Finite Group Databasebeta

← #509×D113 #28403694

Subgroups — #509×D113 #28403694

Order 14464.

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×D113C211yesmaximal, characteristic, complemented
2C2×D8×D113C211yesmaximal, characteristic, complemented
3#20299036C211yesmaximal, characteristic, complemented
4#20299035C212yesmaximal, complemented
5#20299034C211yesmaximal, characteristic, complemented
6#20299033C211yesmaximal, characteristic, complemented
7#20299032C212yesmaximal, complemented
8SD32×D113C212yesmaximal, complemented
9D16×D113C212yesmaximal, complemented
10M32×D113C211yesmaximal, characteristic, complemented
11#509×C113C211yesmaximal, characteristic, complemented
12#41×D113—21no—
13C2×D4×D113—21no—
14#16813645C2211yescharacteristic, complemented
15#16813644C2211yescharacteristic, complemented
16#16813643—21no—
17#16813642—21no—
18#16813641C2211yescharacteristic, complemented
19Q16×D113C2211yescharacteristic, complemented
20SD16×D113—21no—
21C2×D904C2211yescharacteristic, complemented
22C2×#14268632—21no—
23D8×D113—21no—
24D8×D113—21no—
25D8×D113C2212yes—
26D8×D113C2212yescomplemented
27D8×D113C2211yescharacteristic, complemented
28C2×C8×D113C2211yescharacteristic
29#16813567C2212yescomplemented
30#16813566C2212yescomplemented
31#16813565C2212yescomplemented
32D1808C2212yescomplemented
33#16813563C2212yescomplemented
34#16813563C2212yescomplemented
35#16813562C2211yescharacteristic, complemented
36#16813561C2212yescomplemented
37C16×D113C2212yescomplemented
38#135×C113C2211yescharacteristic, complemented
39C2×D8×C113C2211yescharacteristic, complemented
40SD32×C113C2212yescomplemented
41D16×C113C2212yescomplemented
42M32×C113C2211yescharacteristic, complemented
43C23×D113—41no—
44#14268648—21no—
45#14268647—21no—
46#14268647—41no—
47#14268646—21no—
48#14268646—41no—
49Q8×D113—21no—
50C2×D452—21no—
51C2×C22⋊D113—41no—
52D4×D113—22no—
53D4×D113—22no—
54D4×D113—21no—
55D4×D113—41no—
56D4×D113—41no—
57D4×D113—41no—
58C2×C4×D113—41no—
59C2×C4×D113D411yescharacteristic
60Dic452C2311yescharacteristic, complemented
61#14268637—21no—
62#14268636—21no—
63#14268635—21no—
64#14268634—21no—
65D904—21no—
66D904C2312yes—
67D904C2311yescharacteristic, complemented
68#14268632—22no—
69#14268632—22no—
70#14268632—21no—
71C2×#11744751D411yescharacteristic
72C8×D113D411yescharacteristic
73C8×D113D411yescharacteristic
74C8×D113C2311yescharacteristic
75C8×D113C2311yescharacteristic
76#14268622C2312yescomplemented
77#41×C113—21no—
78C2×D4×C113—21no—
79Q16×C113C2311yescharacteristic, complemented
80SD16×C113—21no—
81D8×C113—21no—
82D8×C113C2312yes—
83D8×C113C2311yescharacteristic, complemented
84C2×C8×C113C2311yescharacteristic
85C16×C113C2312yescomplemented
86C23×C113—41no—
87C22×D113—41no—
88C22×D113—41no—
89C22×D113—42no—
90C22×D113—42no—
91C22×D113—84no—
92C22×D113D811yescharacteristic
93Q8×C113—21no—
94D4×C113—22no—
95D4×C113—21no—
96D4×C113—41no—
97D4×C113—41no—
98C2×C4×C113—41no—
99C2×C4×C113C2×D411yescharacteristic
100C22⋊D113—41no—
101C22⋊D113—41no—
102C22⋊D113—42no—
103C22⋊D113—42no—
104C22⋊D113—41no—
105C2×Dic113—41no—
106C2×Dic113D811yescharacteristic
107D452—22no—
108D452—21no—
109D452—41no—
110D452—41no—
111C4×D113—41no—
112C4×D113—41no—
113C4×D113—41no—
114C4×D113D811yescharacteristic
115C4×D113D811yescharacteristic
116C4×D113C2×D411yescharacteristic
117C4×D113C2×D411yescharacteristic
118Dic226—21no—
119Dic226—41no—
120C8×C113C2×D411yescharacteristic
121C8×C113C2411yescharacteristic
122#11744751C2×D411yescharacteristic
123#11744751C2×D411yescharacteristic
124C22×C113—41no—
125C22×C113—42no—
126C22×C113—82no—
127C22×C113C2×D811yescharacteristic
128C2×D113—22no—
129C2×D113—41no—
130C2×D113—42no—
131C2×D113—84no—
132C2×D113—82no—
133C2×D113—82no—
134C2×D113—84no—
135C2×D113C2×D811yescharacteristic
136C2×D113C2×D811yescharacteristic
137C4×C113—41no—
138C4×C113C2×D811yescharacteristic
139C4×C113C22×D411yescharacteristic
140Dic113—41no—
141Dic113C2×D811yescharacteristic
142Dic113C2×D811yescharacteristic
143C2×C113—21no—
144C2×C113—82no—
145C2×C113—81no—
146C2×C113C22×D811yescharacteristic
147D113—21no—
148D113—81no—
149D113—82no—
150D113#50912yesdirect product factor
151C2×#509—1131nomaximal, Sylow
152C113C2×#50911yescharacteristic, complemented, Sylow
153C2×#135—1131no—
154C22×D8—1131no—
155#509—1131no—
156#509—1131no—
157#509—1132no—
158#509—1132no—
159#509—1131no—
160#509D11311yescharacteristic, direct product factor
161C2×SD32—1132no—
162C2×D16—1132no—
163C2×M32—1131no—
164C2×#41—2261no—
165C22×D4—2261no—
166#135—1131no—
167#135—1131no—
168#135—1131no—
169#135—2261no—
170#135—2261no—
171#135C2×D11311yescharacteristic, complemented
172C2×Q16—1131no—
173C2×SD16—2261no—
174C2×D8—1132no—
175C2×D8—1132no—
176C2×D8—1131no—
177C2×D8—1131no—
178C2×D8—2261no—
179C2×D8—2261no—
180C2×D8—2261no—
181C2×D8C2×D11311yescharacteristic, complemented
182C22×C8—1131no—
183SD32—1132no—
184SD32—1132no—
185SD32—1132no—
186SD32C2×D11312yescomplemented
187D16—1132no—
188D16—1132no—
189D16—1132no—
190D16C2×D11312yescomplemented
191M32—1131no—
192M32—1132no—
193M32C2×D11311yescharacteristic, complemented
194C2×C16—1132no—
195C24—4521no—
196#41—21no—
197#41—4521no—
198#41—4521no—
199#41—2261no—
200#41—2261no—
201#41—2261no—
202C2×Q8—2261no—
203C2×D4—21no—
204C2×D4—4521no—
205C2×D4—4521no—
206C2×D4—4521no—
207C2×D4—4521no—
208C2×D4—2262no—
209C2×D4—2262no—
210C2×D4—2261no—
211C2×D4—2261no—
212C22×C4—1131no—
213C22×C4—4521no—
214Q16—1131no—
215Q16—2261no—
216Q16C22×D11311yescharacteristic, complemented
217SD16—21no—
218SD16—2261no—
219SD16—2261no—
220SD16—2261no—
221D8—21no—
222D8—1131no—
223D8—1132no—
224D8—2262no—
225D8—2262no—
226D8—2261no—
227D8—2261no—
228D8C22×D11312yes—
229D8C22×D11311yescharacteristic, complemented
230C2×C8—1131no—
231C2×C8—1131no—
232C2×C8—1131no—
233C2×C8—1131no—
234C2×C8—1131no—
235C2×C8C22×D11311yescharacteristic
236C16—1132no—
237C16C22×D11312yescomplemented
238C23—41no—
239C23—1131no—
240C23—9044no—
241C23—4521no—
242C23—4522no—
243C23—4522no—
244C23—4521no—
245Q8—21no—
246Q8—4521no—
247Q8—2261no—
248D4—22no—
249D4—21no—
250D4—41no—
251D4—41no—
252D4—4521no—
253D4—4522no—
254D4—4522no—
255D4—4521no—
256D4—4521no—
257D4—4521no—
258D4—4521no—
259D4—2262no—
260D4—2261no—
261C2×C4—41no—
262C2×C4—1131no—
263C2×C4—1131no—
264C2×C4—1131no—
265C2×C4—1131no—
266C2×C4—1131no—
267C2×C4—4521no—
268C2×C4—4521no—
269C2×C4—4521no—
270C2×C4—4521no—
271C2×C4D4×D11311yescharacteristic
272C8—1131no—
273C8—1131no—
274C8D4×D11311yescharacteristic
275C8C23×D11311yescharacteristic
276C22—41no—
277C22—42no—
278C22—82no—
279C22—1131no—
280C22—1131no—
281C22—9042no—
282C22—9044no—
283C22—9042no—
284C22—9044no—
285C22—4521no—
286C22—4522no—
287C22—2262no—
288C22D8×D11311yescharacteristic
289C4—41no—
290C4—1131no—
291C4—1131no—
292C4—4521no—
293C4D8×D11311yescharacteristic
294C4C2×D4×D11311yescharacteristic
295C2—21no—
296C2—82no—
297C2—81no—
298C2—1132no—
299C2—9042no—
300C2—9041no—
301C2—2261no—
302C2C2×D8×D11311yescharacteristic
subgroupG/H
centerC2C2×D8×D113
commutator, cosocleC8×C113C24
FrattiniC8C23×D113
Fitting#509×C113C2
socleC2×C113C22×D8
hypercenter#509D113
hypocenterC113C2×#509
LCSG ▷ C8×C113 ▷ C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #135×D113C2semidirect
2. C2×D8×D113C2semidirect
3. #20299036C2semidirect
4. #20299035C2semidirect
5. #20299034C2semidirect
6. #20299033C2semidirect
7. #20299032C2semidirect
8. SD32×D113C2semidirect
9. D16×D113C2semidirect
10. M32×D113C2semidirect
11. #509×C113C2semidirect
14. #16813645C22semidirect
15. #16813644C22semidirect
18. #16813641C22semidirect
19. Q16×D113C22semidirect
21. C2×D904C22semidirect
26. D8×D113C22semidirect
27. D8×D113C22semidirect
29. #16813567C22semidirect
30. #16813566C22semidirect
31. #16813565C22semidirect
32. D1808C22semidirect
33. #16813563C22semidirect
34. #16813563C22semidirect
35. #16813562C22semidirect
36. #16813561C22semidirect
37. C16×D113C22semidirect
38. #135×C113C22semidirect
39. C2×D8×C113C22semidirect
40. SD32×C113C22semidirect
41. D16×C113C22semidirect
42. M32×C113C22semidirect
60. Dic452C23semidirect
67. D904C23semidirect
76. #14268622C23semidirect
79. Q16×C113C23semidirect
83. D8×C113C23semidirect
85. C16×C113C23semidirect
150. D113#509direct (N × H)
152. C113C2×#509semidirect
160. #509D113direct (N × H)
171. #135C2×D113semidirect
181. C2×D8C2×D113semidirect
186. SD32C2×D113semidirect
190. D16C2×D113semidirect
193. M32C2×D113semidirect
216. Q16C22×D113semidirect
229. D8C22×D113semidirect
237. C16C22×D113semidirect

Maximal subgroups

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

subgroupindex
#135×D1132
C2×D8×D1132
#202990362
#202990352
#202990342
#202990332
#202990322
SD32×D1132
D16×D1132
M32×D1132
#509×C1132
C2×#509113