Finite Group Databasebeta

← #509×D103 #27208264

Subgroups — #509×D103 #27208264

Order 13184.

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×D103C211yesmaximal, characteristic, complemented
2C2×D8×D103C211yesmaximal, characteristic, complemented
3#20268229C211yesmaximal, characteristic, complemented
4#20268228C212yesmaximal, complemented
5#20268227C211yesmaximal, characteristic, complemented
6#20268226C211yesmaximal, characteristic, complemented
7#20268225C212yesmaximal, complemented
8SD32×D103C212yesmaximal, complemented
9D16×D103C212yesmaximal, complemented
10M32×D103C211yesmaximal, characteristic, complemented
11#509×C103C211yesmaximal, characteristic, complemented
12#41×D103—21no—
13C2×D4×D103—21no—
14#15693044C2211yescharacteristic, complemented
15#15693043C2211yescharacteristic, complemented
16#15693042—21no—
17#15693041—21no—
18#15693040C2211yescharacteristic, complemented
19Q16×D103C2211yescharacteristic, complemented
20SD16×D103—21no—
21C2×D824C2211yescharacteristic, complemented
22C2×#13110090—21no—
23D8×D103—21no—
24D8×D103—21no—
25D8×D103C2212yes—
26D8×D103C2212yescomplemented
27D8×D103C2211yescharacteristic, complemented
28C2×C8×D103C2211yescharacteristic
29#15692966C2212yescomplemented
30#15692965C2212yescomplemented
31#15692964C2212yescomplemented
32D1648C2212yescomplemented
33#15692962C2212yescomplemented
34#15692962C2212yescomplemented
35#15692961C2211yescharacteristic, complemented
36#15692960C2212yescomplemented
37C16×D103C2212yescomplemented
38#135×C103C2211yescharacteristic, complemented
39C2×D8×C103C2211yescharacteristic, complemented
40SD32×C103C2212yescomplemented
41D16×C103C2212yescomplemented
42M32×C103C2211yescharacteristic, complemented
43C23×D103—41no—
44#13110106—21no—
45#13110105—21no—
46#13110105—41no—
47#13110104—21no—
48#13110104—41no—
49Q8×D103—21no—
50C2×D412—21no—
51C2×C22⋊D103—41no—
52D4×D103—22no—
53D4×D103—22no—
54D4×D103—21no—
55D4×D103—41no—
56D4×D103—41no—
57D4×D103—41no—
58C2×C4×D103—41no—
59C2×C4×D103D411yescharacteristic
60Dic412C2311yescharacteristic, complemented
61#13110095—21no—
62#13110094—21no—
63#13110093—21no—
64#13110092—21no—
65D824—21no—
66D824C2312yes—
67D824C2311yescharacteristic, complemented
68#13110090—22no—
69#13110090—22no—
70#13110090—21no—
71C2×C103⋊C8D411yescharacteristic
72C8×D103D411yescharacteristic
73C8×D103D411yescharacteristic
74C8×D103C2311yescharacteristic
75C8×D103C2311yescharacteristic
76C103⋊C16C2312yescomplemented
77#41×C103—21no—
78C2×D4×C103—21no—
79Q16×C103C2311yescharacteristic, complemented
80SD16×C103—21no—
81D8×C103—21no—
82D8×C103C2312yes—
83D8×C103C2311yescharacteristic, complemented
84C2×C8×C103C2311yescharacteristic
85C16×C103C2312yescomplemented
86C23×C103—41no—
87C22×D103—41no—
88C22×D103—41no—
89C22×D103—42no—
90C22×D103—42no—
91C22×D103—84no—
92C22×D103D811yescharacteristic
93Q8×C103—21no—
94D4×C103—22no—
95D4×C103—21no—
96D4×C103—41no—
97D4×C103—41no—
98C2×C4×C103—41no—
99C2×C4×C103C2×D411yescharacteristic
100C22⋊D103—41no—
101C22⋊D103—41no—
102C22⋊D103—42no—
103C22⋊D103—42no—
104C22⋊D103—41no—
105C2×Dic103—41no—
106C2×Dic103D811yescharacteristic
107D412—22no—
108D412—21no—
109D412—41no—
110D412—41no—
111C4×D103—41no—
112C4×D103—41no—
113C4×D103—41no—
114C4×D103D811yescharacteristic
115C4×D103D811yescharacteristic
116C4×D103C2×D411yescharacteristic
117C4×D103C2×D411yescharacteristic
118Dic206—21no—
119Dic206—41no—
120C8×C103C2×D411yescharacteristic
121C8×C103C2411yescharacteristic
122C103⋊C8C2×D411yescharacteristic
123C103⋊C8C2×D411yescharacteristic
124C22×C103—41no—
125C22×C103—42no—
126C22×C103—82no—
127C22×C103C2×D811yescharacteristic
128C2×D103—22no—
129C2×D103—41no—
130C2×D103—42no—
131C2×D103—84no—
132C2×D103—82no—
133C2×D103—82no—
134C2×D103—84no—
135C2×D103C2×D811yescharacteristic
136C2×D103C2×D811yescharacteristic
137C4×C103—41no—
138C4×C103C2×D811yescharacteristic
139C4×C103C22×D411yescharacteristic
140Dic103—41no—
141Dic103C2×D811yescharacteristic
142Dic103C2×D811yescharacteristic
143C2×C103—21no—
144C2×C103—82no—
145C2×C103—81no—
146C2×C103C22×D811yescharacteristic
147D103—21no—
148D103—81no—
149D103—82no—
150D103#50912yesdirect product factor
151C2×#509—1031nomaximal, Sylow
152C103C2×#50911yescharacteristic, complemented, Sylow
153C2×#135—1031no—
154C22×D8—1031no—
155#509—1031no—
156#509—1031no—
157#509—1032no—
158#509—1032no—
159#509—1031no—
160#509D10311yescharacteristic, direct product factor
161C2×SD32—1032no—
162C2×D16—1032no—
163C2×M32—1031no—
164C2×#41—2061no—
165C22×D4—2061no—
166#135—1031no—
167#135—1031no—
168#135—1031no—
169#135—2061no—
170#135—2061no—
171#135C2×D10311yescharacteristic, complemented
172C2×Q16—1031no—
173C2×SD16—2061no—
174C2×D8—1032no—
175C2×D8—1032no—
176C2×D8—1031no—
177C2×D8—1031no—
178C2×D8—2061no—
179C2×D8—2061no—
180C2×D8—2061no—
181C2×D8C2×D10311yescharacteristic, complemented
182C22×C8—1031no—
183SD32—1032no—
184SD32—1032no—
185SD32—1032no—
186SD32C2×D10312yescomplemented
187D16—1032no—
188D16—1032no—
189D16—1032no—
190D16C2×D10312yescomplemented
191M32—1031no—
192M32—1032no—
193M32C2×D10311yescharacteristic, complemented
194C2×C16—1032no—
195C24—4121no—
196#41—21no—
197#41—4121no—
198#41—4121no—
199#41—2061no—
200#41—2061no—
201#41—2061no—
202C2×Q8—2061no—
203C2×D4—21no—
204C2×D4—4121no—
205C2×D4—4121no—
206C2×D4—4121no—
207C2×D4—4121no—
208C2×D4—2062no—
209C2×D4—2062no—
210C2×D4—2061no—
211C2×D4—2061no—
212C22×C4—1031no—
213C22×C4—4121no—
214Q16—1031no—
215Q16—2061no—
216Q16C22×D10311yescharacteristic, complemented
217SD16—21no—
218SD16—2061no—
219SD16—2061no—
220SD16—2061no—
221D8—21no—
222D8—1032no—
223D8—1031no—
224D8—2062no—
225D8—2062no—
226D8—2061no—
227D8—2061no—
228D8C22×D10312yes—
229D8C22×D10311yescharacteristic, complemented
230C2×C8—1031no—
231C2×C8—1031no—
232C2×C8—1031no—
233C2×C8—1031no—
234C2×C8—1031no—
235C2×C8C22×D10311yescharacteristic
236C16—1032no—
237C16C22×D10312yescomplemented
238C23—41no—
239C23—8244no—
240C23—1031no—
241C23—4121no—
242C23—4122no—
243C23—4122no—
244C23—4121no—
245Q8—21no—
246Q8—4121no—
247Q8—2061no—
248D4—22no—
249D4—21no—
250D4—41no—
251D4—41no—
252D4—4121no—
253D4—4122no—
254D4—4122no—
255D4—4121no—
256D4—4121no—
257D4—4121no—
258D4—4121no—
259D4—2062no—
260D4—2061no—
261C2×C4—41no—
262C2×C4—1031no—
263C2×C4—1031no—
264C2×C4—1031no—
265C2×C4—1031no—
266C2×C4—1031no—
267C2×C4—4121no—
268C2×C4—4121no—
269C2×C4—4121no—
270C2×C4—4121no—
271C2×C4D4×D10311yescharacteristic
272C8—1031no—
273C8—1031no—
274C8D4×D10311yescharacteristic
275C8C23×D10311yescharacteristic
276C22—41no—
277C22—42no—
278C22—82no—
279C22—8242no—
280C22—8244no—
281C22—8242no—
282C22—8244no—
283C22—1031no—
284C22—1031no—
285C22—4121no—
286C22—4122no—
287C22—2062no—
288C22D8×D10311yescharacteristic
289C4—41no—
290C4—1031no—
291C4—1031no—
292C4—4121no—
293C4D8×D10311yescharacteristic
294C4C2×D4×D10311yescharacteristic
295C2—21no—
296C2—82no—
297C2—81no—
298C2—8242no—
299C2—8241no—
300C2—1032no—
301C2—2061no—
302C2C2×D8×D10311yescharacteristic
subgroupG/H
centerC2C2×D8×D103
commutator, cosocleC8×C103C24
FrattiniC8C23×D103
Fitting#509×C103C2
socleC2×C103C22×D8
hypercenter#509D103
hypocenterC103C2×#509
LCSG ▷ C8×C103 ▷ C4×C103 ▷ C2×C103 ▷ C103
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #135×D103C2semidirect
2. C2×D8×D103C2semidirect
3. #20268229C2semidirect
4. #20268228C2semidirect
5. #20268227C2semidirect
6. #20268226C2semidirect
7. #20268225C2semidirect
8. SD32×D103C2semidirect
9. D16×D103C2semidirect
10. M32×D103C2semidirect
11. #509×C103C2semidirect
14. #15693044C22semidirect
15. #15693043C22semidirect
18. #15693040C22semidirect
19. Q16×D103C22semidirect
21. C2×D824C22semidirect
26. D8×D103C22semidirect
27. D8×D103C22semidirect
29. #15692966C22semidirect
30. #15692965C22semidirect
31. #15692964C22semidirect
32. D1648C22semidirect
33. #15692962C22semidirect
34. #15692962C22semidirect
35. #15692961C22semidirect
36. #15692960C22semidirect
37. C16×D103C22semidirect
38. #135×C103C22semidirect
39. C2×D8×C103C22semidirect
40. SD32×C103C22semidirect
41. D16×C103C22semidirect
42. M32×C103C22semidirect
60. Dic412C23semidirect
67. D824C23semidirect
76. C103⋊C16C23semidirect
79. Q16×C103C23semidirect
83. D8×C103C23semidirect
85. C16×C103C23semidirect
150. D103#509direct (N × H)
152. C103C2×#509semidirect
160. #509D103direct (N × H)
171. #135C2×D103semidirect
181. C2×D8C2×D103semidirect
186. SD32C2×D103semidirect
190. D16C2×D103semidirect
193. M32C2×D103semidirect
216. Q16C22×D103semidirect
229. D8C22×D103semidirect
237. C16C22×D103semidirect

Maximal subgroups

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

subgroupindex
#135×D1032
C2×D8×D1032
#202682292
#202682282
#202682272
#202682262
#202682252
SD32×D1032
D16×D1032
M32×D1032
#509×C1032
C2×#509103