Finite Group Databasebeta

← #35226330

Subgroups — #35226330

Order 20864.

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

normal subgroups only

This group has 460 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#23782477C212yesmaximal, complemented
2#23782472C212yesmaximal, complemented
3#23782038C211yesmaximal, characteristic, complemented
4#23781929C212yesmaximal, complemented
5#23781928C212yesmaximal, complemented
6#23781849C212yesmaximal, complemented
7#23781834C212yesmaximal, complemented
8#23781783C211yesmaximal, characteristic, complemented
9#443×C163C211yesmaximal, characteristic, complemented
10#19104782C2212yes—
11#19104781C2212yescomplemented
12#19104768—22no—
13#19104768C2212yescomplemented
14#19104767—22no—
15#19104765C2212yescomplemented
16#19104763—22no—
17#19104763C2212yes—
18#19104760C2212yes—
19#19104759—22no—
20#19104757—22no—
21#19104757C2211yescharacteristic
22#19104756—22no—
23C2×#14579688—22no—
24C2×#14579688C2212yescomplemented
25C2×#14579688C2212yescomplemented
26SD16×D163—22no—
27#19104741C2212yescomplemented
28#19104735C2211yescharacteristic, complemented
29#19104732C2211yescharacteristic
30#19104671C2211yescharacteristic, complemented
31#19104663C2212yescomplemented
32#19104653C2212yescomplemented
33#19104651C2212yescomplemented
34#135×C163C2211yescharacteristic
35#131×C163C2212yescomplemented
36#108×C163C2211yescharacteristic, complemented
37#104×C163C2212yescomplemented
38C4×C8×C163C2211yescharacteristic, complemented
39#14579700—22no—
40#14579699—22no—
41#14579699—22no—
42#14579699C2312yes—
43#14579698—22no—
44#14579698—42no—
45C2×Dic326—22no—
46C2×Dic326C2312yes—
47Q8×D163—22no—
48C2×D652—22no—
49C2×D652—41no—
50C2×D652C2312yes—
51D4×D163—22no—
52D4×D163—42no—
53Dic652—22no—
54Dic652D411yescharacteristic, complemented
55#14579689—22no—
56#14579688—22no—
57#14579688—22no—
58#14579688—22no—
59#14579688—22no—
60#14579688—22no—
61#14579688—22no—
62#14579688D412yescomplemented
63#14579687—22no—
64#14579686—22no—
65D1304—22no—
66D1304D411yescharacteristic, complemented
67#14579684—22no—
68#14579683C2312yes—
69#14579682—22no—
70C8×D163—22no—
71C4⋊Dic163—41no—
72#41×C163C2312yes—
73Q16×C163D411yescharacteristic, complemented
74SD16×C163D412yescomplemented
75D8×C163D411yescharacteristic, complemented
76M16×C163—22no—
77M16×C163C2312yes—
78C2×C8×C163—22no—
79C2×C8×C163C2311yescharacteristic
80C2×C8×C163C2311yescharacteristic
81C42×C163C2311yescharacteristic
82C22×D163—22no—
83C22×D163—42no—
84Dic326—22no—
85Dic326—22no—
86Dic326—42no—
87Dic326C2×D412yes—
88D652—22no—
89D652—22no—
90D652—42no—
91D652—82no—
92D652C2×D412yes—
93C22⋊D163—22no—
94C22⋊D163—42no—
95C22⋊D163—42no—
96C2×Dic163—22no—
97C2×Dic163—42no—
98C4×D163—22no—
99C4×D163—42no—
100C4×D163—42no—
101C163⋊C8—22no—
102Q8×C163C2×D412yes—
103D4×C163—22no—
104D4×C163C2×D412yes—
105C2×C4×C163—21no—
106C2×C4×C163—22no—
107C2×C4×C163C2411yescharacteristic
108C8×C163—22no—
109C8×C163C2×D411yescharacteristic
110C8×C163C2×D411yescharacteristic
111C8×C163C2×D411yescharacteristic
112C8×C163C2×D411yescharacteristic
113C22×C163—22no—
114C22×C163H(5,𝔽2)11yescharacteristic
115C2×D163—22no—
116C2×D163—42no—
117C2×D163—82no—
118C2×D163—162no—
119C4×C163—22no—
120C4×C163—22no—
121C4×C163C22×D411yescharacteristic
122C4×C163C22×D411yescharacteristic
123Dic163—22no—
124Dic163—42no—
125C2×C163—21no—
126C2×C163—42no—
127C2×C163D4211yescharacteristic
128D163—42no—
129D163—82no—
130C163#329111yescharacteristic, complemented, Sylow
131#3291—1631nomaximal, Sylow
132#577—1632no—
133#577—1632no—
134#492—1631no—
135#471—1632no—
136#471—1632no—
137#456—1632no—
138#453—1632no—
139#443—1631no—
140#443D16311yescharacteristic, complemented
1412−1+4—1632no—
142H(5,𝔽2)—1632no—
143#137—3262no—
144#137—3262no—
145#137—1632no—
146#137—1632no—
147AGL(1,ℤ/8)—3262no—
148AGL(1,ℤ/8)—3262no—
149AGL(1,ℤ/8)—1632no—
150AGL(1,ℤ/8)—1632no—
151#135—3262no—
152#135—3262no—
153#135—1631no—
154#135C2×D16311yescharacteristic
155C2×SD16—3262no—
156C2×SD16—3262no—
157C2×SD16—1632no—
158C2×SD16—1632no—
159#131—1632no—
160#131C2×D16312yescomplemented
161C4⋊Q8—1631no—
162#127—1631no—
163#108—1631no—
164#108C2×D16311yescharacteristic, complemented
165#104—1632no—
166#104C2×D16312yescomplemented
167#101—1632no—
168#100—1632no—
169C4×C8C2×D16311yescharacteristic, complemented
170#41—3262no—
171#41—3262no—
172#41—3262no—
173#41—3262no—
174#41—6522no—
175#41—1632no—
176#41C22×D16312yes—
177C2×Q8—3262no—
178C2×Q8—3262no—
179C2×Q8—1632no—
180C2×D4—3262no—
181C2×D4—3262no—
182C2×D4—6521no—
183C2×D4—6522no—
184C2×D4—1632no—
185Q16—3262no—
186Q16—3262no—
187Q16—1631no—
188Q16D65211yescharacteristic, complemented
189SD16—3262no—
190SD16—3262no—
191SD16—3262no—
192SD16—3262no—
193SD16—3262no—
194SD16—3262no—
195SD16—3262no—
196SD16—3262no—
197SD16—1632no—
198SD16D65212yescomplemented
199D8—3262no—
200D8—3262no—
201D8—1631no—
202D8D65211yescharacteristic, complemented
203M16—22no—
204M16—3262no—
205M16—1632no—
206M16C22×D16312yes—
207C2×C8—22no—
208C2×C8—3262no—
209C2×C8C22×D16311yescharacteristic
210C2×C8C22×D16311yescharacteristic
211C4⋊C4—6521no—
212C42C22×D16311yescharacteristic
213C23—3262no—
214C23—6522no—
215Q8—3262no—
216Q8—3262no—
217Q8—6522no—
218Q8—1632no—
219Q8C2×D65212yes—
220D4—22no—
221D4—13042no—
222D4—3262no—
223D4—3262no—
224D4—3262no—
225D4—6522no—
226D4—6522no—
227D4—6522no—
228D4—1632no—
229D4C2×D65212yes—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—3262no—
233C2×C4—3262no—
234C2×C4—6522no—
235C2×C4—6522no—
236C2×C4—6522no—
237C2×C4C23×D16311yescharacteristic
238C8—22no—
239C8—3262no—
240C8D4×D16311yescharacteristic
241C8D4×D16311yescharacteristic
242C8C2×D65211yescharacteristic
243C8C2×D65211yescharacteristic
244C22—22no—
245C22—13042no—
246C22—26082no—
247C22—3262no—
248C22—6522no—
249C22#1910478111yescharacteristic
250C4—22no—
251C4—22no—
252C4—3262no—
253C4—6522no—
254C4C2×D4×D16311yescharacteristic
255C4C22×D65211yescharacteristic
256C2—21no—
257C2—42no—
258C2—13042no—
259C2—6522no—
260C2D4×D65211yescharacteristic
subgroupG/H
centerC2D4×D652
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#443×C163C2
socleC2×C163D42
hypercenter#443D163
hypocenterC163#3291
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C2 ◁ C2×C4 ◁ #443

Complements

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

NH ≅ G/Ntype
1. #23782477C2semidirect
2. #23782472C2semidirect
3. #23782038C2semidirect
4. #23781929C2semidirect
5. #23781928C2semidirect
6. #23781849C2semidirect
7. #23781834C2semidirect
8. #23781783C2semidirect
9. #443×C163C2semidirect
11. #19104781C22semidirect
13. #19104768C22semidirect
15. #19104765C22semidirect
24. C2×#14579688C22semidirect
25. C2×#14579688C22semidirect
27. #19104741C22semidirect
28. #19104735C22semidirect
30. #19104671C22semidirect
31. #19104663C22semidirect
32. #19104653C22semidirect
33. #19104651C22semidirect
35. #131×C163C22semidirect
36. #108×C163C22semidirect
37. #104×C163C22semidirect
38. C4×C8×C163C22semidirect
54. Dic652D4semidirect
62. #14579688D4semidirect
66. D1304D4semidirect
73. Q16×C163D4semidirect
74. SD16×C163D4semidirect
75. D8×C163D4semidirect
130. C163#3291semidirect
140. #443D163semidirect
160. #131C2×D163semidirect
164. #108C2×D163semidirect
166. #104C2×D163semidirect
169. C4×C8C2×D163semidirect
188. Q16D652semidirect
198. SD16D652semidirect
202. D8D652semidirect

Maximal subgroups

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

subgroupindex
#237824772
#237824722
#237820382
#237819292
#237819282
#237818492
#237818342
#237817832
#443×C1632
#3291163