Finite Group Databasebeta

← #509×D97 #27154856

Subgroups — #509×D97 #27154856

Order 12416.

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×D97C211yesmaximal, characteristic, complemented
2C2×D8×D97C211yesmaximal, characteristic, complemented
3#20261173C211yesmaximal, characteristic, complemented
4#20261172C212yesmaximal, complemented
5#20261171C211yesmaximal, characteristic, complemented
6#20261170C211yesmaximal, characteristic, complemented
7#20261169C212yesmaximal, complemented
8SD32×D97C212yesmaximal, complemented
9D16×D97C212yesmaximal, complemented
10M32×D97C211yesmaximal, characteristic, complemented
11#509×C97C211yesmaximal, characteristic, complemented
12#41×D97—21no—
13C2×D4×D97—21no—
14#15690833C2211yescharacteristic, complemented
15#15690832C2211yescharacteristic, complemented
16#15690831—21no—
17#15690830—21no—
18#15690829C2211yescharacteristic, complemented
19Q16×D97C2211yescharacteristic, complemented
20SD16×D97—21no—
21C2×D776C2211yescharacteristic, complemented
22C2×#13095496—21no—
23D8×D97—21no—
24D8×D97—21no—
25D8×D97C2212yes—
26D8×D97C2212yescomplemented
27D8×D97C2211yescharacteristic, complemented
28C2×C8×D97C2211yescharacteristic
29#15690755C2212yescomplemented
30#15690754C2212yescomplemented
31#15690753C2212yescomplemented
32D1552C2212yescomplemented
33#15690751C2212yescomplemented
34#15690751C2212yescomplemented
35#15690750C2211yescharacteristic, complemented
36#15690749C2212yescomplemented
37C16×D97C2212yescomplemented
38#135×C97C2211yescharacteristic, complemented
39C2×D8×C97C2211yescharacteristic, complemented
40SD32×C97C2212yescomplemented
41D16×C97C2212yescomplemented
42M32×C97C2211yescharacteristic, complemented
43C23×D97—41no—
44#13095512—21no—
45#13095511—21no—
46#13095511—41no—
47#13095510—21no—
48#13095510—41no—
49Q8×D97—21no—
50C2×D388—21no—
51C2×C22⋊D97—41no—
52D4×D97—22no—
53D4×D97—22no—
54D4×D97—21no—
55D4×D97—41no—
56D4×D97—41no—
57D4×D97—41no—
58C2×C4×D97—41no—
59C2×C4×D97D411yescharacteristic
60Dic388C2311yescharacteristic, complemented
61#13095501—21no—
62#13095500—21no—
63#13095499—21no—
64#13095498—21no—
65D776—21no—
66D776C2312yes—
67D776C2311yescharacteristic, complemented
68#13095496—22no—
69#13095496—22no—
70#13095496—21no—
71C2×#11715895D411yescharacteristic
72C8×D97D411yescharacteristic
73C8×D97D411yescharacteristic
74C8×D97C2311yescharacteristic
75C8×D97C2311yescharacteristic
76#13095486C2312yescomplemented
77#41×C97—21no—
78C2×D4×C97—21no—
79Q16×C97C2311yescharacteristic, complemented
80SD16×C97—21no—
81D8×C97—21no—
82D8×C97C2312yes—
83D8×C97C2311yescharacteristic, complemented
84C2×C8×C97C2311yescharacteristic
85C16×C97C2312yescomplemented
86C23×C97—41no—
87C22×D97—41no—
88C22×D97—41no—
89C22×D97—42no—
90C22×D97—42no—
91C22×D97—84no—
92C22×D97D811yescharacteristic
93Q8×C97—21no—
94D4×C97—22no—
95D4×C97—21no—
96D4×C97—41no—
97D4×C97—41no—
98C2×C4×C97—41no—
99C2×C4×C97C2×D411yescharacteristic
100C22⋊D97—41no—
101C22⋊D97—41no—
102C22⋊D97—42no—
103C22⋊D97—42no—
104C22⋊D97—41no—
105C2×Dic97—41no—
106C2×Dic97D811yescharacteristic
107D388—22no—
108D388—21no—
109D388—41no—
110D388—41no—
111C4×D97—41no—
112C4×D97—41no—
113C4×D97—41no—
114C4×D97D811yescharacteristic
115C4×D97D811yescharacteristic
116C4×D97C2×D411yescharacteristic
117C4×D97C2×D411yescharacteristic
118Dic194—21no—
119Dic194—41no—
120C8×C97C2×D411yescharacteristic
121C8×C97C2411yescharacteristic
122#11715895C2×D411yescharacteristic
123#11715895C2×D411yescharacteristic
124C22×C97—41no—
125C22×C97—42no—
126C22×C97—82no—
127C22×C97C2×D811yescharacteristic
128C2×D97—22no—
129C2×D97—41no—
130C2×D97—42no—
131C2×D97—84no—
132C2×D97—82no—
133C2×D97—82no—
134C2×D97—84no—
135C2×D97C2×D811yescharacteristic
136C2×D97C2×D811yescharacteristic
137C4×C97—41no—
138C4×C97C2×D811yescharacteristic
139C4×C97C22×D411yescharacteristic
140Dic97—41no—
141Dic97C2×D811yescharacteristic
142Dic97C2×D811yescharacteristic
143C2×C97—21no—
144C2×C97—82no—
145C2×C97—81no—
146C2×C97C22×D811yescharacteristic
147D97—21no—
148D97—81no—
149D97—82no—
150D97#50912yesdirect product factor
151C2×#509—971nomaximal, Sylow
152C97C2×#50911yescharacteristic, complemented, Sylow
153C2×#135—971no—
154C22×D8—971no—
155#509—971no—
156#509—971no—
157#509—972no—
158#509—972no—
159#509—971no—
160#509D9711yescharacteristic, direct product factor
161C2×SD32—972no—
162C2×D16—972no—
163C2×M32—971no—
164C2×#41—1941no—
165C22×D4—1941no—
166#135—971no—
167#135—971no—
168#135—971no—
169#135—1941no—
170#135—1941no—
171#135C2×D9711yescharacteristic, complemented
172C2×Q16—971no—
173C2×SD16—1941no—
174C2×D8—972no—
175C2×D8—972no—
176C2×D8—971no—
177C2×D8—971no—
178C2×D8—1941no—
179C2×D8—1941no—
180C2×D8—1941no—
181C2×D8C2×D9711yescharacteristic, complemented
182C22×C8—971no—
183SD32—972no—
184SD32—972no—
185SD32—972no—
186SD32C2×D9712yescomplemented
187D16—972no—
188D16—972no—
189D16—972no—
190D16C2×D9712yescomplemented
191M32—971no—
192M32—972no—
193M32C2×D9711yescharacteristic, complemented
194C2×C16—972no—
195C24—3881no—
196#41—21no—
197#41—3881no—
198#41—3881no—
199#41—1941no—
200#41—1941no—
201#41—1941no—
202C2×Q8—1941no—
203C2×D4—21no—
204C2×D4—3881no—
205C2×D4—3881no—
206C2×D4—3881no—
207C2×D4—3881no—
208C2×D4—1942no—
209C2×D4—1942no—
210C2×D4—1941no—
211C2×D4—1941no—
212C22×C4—971no—
213C22×C4—3881no—
214Q16—971no—
215Q16—1941no—
216Q16C22×D9711yescharacteristic, complemented
217SD16—21no—
218SD16—1941no—
219SD16—1941no—
220SD16—1941no—
221D8—21no—
222D8—972no—
223D8—971no—
224D8—1942no—
225D8—1942no—
226D8—1941no—
227D8—1941no—
228D8C22×D9712yes—
229D8C22×D9711yescharacteristic, complemented
230C2×C8—971no—
231C2×C8—971no—
232C2×C8—971no—
233C2×C8—971no—
234C2×C8—971no—
235C2×C8C22×D9711yescharacteristic
236C16—972no—
237C16C22×D9712yescomplemented
238C23—41no—
239C23—7764no—
240C23—971no—
241C23—3881no—
242C23—3882no—
243C23—3882no—
244C23—3881no—
245Q8—21no—
246Q8—3881no—
247Q8—1941no—
248D4—22no—
249D4—21no—
250D4—41no—
251D4—41no—
252D4—3881no—
253D4—3882no—
254D4—3882no—
255D4—3881no—
256D4—3881no—
257D4—3881no—
258D4—3881no—
259D4—1942no—
260D4—1941no—
261C2×C4—41no—
262C2×C4—971no—
263C2×C4—971no—
264C2×C4—971no—
265C2×C4—971no—
266C2×C4—971no—
267C2×C4—3881no—
268C2×C4—3881no—
269C2×C4—3881no—
270C2×C4—3881no—
271C2×C4D4×D9711yescharacteristic
272C8—971no—
273C8—971no—
274C8D4×D9711yescharacteristic
275C8C23×D9711yescharacteristic
276C22—41no—
277C22—42no—
278C22—82no—
279C22—7762no—
280C22—7764no—
281C22—7762no—
282C22—7764no—
283C22—971no—
284C22—971no—
285C22—3881no—
286C22—3882no—
287C22—1942no—
288C22D8×D9711yescharacteristic
289C4—41no—
290C4—971no—
291C4—971no—
292C4—3881no—
293C4D8×D9711yescharacteristic
294C4C2×D4×D9711yescharacteristic
295C2—21no—
296C2—82no—
297C2—81no—
298C2—7762no—
299C2—7761no—
300C2—972no—
301C2—1941no—
302C2C2×D8×D9711yescharacteristic
subgroupG/H
centerC2C2×D8×D97
commutator, cosocleC8×C97C24
FrattiniC8C23×D97
Fitting#509×C97C2
socleC2×C97C22×D8
hypercenter#509D97
hypocenterC97C2×#509
LCSG ▷ C8×C97 ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #135×D97C2semidirect
2. C2×D8×D97C2semidirect
3. #20261173C2semidirect
4. #20261172C2semidirect
5. #20261171C2semidirect
6. #20261170C2semidirect
7. #20261169C2semidirect
8. SD32×D97C2semidirect
9. D16×D97C2semidirect
10. M32×D97C2semidirect
11. #509×C97C2semidirect
14. #15690833C22semidirect
15. #15690832C22semidirect
18. #15690829C22semidirect
19. Q16×D97C22semidirect
21. C2×D776C22semidirect
26. D8×D97C22semidirect
27. D8×D97C22semidirect
29. #15690755C22semidirect
30. #15690754C22semidirect
31. #15690753C22semidirect
32. D1552C22semidirect
33. #15690751C22semidirect
34. #15690751C22semidirect
35. #15690750C22semidirect
36. #15690749C22semidirect
37. C16×D97C22semidirect
38. #135×C97C22semidirect
39. C2×D8×C97C22semidirect
40. SD32×C97C22semidirect
41. D16×C97C22semidirect
42. M32×C97C22semidirect
60. Dic388C23semidirect
67. D776C23semidirect
76. #13095486C23semidirect
79. Q16×C97C23semidirect
83. D8×C97C23semidirect
85. C16×C97C23semidirect
150. D97#509direct (N × H)
152. C97C2×#509semidirect
160. #509D97direct (N × H)
171. #135C2×D97semidirect
181. C2×D8C2×D97semidirect
186. SD32C2×D97semidirect
190. D16C2×D97semidirect
193. M32C2×D97semidirect
216. Q16C22×D97semidirect
229. D8C22×D97semidirect
237. C16C22×D97semidirect

Maximal subgroups

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

subgroupindex
#135×D972
C2×D8×D972
#202611732
#202611722
#202611712
#202611702
#202611692
SD32×D972
D16×D972
M32×D972
#509×C972
C2×#50997