Finite Group Databasebeta

← #509×D157 #34113256

Subgroups — #509×D157 #34113256

Order 20096.

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×D157C211yesmaximal, characteristic, complemented
2C2×D8×D157C211yesmaximal, characteristic, complemented
3#23757711C211yesmaximal, characteristic, complemented
4#23757710C212yesmaximal, complemented
5#23757709C211yesmaximal, characteristic, complemented
6#23757708C211yesmaximal, characteristic, complemented
7#23757707C212yesmaximal, complemented
8SD32×D157C212yesmaximal, complemented
9D16×D157C212yesmaximal, complemented
10M32×D157C211yesmaximal, characteristic, complemented
11#509×C157C211yesmaximal, characteristic, complemented
12#41×D157—21no—
13C2×D4×D157—21no—
14#19097794C2211yescharacteristic, complemented
15#19097793C2211yescharacteristic, complemented
16#19097792—21no—
17#19097791—21no—
18#19097790C2211yescharacteristic, complemented
19Q16×D157C2211yescharacteristic, complemented
20SD16×D157—21no—
21C2×D1256C2211yescharacteristic, complemented
22C2×#14578939—21no—
23D8×D157—21no—
24D8×D157—21no—
25D8×D157C2212yes—
26D8×D157C2212yescomplemented
27D8×D157C2211yescharacteristic, complemented
28C2×C8×D157C2211yescharacteristic
29#19097716C2212yescomplemented
30#19097715C2212yescomplemented
31#19097714C2212yescomplemented
32D2512C2212yescomplemented
33#19097712C2212yescomplemented
34#19097712C2212yescomplemented
35#19097711C2211yescharacteristic, complemented
36#19097710C2212yescomplemented
37C16×D157C2212yescomplemented
38#135×C157C2211yescharacteristic, complemented
39C2×D8×C157C2211yescharacteristic, complemented
40SD32×C157C2212yescomplemented
41D16×C157C2212yescomplemented
42M32×C157C2211yescharacteristic, complemented
43C23×D157—41no—
44#14578955—21no—
45#14578954—21no—
46#14578954—41no—
47#14578953—21no—
48#14578953—41no—
49Q8×D157—21no—
50C2×D628—21no—
51C2×C22⋊D157—41no—
52D4×D157—22no—
53D4×D157—22no—
54D4×D157—21no—
55D4×D157—41no—
56D4×D157—41no—
57D4×D157—41no—
58C2×C4×D157—41no—
59C2×C4×D157D411yescharacteristic
60Dic628C2311yescharacteristic, complemented
61#14578944—21no—
62#14578943—21no—
63#14578942—21no—
64#14578941—21no—
65D1256—21no—
66D1256C2312yes—
67D1256C2311yescharacteristic, complemented
68#14578939—22no—
69#14578939—22no—
70#14578939—21no—
71C2×#11929434D411yescharacteristic
72C8×D157D411yescharacteristic
73C8×D157D411yescharacteristic
74C8×D157C2311yescharacteristic
75C8×D157C2311yescharacteristic
76#14578929C2312yescomplemented
77#41×C157—21no—
78C2×D4×C157—21no—
79Q16×C157C2311yescharacteristic, complemented
80SD16×C157—21no—
81D8×C157—21no—
82D8×C157C2312yes—
83D8×C157C2311yescharacteristic, complemented
84C2×C8×C157C2311yescharacteristic
85C16×C157C2312yescomplemented
86C22×D157—41no—
87C22×D157—41no—
88C22×D157—42no—
89C22×D157—42no—
90C22×D157—84no—
91C22×D157D811yescharacteristic
92Dic314—21no—
93Dic314—41no—
94D628—22no—
95D628—21no—
96D628—41no—
97D628—41no—
98C22⋊D157—41no—
99C22⋊D157—41no—
100C22⋊D157—42no—
101C22⋊D157—42no—
102C22⋊D157—41no—
103C2×Dic157—41no—
104C2×Dic157D811yescharacteristic
105C4×D157—41no—
106C4×D157—41no—
107C4×D157—41no—
108C4×D157D811yescharacteristic
109C4×D157D811yescharacteristic
110C4×D157C2×D411yescharacteristic
111C4×D157C2×D411yescharacteristic
112#11929434C2×D411yescharacteristic
113#11929434C2×D411yescharacteristic
114C23×C157—41no—
115Q8×C157—21no—
116D4×C157—22no—
117D4×C157—21no—
118D4×C157—41no—
119D4×C157—41no—
120C2×C4×C157—41no—
121C2×C4×C157C2×D411yescharacteristic
122C8×C157C2×D411yescharacteristic
123C8×C157C2411yescharacteristic
124C22×C157—41no—
125C22×C157—42no—
126C22×C157—82no—
127C22×C157C2×D811yescharacteristic
128C2×D157—22no—
129C2×D157—41no—
130C2×D157—42no—
131C2×D157—84no—
132C2×D157—82no—
133C2×D157—82no—
134C2×D157—84no—
135C2×D157C2×D811yescharacteristic
136C2×D157C2×D811yescharacteristic
137C4×C157—41no—
138C4×C157C2×D811yescharacteristic
139C4×C157C22×D411yescharacteristic
140Dic157—41no—
141Dic157C2×D811yescharacteristic
142Dic157C2×D811yescharacteristic
143C2×C157—21no—
144C2×C157—82no—
145C2×C157—81no—
146C2×C157C22×D811yescharacteristic
147D157—21no—
148D157—81no—
149D157—82no—
150D157#50912yesdirect product factor
151C157C2×#50911yescharacteristic, complemented, Sylow
152C2×#509—1571nomaximal, Sylow
153C2×#135—1571no—
154C22×D8—1571no—
155#509—1571no—
156#509—1571no—
157#509—1572no—
158#509—1572no—
159#509—1571no—
160#509D15711yescharacteristic, direct product factor
161C2×SD32—1572no—
162C2×D16—1572no—
163C2×M32—1571no—
164C2×#41—3141no—
165C22×D4—3141no—
166#135—3141no—
167#135—3141no—
168#135—1571no—
169#135—1571no—
170#135—1571no—
171#135C2×D15711yescharacteristic, complemented
172C2×Q16—1571no—
173C2×SD16—3141no—
174C2×D8—3141no—
175C2×D8—3141no—
176C2×D8—3141no—
177C2×D8—1572no—
178C2×D8—1572no—
179C2×D8—1571no—
180C2×D8—1571no—
181C2×D8C2×D15711yescharacteristic, complemented
182C22×C8—1571no—
183SD32—1572no—
184SD32—1572no—
185SD32—1572no—
186SD32C2×D15712yescomplemented
187D16—1572no—
188D16—1572no—
189D16—1572no—
190D16C2×D15712yescomplemented
191M32—1571no—
192M32—1572no—
193M32C2×D15711yescharacteristic, complemented
194C2×C16—1572no—
195C24—6281no—
196#41—21no—
197#41—3141no—
198#41—3141no—
199#41—3141no—
200#41—6281no—
201#41—6281no—
202C2×Q8—3141no—
203C2×D4—21no—
204C2×D4—3142no—
205C2×D4—3142no—
206C2×D4—3141no—
207C2×D4—3141no—
208C2×D4—6281no—
209C2×D4—6281no—
210C2×D4—6281no—
211C2×D4—6281no—
212C22×C4—6281no—
213C22×C4—1571no—
214Q16—3141no—
215Q16—1571no—
216Q16C22×D15711yescharacteristic, complemented
217SD16—21no—
218SD16—3141no—
219SD16—3141no—
220SD16—3141no—
221D8—21no—
222D8—3142no—
223D8—3142no—
224D8—3141no—
225D8—3141no—
226D8—1571no—
227D8—1572no—
228D8C22×D15712yes—
229D8C22×D15711yescharacteristic, complemented
230C2×C8—1571no—
231C2×C8—1571no—
232C2×C8—1571no—
233C2×C8—1571no—
234C2×C8—1571no—
235C2×C8C22×D15711yescharacteristic
236C16—1572no—
237C16C22×D15712yescomplemented
238C23—41no—
239C23—6281no—
240C23—6282no—
241C23—6282no—
242C23—6281no—
243C23—1571no—
244C23—12564no—
245Q8—21no—
246Q8—3141no—
247Q8—6281no—
248D4—22no—
249D4—21no—
250D4—41no—
251D4—41no—
252D4—3142no—
253D4—3141no—
254D4—6281no—
255D4—6281no—
256D4—6281no—
257D4—6282no—
258D4—6282no—
259D4—6281no—
260D4—6281no—
261C2×C4—41no—
262C2×C4—6281no—
263C2×C4—6281no—
264C2×C4—6281no—
265C2×C4—6281no—
266C2×C4—1571no—
267C2×C4—1571no—
268C2×C4—1571no—
269C2×C4—1571no—
270C2×C4—1571no—
271C2×C4D4×D15711yescharacteristic
272C8—1571no—
273C8—1571no—
274C8D4×D15711yescharacteristic
275C8C23×D15711yescharacteristic
276C22—41no—
277C22—42no—
278C22—82no—
279C22—3142no—
280C22—6281no—
281C22—6282no—
282C22—1571no—
283C22—1571no—
284C22—12562no—
285C22—12564no—
286C22—12564no—
287C22—12562no—
288C22D8×D15711yescharacteristic
289C4—41no—
290C4—6281no—
291C4—1571no—
292C4—1571no—
293C4D8×D15711yescharacteristic
294C4C2×D4×D15711yescharacteristic
295C2—21no—
296C2—82no—
297C2—81no—
298C2—3141no—
299C2—1572no—
300C2—12562no—
301C2—12561no—
302C2C2×D8×D15711yescharacteristic
subgroupG/H
centerC2C2×D8×D157
commutator, cosocleC8×C157C24
FrattiniC8C23×D157
Fitting#509×C157C2
socleC2×C157C22×D8
hypercenter#509D157
hypocenterC157C2×#509
LCSG ▷ C8×C157 ▷ C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #135×D157C2semidirect
2. C2×D8×D157C2semidirect
3. #23757711C2semidirect
4. #23757710C2semidirect
5. #23757709C2semidirect
6. #23757708C2semidirect
7. #23757707C2semidirect
8. SD32×D157C2semidirect
9. D16×D157C2semidirect
10. M32×D157C2semidirect
11. #509×C157C2semidirect
14. #19097794C22semidirect
15. #19097793C22semidirect
18. #19097790C22semidirect
19. Q16×D157C22semidirect
21. C2×D1256C22semidirect
26. D8×D157C22semidirect
27. D8×D157C22semidirect
29. #19097716C22semidirect
30. #19097715C22semidirect
31. #19097714C22semidirect
32. D2512C22semidirect
33. #19097712C22semidirect
34. #19097712C22semidirect
35. #19097711C22semidirect
36. #19097710C22semidirect
37. C16×D157C22semidirect
38. #135×C157C22semidirect
39. C2×D8×C157C22semidirect
40. SD32×C157C22semidirect
41. D16×C157C22semidirect
42. M32×C157C22semidirect
60. Dic628C23semidirect
67. D1256C23semidirect
76. #14578929C23semidirect
79. Q16×C157C23semidirect
83. D8×C157C23semidirect
85. C16×C157C23semidirect
150. D157#509direct (N × H)
151. C157C2×#509semidirect
160. #509D157direct (N × H)
171. #135C2×D157semidirect
181. C2×D8C2×D157semidirect
186. SD32C2×D157semidirect
190. D16C2×D157semidirect
193. M32C2×D157semidirect
216. Q16C22×D157semidirect
229. D8C22×D157semidirect
237. C16C22×D157semidirect

Maximal subgroups

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

subgroupindex
#135×D1572
C2×D8×D1572
#237577112
#237577102
#237577092
#237577082
#237577072
SD32×D1572
D16×D1572
M32×D1572
#509×C1572
C2×#509157