Finite Group Databasebeta

← #509×D101 #27184252

Subgroups — #509×D101 #27184252

Order 12928.

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×D101C211yesmaximal, characteristic, complemented
2C2×D8×D101C211yesmaximal, characteristic, complemented
3#20265615C211yesmaximal, characteristic, complemented
4#20265614C212yesmaximal, complemented
5#20265613C211yesmaximal, characteristic, complemented
6#20265612C211yesmaximal, characteristic, complemented
7#20265611C212yesmaximal, complemented
8SD32×D101C212yesmaximal, complemented
9D16×D101C212yesmaximal, complemented
10M32×D101C211yesmaximal, characteristic, complemented
11#509×C101C211yesmaximal, characteristic, complemented
12#41×D101—21no—
13C2×D4×D101—21no—
14#15692278C2211yescharacteristic, complemented
15#15692277C2211yescharacteristic, complemented
16#15692276—21no—
17#15692275—21no—
18#15692274C2211yescharacteristic, complemented
19Q16×D101C2211yescharacteristic, complemented
20SD16×D101—21no—
21C2×D808C2211yescharacteristic, complemented
22C2×#13108059—21no—
23D8×D101—21no—
24D8×D101—21no—
25D8×D101C2212yes—
26D8×D101C2212yescomplemented
27D8×D101C2211yescharacteristic, complemented
28C2×C8×D101C2211yescharacteristic
29#15692200C2212yescomplemented
30#15692199C2212yescomplemented
31#15692198C2212yescomplemented
32D1616C2212yescomplemented
33#15692196C2212yescomplemented
34#15692196C2212yescomplemented
35#15692195C2211yescharacteristic, complemented
36#15692194C2212yescomplemented
37C16×D101C2212yescomplemented
38#135×C101C2211yescharacteristic, complemented
39C2×D8×C101C2211yescharacteristic, complemented
40SD32×C101C2212yescomplemented
41D16×C101C2212yescomplemented
42M32×C101C2211yescharacteristic, complemented
43C23×D101—41no—
44#13108075—21no—
45#13108074—21no—
46#13108074—41no—
47#13108073—21no—
48#13108073—41no—
49Q8×D101—21no—
50C2×D404—21no—
51C2×C22⋊D101—41no—
52D4×D101—22no—
53D4×D101—22no—
54D4×D101—21no—
55D4×D101—41no—
56D4×D101—41no—
57D4×D101—41no—
58C2×C4×D101—41no—
59C2×C4×D101D411yescharacteristic
60Dic404C2311yescharacteristic, complemented
61#13108064—21no—
62#13108063—21no—
63#13108062—21no—
64#13108061—21no—
65D808—21no—
66D808C2312yes—
67D808C2311yescharacteristic, complemented
68#13108059—22no—
69#13108059—22no—
70#13108059—21no—
71C2×#11717579D411yescharacteristic
72C8×D101D411yescharacteristic
73C8×D101D411yescharacteristic
74C8×D101C2311yescharacteristic
75C8×D101C2311yescharacteristic
76#13108049C2312yescomplemented
77#41×C101—21no—
78C2×D4×C101—21no—
79Q16×C101C2311yescharacteristic, complemented
80SD16×C101—21no—
81D8×C101—21no—
82D8×C101C2312yes—
83D8×C101C2311yescharacteristic, complemented
84C2×C8×C101C2311yescharacteristic
85C16×C101C2312yescomplemented
86C23×C101—41no—
87C22×D101—41no—
88C22×D101—41no—
89C22×D101—42no—
90C22×D101—42no—
91C22×D101—84no—
92C22×D101D811yescharacteristic
93Q8×C101—21no—
94D4×C101—22no—
95D4×C101—21no—
96D4×C101—41no—
97D4×C101—41no—
98C2×C4×C101—41no—
99C2×C4×C101C2×D411yescharacteristic
100C22⋊D101—41no—
101C22⋊D101—41no—
102C22⋊D101—42no—
103C22⋊D101—42no—
104C22⋊D101—41no—
105C2×Dic101—41no—
106C2×Dic101D811yescharacteristic
107D404—22no—
108D404—21no—
109D404—41no—
110D404—41no—
111C4×D101—41no—
112C4×D101—41no—
113C4×D101—41no—
114C4×D101D811yescharacteristic
115C4×D101D811yescharacteristic
116C4×D101C2×D411yescharacteristic
117C4×D101C2×D411yescharacteristic
118Dic202—21no—
119Dic202—41no—
120C8×C101C2×D411yescharacteristic
121C8×C101C2411yescharacteristic
122#11717579C2×D411yescharacteristic
123#11717579C2×D411yescharacteristic
124C22×C101—41no—
125C22×C101—42no—
126C22×C101—82no—
127C22×C101C2×D811yescharacteristic
128C2×D101—22no—
129C2×D101—41no—
130C2×D101—42no—
131C2×D101—84no—
132C2×D101—82no—
133C2×D101—82no—
134C2×D101—84no—
135C2×D101C2×D811yescharacteristic
136C2×D101C2×D811yescharacteristic
137C4×C101—41no—
138C4×C101C2×D811yescharacteristic
139C4×C101C22×D411yescharacteristic
140Dic101—41no—
141Dic101C2×D811yescharacteristic
142Dic101C2×D811yescharacteristic
143C2×C101—21no—
144C2×C101—82no—
145C2×C101—81no—
146C2×C101C22×D811yescharacteristic
147D101—21no—
148D101—81no—
149D101—82no—
150D101#50912yesdirect product factor
151C2×#509—1011nomaximal, Sylow
152C101C2×#50911yescharacteristic, complemented, Sylow
153C2×#135—1011no—
154C22×D8—1011no—
155#509—1011no—
156#509—1011no—
157#509—1012no—
158#509—1012no—
159#509—1011no—
160#509D10111yescharacteristic, direct product factor
161C2×SD32—1012no—
162C2×D16—1012no—
163C2×M32—1011no—
164C2×#41—2021no—
165C22×D4—2021no—
166#135—1011no—
167#135—1011no—
168#135—1011no—
169#135—2021no—
170#135—2021no—
171#135C2×D10111yescharacteristic, complemented
172C2×Q16—1011no—
173C2×SD16—2021no—
174C2×D8—1012no—
175C2×D8—1012no—
176C2×D8—1011no—
177C2×D8—1011no—
178C2×D8—2021no—
179C2×D8—2021no—
180C2×D8—2021no—
181C2×D8C2×D10111yescharacteristic, complemented
182C22×C8—1011no—
183SD32—1012no—
184SD32—1012no—
185SD32—1012no—
186SD32C2×D10112yescomplemented
187D16—1012no—
188D16—1012no—
189D16—1012no—
190D16C2×D10112yescomplemented
191M32—1011no—
192M32—1012no—
193M32C2×D10111yescharacteristic, complemented
194C2×C16—1012no—
195C24—4041no—
196#41—21no—
197#41—4041no—
198#41—4041no—
199#41—2021no—
200#41—2021no—
201#41—2021no—
202C2×Q8—2021no—
203C2×D4—21no—
204C2×D4—4041no—
205C2×D4—4041no—
206C2×D4—4041no—
207C2×D4—4041no—
208C2×D4—2022no—
209C2×D4—2022no—
210C2×D4—2021no—
211C2×D4—2021no—
212C22×C4—1011no—
213C22×C4—4041no—
214Q16—1011no—
215Q16—2021no—
216Q16C22×D10111yescharacteristic, complemented
217SD16—21no—
218SD16—2021no—
219SD16—2021no—
220SD16—2021no—
221D8—21no—
222D8—1012no—
223D8—1011no—
224D8—2022no—
225D8—2022no—
226D8—2021no—
227D8—2021no—
228D8C22×D10112yes—
229D8C22×D10111yescharacteristic, complemented
230C2×C8—1011no—
231C2×C8—1011no—
232C2×C8—1011no—
233C2×C8—1011no—
234C2×C8—1011no—
235C2×C8C22×D10111yescharacteristic
236C16—1012no—
237C16C22×D10112yescomplemented
238C23—41no—
239C23—8084no—
240C23—1011no—
241C23—4041no—
242C23—4042no—
243C23—4042no—
244C23—4041no—
245Q8—21no—
246Q8—4041no—
247Q8—2021no—
248D4—22no—
249D4—21no—
250D4—41no—
251D4—41no—
252D4—4041no—
253D4—4042no—
254D4—4042no—
255D4—4041no—
256D4—4041no—
257D4—4041no—
258D4—4041no—
259D4—2022no—
260D4—2021no—
261C2×C4—41no—
262C2×C4—1011no—
263C2×C4—1011no—
264C2×C4—1011no—
265C2×C4—1011no—
266C2×C4—1011no—
267C2×C4—4041no—
268C2×C4—4041no—
269C2×C4—4041no—
270C2×C4—4041no—
271C2×C4D4×D10111yescharacteristic
272C8—1011no—
273C8—1011no—
274C8D4×D10111yescharacteristic
275C8C23×D10111yescharacteristic
276C22—41no—
277C22—42no—
278C22—82no—
279C22—8082no—
280C22—8084no—
281C22—8082no—
282C22—8084no—
283C22—1011no—
284C22—1011no—
285C22—4041no—
286C22—4042no—
287C22—2022no—
288C22D8×D10111yescharacteristic
289C4—41no—
290C4—1011no—
291C4—1011no—
292C4—4041no—
293C4D8×D10111yescharacteristic
294C4C2×D4×D10111yescharacteristic
295C2—21no—
296C2—82no—
297C2—81no—
298C2—8082no—
299C2—8081no—
300C2—1012no—
301C2—2021no—
302C2C2×D8×D10111yescharacteristic
subgroupG/H
centerC2C2×D8×D101
commutator, cosocleC8×C101C24
FrattiniC8C23×D101
Fitting#509×C101C2
socleC2×C101C22×D8
hypercenter#509D101
hypocenterC101C2×#509
LCSG ▷ C8×C101 ▷ C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #135×D101C2semidirect
2. C2×D8×D101C2semidirect
3. #20265615C2semidirect
4. #20265614C2semidirect
5. #20265613C2semidirect
6. #20265612C2semidirect
7. #20265611C2semidirect
8. SD32×D101C2semidirect
9. D16×D101C2semidirect
10. M32×D101C2semidirect
11. #509×C101C2semidirect
14. #15692278C22semidirect
15. #15692277C22semidirect
18. #15692274C22semidirect
19. Q16×D101C22semidirect
21. C2×D808C22semidirect
26. D8×D101C22semidirect
27. D8×D101C22semidirect
29. #15692200C22semidirect
30. #15692199C22semidirect
31. #15692198C22semidirect
32. D1616C22semidirect
33. #15692196C22semidirect
34. #15692196C22semidirect
35. #15692195C22semidirect
36. #15692194C22semidirect
37. C16×D101C22semidirect
38. #135×C101C22semidirect
39. C2×D8×C101C22semidirect
40. SD32×C101C22semidirect
41. D16×C101C22semidirect
42. M32×C101C22semidirect
60. Dic404C23semidirect
67. D808C23semidirect
76. #13108049C23semidirect
79. Q16×C101C23semidirect
83. D8×C101C23semidirect
85. C16×C101C23semidirect
150. D101#509direct (N × H)
152. C101C2×#509semidirect
160. #509D101direct (N × H)
171. #135C2×D101semidirect
181. C2×D8C2×D101semidirect
186. SD32C2×D101semidirect
190. D16C2×D101semidirect
193. M32C2×D101semidirect
216. Q16C22×D101semidirect
229. D8C22×D101semidirect
237. C16C22×D101semidirect

Maximal subgroups

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

subgroupindex
#135×D1012
C2×D8×D1012
#202656152
#202656142
#202656132
#202656122
#202656112
SD32×D1012
D16×D1012
M32×D1012
#509×C1012
C2×#509101