1632no—
196#102—1632no—
197C22⋊C8—1632no—
198C22⋊C8—1632no—
199C24—3261no—
200C24—1631no—
201C2×D4—3262no—
202C2×D4—3262no—
203C2×D4—3261no—
204C2×D4—3262no—
205C2×D4—3262no—
206C2×D4—6524no—
207C2×D4—1632no—
208C22×C4—21no—
209C22×C4—3261no—
210C22×C4—3261no—
211C22×C4—3262no—
212C22×C4—3262no—
213C22×C4—1631no—
214C22×C4—1631no—
215C22×C4—1631no—
216C22×C4—1631no—
217C22×C4—1631no—
218C22×C4C22×D16311yescharacteristic
219M16—22no—
220M16—3262no—
221C2×C8—1632no—
222C2×C8C22×D16312yes—
223C4⋊C4—21no—
224C4⋊C4—3261no—
225C4⋊C4—3261no—
226C4⋊C4—1631no—
227C4⋊C4—1631no—
228C4⋊C4C22×D16311yescharacteristic
229C4⋊C4C22×D16311yescharacteristic
230C4⋊C4C22×D16312yes—
231C22⋊C4—21no—
232C22⋊C4—3261no—
233C22⋊C4—3261no—
234C22⋊C4—3262no—
235C22⋊C4—3262no—
236C22⋊C4—3261no—
237C42—21no—
238C42—3261no—
239C42—3261no—
240C42—3261no—
241C23—3262no—
242C23—3262no—
243C23—3262no—
244C23—6522no—
245C23—6522no—
246C23—6521no—
247C23—6522no—
248C23—1631no—
249C23—1631no—
250C23D4×D16311yescharacteristic
251D4—3262no—
252D4—3262no—
253D4—6524no—
254D4—6524no—
255D4—6522no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—42no—
260C2×C4—3262no—
261C2×C4—3261no—
262C2×C4—3261no—
263C2×C4—3261no—
264C2×C4—3261no—
265C2×C4—3262no—
266C2×C4—3262no—
267C2×C4—3261no—
268C2×C4—3261no—
269C2×C4—6522no—
270C2×C4—6522no—
271C2×C4—1631no—
272C2×C4—1631no—
273C2×C4D4×D16311yescharacteristic
274C2×C4#1457969912yes—
275C2×C4#1457970012yes—
276C2×C4C23×D16311yescharacteristic
277C8—22no—
278C8—3262no—
279C22—21no—
280C22—21no—
281C22—13044no—
282C22—3261no—
283C22—3261no—
284C22—3261no—
285C22—3261no—
286C22—3261no—
287C22—6524no—
288C22—6522no—
289C22—6522no—
290C22—6522no—
291C22—1632no—
292C22#1910472012yes—
293C22C2×D4×D16311yescharacteristic
294C4—42no—
295C4—41no—
296C4—41no—
297C4—3261no—
298C4—3261no—
299C4—6521no—
300C4—6521no—
301C4#1910472012yes—
302C4C2×#1457969911yescharacteristic
303C4C2×#1457970011yescharacteristic
304C2—22no—
305C2—13042no—
306C2—3262no—
307C2—6521no—
308C2C2×#1910472011yescharacteristic
309C2AGL(1,ℤ/8)×D16311yescharacteristic
310C2#2378246411yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#428×C163C2
socleC22×C163C22×D4
prime radicalC2×#19104710C2
hypercenter#428D163
hypocenterC163#3094
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
2. #117×D163C2semidirect
3. #23781992C2semidirect
4. #23781991C2semidirect
5. #23781985C2semidirect
6. #23781980C2semidirect
7. #23781705C2semidirect
8. #23781671C2semidirect
9. #23781669C2semidirect
10. #23781628C2semidirect
11. #428×C163C2semidirect
21. #19104693C22semidirect
22. #19104693C22semidirect
25. C4⋊C4×D163C22semidirect
26. C4⋊C4×D163C22semidirect
30. #19104668C22semidirect
31. #19104666C22semidirect
32. #19104657C22semidirect
33. #19104657C22semidirect
34. #19104655C22semidirect
35. #19104655C22semidirect
36. (C2×D163)⋊C8C22semidirect
37. (C2×D163)⋊C8C22semidirect
41. #107×C163C22semidirect
42. #106×C163C22semidirect
155. C163#3094semidirect
164. #428D163semidirect
190. #107C2×D163semidirect
192. #106C2×D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047102
#117×D1632
#237819922
#237819912
#237819852
#237819802
#237817052
#237816712
#237816692
#237816282
#428×C1632
#3094163