Finite Group Databasebeta

← #36712571

Subgroups — #36712571

Order 22784.

466 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 127 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#508×D89C212yesmaximal, complemented
2#26022571C212yesmaximal, complemented
3#26022568C212yesmaximal, complemented
4#26022566C212yesmaximal, complemented
5#26022563C212yesmaximal, complemented
6#26022561C211yesmaximal, characteristic, complemented
7#26022559C212yesmaximal, complemented
8C2×#19151431C211yesmaximal, characteristic, complemented
9C89×#2262C211yesmaximal, characteristic, complemented
10#135×D89—22no—
11#19152031C2212yes—
12#19152030C2212yescomplemented
13#19152029—22no—
14#19152028—22no—
15#19152027C2212yes—
16Q32×D89C2212yescomplemented
17SD32×D89—22no—
18D16×D89C2212yescomplemented
19C2×C16×D89C2211yescharacteristic
20Dic1424C2211yescharacteristic, complemented
21#19151438C2212yescomplemented
22#19151437C2212yescomplemented
23#19151436C2212yescomplemented
24#19151435C2212yescomplemented
25D2848C2211yescharacteristic, complemented
26#19151433C2212yescomplemented
27#19151431C2212yescomplemented
28#19151431C2212yescomplemented
29C2×#15667918C2211yescharacteristic, complemented
30#508×C89C2212yescomplemented
31Q64×C89C2211yescharacteristic, complemented
32SD64×C89C2212yescomplemented
33D32×C89C2211yescharacteristic, complemented
34C2×C32×C89C2211yescharacteristic, complemented
35#41×D89—42no—
36#15668035—22no—
37#15668034—22no—
38#15668033—42no—
39#15668032—42no—
40#15668031—22no—
41Q16×D89—22no—
42SD16×D89—42no—
43D8×D89—22no—
44C2×C8×D89D411yescharacteristic
45Dic712C2312yes—
46#15667958—22no—
47#15667957—22no—
48#15667956—22no—
49#15667955—22no—
50D1424C2312yes—
51#15667953—22no—
52C2×#13083549D411yescharacteristic
53C16×D89D412yes—
54C16×D89C2312yes—
55#15667918C2312yes—
56#135×C89—22no—
57Q32×C89C2312yes—
58SD32×C89—22no—
59D16×C89C2312yes—
60C2×C16×C89C2311yescharacteristic
61C32×C89C2311yescharacteristic
62C32×C89C2311yescharacteristic
63#13083575—42no—
64#13083574—42no—
65#13083574—82no—
66#13083573—42no—
67#13083573—82no—
68Q8×D89—42no—
69D4×D89—42no—
70D4×D89—82no—
71C2×C4×D89—82no—
72C2×C4×D89D811yescharacteristic
73Dic356—22no—
74#13083564—42no—
75#13083563—42no—
76#13083562—42no—
77#13083561—42no—
78D712—22no—
79#13083559—42no—
80C2×#10623494D811yescharacteristic
81C8×D89D812yes—
82C8×D89C2×D412yes—
83#13083549C2×D412yes—
84#41×C89—42no—
85Q16×C89—22no—
86SD16×C89—42no—
87D8×C89—22no—
88C2×C8×C89C2×D411yescharacteristic
89C16×C89C2×D411yescharacteristic
90C16×C89C2411yescharacteristic
91C22×D89—82no—
92C22×D89D1611yescharacteristic
93Q8×C89—42no—
94D4×C89—42no—
95D4×C89—82no—
96C2×C4×C89—82no—
97C2×C4×C89C2×D811yescharacteristic
98C22⋊D89—82no—
99C22⋊D89—82no—
100C22⋊D89—82no—
101C2×Dic89—82no—
102C2×Dic89D1611yescharacteristic
103D356—42no—
104D356—82no—
105C4×D89—82no—
106C4×D89—82no—
107C4×D89—82no—
108C4×D89D1612yes—
109C4×D89C2×D812yes—
110Dic178—42no—
111Dic178—82no—
112C8×C89C2×D811yescharacteristic
113C8×C89C22×D411yescharacteristic
114#10623494C2×D812yes—
115C22×C89—82no—
116C22×C89C2×D1611yescharacteristic
117C2×D89—41no—
118C2×D89—82no—
119C2×D89—164no—
120C2×D89C2×D1612yes—
121C4×C89—82no—
122C4×C89C2×D1611yescharacteristic
123C4×C89C22×D811yescharacteristic
124Dic89—82no—
125Dic89C2×D1612yes—
126#33981—891nomaximal, Sylow
127C2×C89—21no—
128C2×C89—162no—
129C2×C89C22×D1611yescharacteristic
130D89—22no—
131D89—162no—
132C2×#508—892no—
133#2264—892no—
134#2264—892no—
135#2263—892no—
136#2263—892no—
137#2262—892no—
138#2262—891no—
139#2262D8911yescharacteristic, complemented
140C2×M64—891no—
141C89#3398111yescharacteristic, complemented, Sylow
142C2×#135—1782no—
143#508—892no—
144#508—892no—
145#508—892no—
146#508—1782no—
147#508—1782no—
148#508C2×D8912yescomplemented
149C2×Q32—892no—
150C2×SD32—1782no—
151C2×D16—892no—
152C22×C16—891no—
153Q64—892no—
154Q64—891no—
155Q64C2×D8911yescharacteristic, complemented
156SD64—892no—
157SD64—892no—
158SD64—892no—
159SD64C2×D8912yescomplemented
160D32—892no—
161D32—891no—
162D32C2×D8911yescharacteristic, complemented
163M64—892no—
164M64—892no—
165C2×C32—891no—
166C2×C32C2×D8911yescharacteristic, complemented
167C2×#41—3562no—
168#135—22no—
169#135—3562no—
170#135—3562no—
171#135—1782no—
172#135—1782no—
173#135—1782no—
174C2×Q16—1782no—
175C2×SD16—3562no—
176C2×D8—1782no—
177C22×C8—891no—
178Q32—892no—
179Q32—1782no—
180Q32C22×D8912yes—
181SD32—22no—
182SD32—1782no—
183SD32—1782no—
184SD32—1782no—
185D16—892no—
186D16—1782no—
187D16C22×D8912yes—
188C2×C16—891no—
189C2×C16—892no—
190C2×C16—892no—
191C2×C16C22×D8911yescharacteristic
192C32—892no—
193C32C22×D8911yescharacteristic
194C32C22×D8911yescharacteristic
195#41—42no—
196#41—3562no—
197#41—3562no—
198#41—3562no—
199#41—7122no—
200#41—7122no—
201C2×Q8—3562no—
202C2×D4—3562no—
203C2×D4—7122no—
204C22×C4—891no—
205C22×C4—7122no—
206Q16—22no—
207Q16—3562no—
208Q16—1782no—
209SD16—42no—
210SD16—3562no—
211SD16—3562no—
212SD16—3562no—
213D8—22no—
214D8—3562no—
215D8—1782no—
216C2×C8—891no—
217C2×C8—892no—
218C2×C8—892no—
219C2×C8D4×D8911yescharacteristic
220C16—892no—
221C16D4×D8911yescharacteristic
222C16C23×D8911yescharacteristic
223C23—891no—
224C23—7122no—
225Q8—42no—
226Q8—3562no—
227Q8—7122no—
228D4—42no—
229D4—82no—
230D4—3562no—
231D4—7122no—
232D4—7122no—
233D4—7122no—
234D4—7122no—
235C2×C4—82no—
236C2×C4—891no—
237C2×C4—892no—
238C2×C4—892no—
239C2×C4—7122no—
240C2×C4—7122no—
241C2×C4—7122no—
242C2×C4—7122no—
243C2×C4D8×D8911yescharacteristic
244C8—892no—
245C8D8×D8911yescharacteristic
246C8C2×D4×D8911yescharacteristic
247C22—82no—
248C22—892no—
249C22—3561no—
250C22—14244no—
251C22—7122no—
252C22D16×D8911yescharacteristic
253C4—82no—
254C4—892no—
255C4—7122no—
256C4D16×D8911yescharacteristic
257C4C2×D8×D8911yescharacteristic
258C2—21no—
259C2—162no—
260C2—14242no—
261C2—1782no—
262C2C2×D16×D8911yescharacteristic
subgroupG/H
centerC4D16×D89
commutator, cosocleC16×C89C24
FrattiniC16C23×D89
FittingC89×#2262C2
socleC2×C89C22×D16
hypercenter#2262D89
hypocenterC89#33981
LCSG ▷ C16×C89 ▷ C8×C89 ▷ C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #2262

Complements

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

NH ≅ G/Ntype
1. #508×D89C2semidirect
2. #26022571C2semidirect
3. #26022568C2semidirect
4. #26022566C2semidirect
5. #26022563C2semidirect
6. #26022561C2semidirect
7. #26022559C2semidirect
8. C2×#19151431C2semidirect
9. C89×#2262C2semidirect
12. #19152030C22semidirect
16. Q32×D89C22semidirect
18. D16×D89C22semidirect
20. Dic1424C22semidirect
21. #19151438C22semidirect
22. #19151437C22semidirect
23. #19151436C22semidirect
24. #19151435C22semidirect
25. D2848C22semidirect
26. #19151433C22semidirect
27. #19151431C22semidirect
28. #19151431C22semidirect
29. C2×#15667918C22semidirect
30. #508×C89C22semidirect
31. Q64×C89C22semidirect
32. SD64×C89C22semidirect
33. D32×C89C22semidirect
34. C2×C32×C89C22semidirect
139. #2262D89semidirect
141. C89#33981semidirect
148. #508C2×D89semidirect
155. Q64C2×D89semidirect
159. SD64C2×D89semidirect
162. D32C2×D89semidirect
166. C2×C32C2×D89semidirect

Maximal subgroups

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

subgroupindex
#508×D892
#260225712
#260225682
#260225662
#260225632
#260225612
#260225592
C2×#191514312
C89×#22622
#3398189