Finite Group Databasebeta

← #35227681

Subgroups — #35227681

Order 20864.

360 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 113 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#23782478C211yesmaximal, characteristic, complemented
2#23782467C211yesmaximal, characteristic, complemented
3#23782107C212yesmaximal, complemented
4#23782103C211yesmaximal, characteristic, complemented
5#23782102C212yesmaximal, complemented
6#23782098C212yesmaximal, complemented
7#23782095C211yesmaximal, characteristic, complemented
8C2×#19104678C211yesmaximal, characteristic, complemented
9SD32×D163C212yesmaximal, complemented
10#23782083C211yesmaximal, characteristic, complemented
11#510×C163C211yesmaximal, characteristic, complemented
12#19104783—21no—
13#19104781—21no—
14#19104767—22no—
15#19104766—21no—
16#19104762—22no—
17#19104760—21no—
18#19104758C2212yes—
19#19104758C2212yescomplemented
20#19104757C2211yescharacteristic, complemented
21Q16×D163C2212yescomplemented
22C2×D1304C2211yescharacteristic
23D8×D163C2212yes—
24#19104741C2211yescharacteristic
25#19104681C2211yescharacteristic, complemented
26#19104680C2212yescomplemented
27#19104679C2211yescharacteristic, complemented
28#19104678C2212yescomplemented
29#19104678C2212yescomplemented
30#19104678C2211yescharacteristic, complemented
31D2608C2212yescomplemented
32#19104676C2211yescharacteristic, complemented
33#19104674C2212yescomplemented
34C2×C163⋊C16C2211yescharacteristic, complemented
35C16×D163C2212yescomplemented
36#135×C163C2211yescharacteristic
37C2×Q16×C163C2211yescharacteristic, complemented
38Q32×C163C2212yescomplemented
39SD32×C163C2212yescomplemented
40M32×C163C2211yescharacteristic, complemented
41#14579700—22no—
42#14579700—22no—
43#14579700—41no—
44#14579699—21no—
45#14579699—41no—
46#14579699—42no—
47#14579699D411yescharacteristic
48Q8×D163—22no—
49Q8×D163—41no—
50C2×D652—21no—
51C2×D652—41no—
52D4×D163—22no—
53D4×D163—42no—
54Dic652C2311yescharacteristic
55#14579689—22no—
56#14579688—21no—
57#14579686—22no—
58#14579686—22no—
59D1304—21no—
60D1304C2311yescharacteristic
61D1304C2312yes—
62#14579684—22no—
63#14579683D411yescharacteristic
64#14579682D412yes—
65C8×D163C2312yes—
66C163⋊C16C2311yescharacteristic
67C163⋊C16C2311yescharacteristic
68#41×C163—21no—
69C2×Q8×C163—21no—
70Q16×C163—21no—
71Q16×C163C2312yes—
72Q16×C163C2311yescharacteristic
73SD16×C163—21no—
74D8×C163C2311yescharacteristic
75C2×C8×C163C2311yescharacteristic
76C16×C163C2312yes—
77C22×D163—41no—
78C22×D163—42no—
79Dic326—21no—
80Dic326—42no—
81Dic326D811yescharacteristic
82D652—22no—
83D652—21no—
84D652—42no—
85D652—42no—
86D652—41no—
87D652D811yescharacteristic
88C22⋊D163—41no—
89C22⋊D163—42no—
90C22⋊D163D812yes—
91C4×D163—42no—
92C4×D163—41no—
93C4×D163—42no—
94C4×D163—42no—
95C4×D163C2×D412yes—
96C163⋊C8C2×D412yes—
97Q8×C163—21no—
98Q8×C163—22no—
99Q8×C163—41no—
100D4×C163—21no—
101D4×C163—41no—
102C2×C4×C163—41no—
103C2×C4×C163—41no—
104C2×C4×C163C2×D411yescharacteristic
105C8×C163C2×D411yescharacteristic
106C8×C163C2411yescharacteristic
107C22×C163—41no—
108C22×C163C2×D811yescharacteristic
109C2×D163—41no—
110C2×D163—42no—
111C2×D163—162no—
112C2×D163—161no—
113C2×D163C2×D812yes—
114C4×C163—41no—
115C4×C163—42no—
116C4×C163C2×D811yescharacteristic
117C4×C163C22×D411yescharacteristic
118Dic163—41no—
119Dic163C2×D812yes—
120C2×C163—21no—
121C2×C163—81no—
122C2×C163C22×D811yescharacteristic
123D163—22no—
124D163—81no—
125D163—82no—
126C163#341611yescharacteristic, complemented, Sylow
127#3416—1631nomaximal, Sylow
128#578—1631no—
129#576—1631no—
130#510—1632no—
131#510D16311yescharacteristic, complemented
132#509—1632no—
133#509—1631no—
134#508—1631no—
135#508—1632no—
136C2×SD32—1631no—
137C2×SD32—1632no—
138#504—1631no—
1392−1+4—3261no—
140H(5,𝔽2)—3261no—
141#137—3262no—
142#137—3261no—
143AGL(1,ℤ/8)—3262no—
144AGL(1,ℤ/8)—3261no—
145#135—1632no—
146#135—1632no—
147#135—1631no—
148#135C2×D16311yescharacteristic
149C2×Q16—1632no—
150C2×Q16C2×D16311yescharacteristic, complemented
151C2×D8—1632no—
152C2×D8—1631no—
153#131—1631no—
154Q32—1631no—
155Q32C2×D16312yescomplemented
156SD32—1632no—
157SD32—1632no—
158SD32—1631no—
159SD32—1631no—
160SD32—1632no—
161SD32C2×D16312yescomplemented
162D16—1631no—
163D16—1632no—
164M32—1632no—
165M32C2×D16311yescharacteristic, complemented
166C2×C16—1631no—
167C2×C16—1632no—
168#41—21no—
169#41—3262no—
170#41—3262no—
171#41—3261no—
172#41—6521no—
173#41—6522no—
174#41—6521no—
175#41—1631no—
176C2×Q8—21no—
177C2×Q8—3262no—
178C2×Q8—6521no—
179C2×D4—3262no—
180C2×D4—3261no—
181C2×D4—6521no—
182C2×D4—6522no—
183Q16—21no—
184Q16—3262no—
185Q16—1631no—
186Q16C22×D16312yes—
187Q16C22×D16311yescharacteristic
188SD16—21no—
189SD16—3262no—
190SD16—3262no—
191SD16—3261no—
192D8—3262no—
193D8—3261no—
194D8—1631no—
195D8—1632no—
196D8C22×D16311yescharacteristic
197M16—1631no—
198M16—1632no—
199C2×C8—1632no—
200C2×C8C22×D16311yescharacteristic
201C16—1631no—
202C16—1631no—
203C16C22×D16312yes—
204C23—6522no—
205C23—6521no—
206Q8—21no—
207Q8—22no—
208Q8—41no—
209Q8—3261no—
210Q8—6522no—
211Q8—1631no—
212D4—21no—
213D4—41no—
214D4—3262no—
215D4—3261no—
216D4—6522no—
217D4—6522no—
218D4—6521no—
219D4—6522no—
220D4—6521no—
221D4—1632no—
222D4—1631no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—6522no—
226C2×C4—6522no—
227C2×C4—6522no—
228C2×C4—6521no—
229C2×C4—1632no—
230C2×C4D4×D16311yescharacteristic
231C8—1632no—
232C8D4×D16311yescharacteristic
233C8C23×D16311yescharacteristic
234C22—41no—
235C22—26081no—
236C22—26082no—
237C22—6521no—
238C22—6522no—
239C22—1632no—
240C22D8×D16311yescharacteristic
241C4—41no—
242C4—42no—
243C4—6521no—
244C4—1632no—
245C4D8×D16311yescharacteristic
246C4C2×D4×D16311yescharacteristic
247C2—21no—
248C2—81no—
249C2—13042no—
250C2—13041no—
251C2—3262no—
252C2C2×D8×D16311yescharacteristic
subgroupG/H
centerC2C2×D8×D163
commutator, cosocleC8×C163C24
FrattiniC8C23×D163
Fitting#510×C163C2
socleC2×C163C22×D8
hypercenter#510D163
hypocenterC163#3416
LCSG ▷ C8×C163 ▷ C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #510

Complements

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

NH ≅ G/Ntype
1. #23782478C2semidirect
2. #23782467C2semidirect
3. #23782107C2semidirect
4. #23782103C2semidirect
5. #23782102C2semidirect
6. #23782098C2semidirect
7. #23782095C2semidirect
8. C2×#19104678C2semidirect
9. SD32×D163C2semidirect
10. #23782083C2semidirect
11. #510×C163C2semidirect
19. #19104758C22semidirect
20. #19104757C22semidirect
21. Q16×D163C22semidirect
25. #19104681C22semidirect
26. #19104680C22semidirect
27. #19104679C22semidirect
28. #19104678C22semidirect
29. #19104678C22semidirect
30. #19104678C22semidirect
31. D2608C22semidirect
32. #19104676C22semidirect
33. #19104674C22semidirect
34. C2×C163⋊C16C22semidirect
35. C16×D163C22semidirect
37. C2×Q16×C163C22semidirect
38. Q32×C163C22semidirect
39. SD32×C163C22semidirect
40. M32×C163C22semidirect
126. C163#3416semidirect
131. #510D163semidirect
150. C2×Q16C2×D163semidirect
155. Q32C2×D163semidirect
161. SD32C2×D163semidirect
165. M32C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237824782
#237824672
#237821072
#237821032
#237821022
#237820982
#237820952
C2×#191046782
SD32×D1632
#237820832
#510×C1632
#3416163