Finite Group Databasebeta

← #509×D163 #35227656

Subgroups — #509×D163 #35227656

Order 20864.

400 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 115 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#135×D163C211yesmaximal, characteristic, complemented
2C2×D8×D163C211yesmaximal, characteristic, complemented
3#23782103C211yesmaximal, characteristic, complemented
4#23782102C212yesmaximal, complemented
5#23782101C211yesmaximal, characteristic, complemented
6#23782100C211yesmaximal, characteristic, complemented
7#23782099C212yesmaximal, complemented
8SD32×D163C212yesmaximal, complemented
9D16×D163C212yesmaximal, complemented
10M32×D163C211yesmaximal, characteristic, complemented
11#509×C163C211yesmaximal, characteristic, complemented
12#41×D163—21no—
13C2×D4×D163—21no—
14#19104758C2211yescharacteristic, complemented
15#19104757C2211yescharacteristic, complemented
16#19104756—21no—
17#19104755—21no—
18#19104754C2211yescharacteristic, complemented
19Q16×D163C2211yescharacteristic, complemented
20SD16×D163—21no—
21C2×D1304C2211yescharacteristic, complemented
22C2×#14579684—21no—
23D8×D163—21no—
24D8×D163—21no—
25D8×D163C2212yes—
26D8×D163C2212yescomplemented
27D8×D163C2211yescharacteristic, complemented
28C2×C8×D163C2211yescharacteristic
29#19104680C2212yescomplemented
30#19104679C2212yescomplemented
31#19104678C2212yescomplemented
32D2608C2212yescomplemented
33#19104676C2212yescomplemented
34#19104676C2212yescomplemented
35#19104675C2211yescharacteristic, complemented
36#19104674C2212yescomplemented
37C16×D163C2212yescomplemented
38#135×C163C2211yescharacteristic, complemented
39C2×D8×C163C2211yescharacteristic, complemented
40SD32×C163C2212yescomplemented
41D16×C163C2212yescomplemented
42M32×C163C2211yescharacteristic, complemented
43C23×D163—41no—
44#14579700—21no—
45#14579699—21no—
46#14579699—41no—
47#14579698—21no—
48#14579698—41no—
49Q8×D163—21no—
50C2×D652—21no—
51C2×C22⋊D163—41no—
52D4×D163—22no—
53D4×D163—22no—
54D4×D163—21no—
55D4×D163—41no—
56D4×D163—41no—
57D4×D163—41no—
58C2×C4×D163—41no—
59C2×C4×D163D411yescharacteristic
60Dic652C2311yescharacteristic, complemented
61#14579689—21no—
62#14579688—21no—
63#14579687—21no—
64#14579686—21no—
65D1304—21no—
66D1304C2312yes—
67D1304C2311yescharacteristic, complemented
68#14579684—22no—
69#14579684—22no—
70#14579684—21no—
71C2×C163⋊C8D411yescharacteristic
72C8×D163D411yescharacteristic
73C8×D163D411yescharacteristic
74C8×D163C2311yescharacteristic
75C8×D163C2311yescharacteristic
76C163⋊C16C2312yescomplemented
77#41×C163—21no—
78C2×D4×C163—21no—
79Q16×C163C2311yescharacteristic, complemented
80SD16×C163—21no—
81D8×C163—21no—
82D8×C163C2312yes—
83D8×C163C2311yescharacteristic, complemented
84C2×C8×C163C2311yescharacteristic
85C16×C163C2312yescomplemented
86C22×D163—41no—
87C22×D163—41no—
88C22×D163—42no—
89C22×D163—42no—
90C22×D163—84no—
91C22×D163D811yescharacteristic
92Dic326—21no—
93Dic326—41no—
94D652—22no—
95D652—21no—
96D652—41no—
97D652—41no—
98C22⋊D163—41no—
99C22⋊D163—41no—
100C22⋊D163—42no—
101C22⋊D163—42no—
102C22⋊D163—41no—
103C2×Dic163—41no—
104C2×Dic163D811yescharacteristic
105C4×D163—41no—
106C4×D163—41no—
107C4×D163—41no—
108C4×D163D811yescharacteristic
109C4×D163D811yescharacteristic
110C4×D163C2×D411yescharacteristic
111C4×D163C2×D411yescharacteristic
112C163⋊C8C2×D411yescharacteristic
113C163⋊C8C2×D411yescharacteristic
114C23×C163—41no—
115Q8×C163—21no—
116D4×C163—22no—
117D4×C163—21no—
118D4×C163—41no—
119D4×C163—41no—
120C2×C4×C163—41no—
121C2×C4×C163C2×D411yescharacteristic
122C8×C163C2×D411yescharacteristic
123C8×C163C2411yescharacteristic
124C22×C163—41no—
125C22×C163—42no—
126C22×C163—82no—
127C22×C163C2×D811yescharacteristic
128C2×D163—22no—
129C2×D163—41no—
130C2×D163—42no—
131C2×D163—84no—
132C2×D163—82no—
133C2×D163—82no—
134C2×D163—84no—
135C2×D163C2×D811yescharacteristic
136C2×D163C2×D811yescharacteristic
137C4×C163—41no—
138C4×C163C2×D811yescharacteristic
139C4×C163C22×D411yescharacteristic
140Dic163—41no—
141Dic163C2×D811yescharacteristic
142Dic163C2×D811yescharacteristic
143C2×C163—21no—
144C2×C163—82no—
145C2×C163—81no—
146C2×C163C22×D811yescharacteristic
147D163—21no—
148D163—81no—
149D163—82no—
150D163#50912yesdirect product factor
151C163C2×#50911yescharacteristic, complemented, Sylow
152C2×#509—1631nomaximal, Sylow
153C2×#135—1631no—
154C22×D8—1631no—
155#509—1631no—
156#509—1631no—
157#509—1632no—
158#509—1632no—
159#509—1631no—
160#509D16311yescharacteristic, direct product factor
161C2×SD32—1632no—
162C2×D16—1632no—
163C2×M32—1631no—
164C2×#41—3261no—
165C22×D4—3261no—
166#135—3261no—
167#135—3261no—
168#135—1631no—
169#135—1631no—
170#135—1631no—
171#135C2×D16311yescharacteristic, complemented
172C2×Q16—1631no—
173C2×SD16—3261no—
174C2×D8—3261no—
175C2×D8—3261no—
176C2×D8—3261no—
177C2×D8—1632no—
178C2×D8—1632no—
179C2×D8—1631no—
180C2×D8—1631no—
181C2×D8C2×D16311yescharacteristic, complemented
182C22×C8—1631no—
183SD32—1632no—
184SD32—1632no—
185SD32—1632no—
186SD32C2×D16312yescomplemented
187D16—1632no—
188D16—1632no—
189D16—1632no—
190D16C2×D16312yescomplemented
191M32—1631no—
192M32—1632no—
193M32C2×D16311yescharacteristic, complemented
194C2×C16—1632no—
195C24—6521no—
196#41—21no—
197#41—3261no—
198#41—3261no—
199#41—3261no—
200#41—6521no—
201#41—6521no—
202C2×Q8—3261no—
203C2×D4—21no—
204C2×D4—3262no—
205C2×D4—3262no—
206C2×D4—3261no—
207C2×D4—3261no—
208C2×D4—6521no—
209C2×D4—6521no—
210C2×D4—6521no—
211C2×D4—6521no—
212C22×C4—6521no—
213C22×C4—1631no—
214Q16—3261no—
215Q16—1631no—
216Q16C22×D16311yescharacteristic, complemented
217SD16—21no—
218SD16—3261no—
219SD16—3261no—
220SD16—3261no—
221D8—21no—
222D8—3262no—
223D8—3262no—
224D8—3261no—
225D8—3261no—
226D8—1631no—
227D8—1632no—
228D8C22×D16312yes—
229D8C22×D16311yescharacteristic, complemented
230C2×C8—1631no—
231C2×C8—1631no—
232C2×C8—1631no—
233C2×C8—1631no—
234C2×C8—1631no—
235C2×C8C22×D16311yescharacteristic
236C16—1632no—
237C16C22×D16312yescomplemented
238C23—41no—
239C23—13044no—
240C23—6521no—
241C23—6522no—
242C23—6522no—
243C23—6521no—
244C23—1631no—
245Q8—21no—
246Q8—3261no—
247Q8—6521no—
248D4—22no—
249D4—21no—
250D4—41no—
251D4—41no—
252D4—3262no—
253D4—3261no—
254D4—6521no—
255D4—6521no—
256D4—6521no—
257D4—6522no—
258D4—6522no—
259D4—6521no—
260D4—6521no—
261C2×C4—41no—
262C2×C4—6521no—
263C2×C4—6521no—
264C2×C4—6521no—
265C2×C4—6521no—
266C2×C4—1631no—
267C2×C4—1631no—
268C2×C4—1631no—
269C2×C4—1631no—
270C2×C4—1631no—
271C2×C4D4×D16311yescharacteristic
272C8—1631no—
273C8—1631no—
274C8D4×D16311yescharacteristic
275C8C23×D16311yescharacteristic
276C22—41no—
277C22—42no—
278C22—82no—
279C22—13042no—
280C22—13044no—
281C22—13044no—
282C22—13042no—
283C22—3262no—
284C22—6521no—
285C22—6522no—
286C22—1631no—
287C22—1631no—
288C22D8×D16311yescharacteristic
289C4—41no—
290C4—6521no—
291C4—1631no—
292C4—1631no—
293C4D8×D16311yescharacteristic
294C4C2×D4×D16311yescharacteristic
295C2—21no—
296C2—82no—
297C2—81no—
298C2—13042no—
299C2—13041no—
300C2—3261no—
301C2—1632no—
302C2C2×D8×D16311yescharacteristic
subgroupG/H
centerC2C2×D8×D163
commutator, cosocleC8×C163C24
FrattiniC8C23×D163
Fitting#509×C163C2
socleC2×C163C22×D8
hypercenter#509D163
hypocenterC163C2×#509
LCSG ▷ C8×C163 ▷ C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #135×D163C2semidirect
2. C2×D8×D163C2semidirect
3. #23782103C2semidirect
4. #23782102C2semidirect
5. #23782101C2semidirect
6. #23782100C2semidirect
7. #23782099C2semidirect
8. SD32×D163C2semidirect
9. D16×D163C2semidirect
10. M32×D163C2semidirect
11. #509×C163C2semidirect
14. #19104758C22semidirect
15. #19104757C22semidirect
18. #19104754C22semidirect
19. Q16×D163C22semidirect
21. C2×D1304C22semidirect
26. D8×D163C22semidirect
27. D8×D163C22semidirect
29. #19104680C22semidirect
30. #19104679C22semidirect
31. #19104678C22semidirect
32. D2608C22semidirect
33. #19104676C22semidirect
34. #19104676C22semidirect
35. #19104675C22semidirect
36. #19104674C22semidirect
37. C16×D163C22semidirect
38. #135×C163C22semidirect
39. C2×D8×C163C22semidirect
40. SD32×C163C22semidirect
41. D16×C163C22semidirect
42. M32×C163C22semidirect
60. Dic652C23semidirect
67. D1304C23semidirect
76. C163⋊C16C23semidirect
79. Q16×C163C23semidirect
83. D8×C163C23semidirect
85. C16×C163C23semidirect
150. D163#509direct (N × H)
151. C163C2×#509semidirect
160. #509D163direct (N × H)
171. #135C2×D163semidirect
181. C2×D8C2×D163semidirect
186. SD32C2×D163semidirect
190. D16C2×D163semidirect
193. M32C2×D163semidirect
216. Q16C22×D163semidirect
229. D8C22×D163semidirect
237. C16C22×D163semidirect

Maximal subgroups

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

subgroupindex
#135×D1632
C2×D8×D1632
#237821032
#237821022
#237821012
#237821002
#237820992
SD32×D1632
D16×D1632
M32×D1632
#509×C1632
C2×#509163