Finite Group Databasebeta

← #36712575

Subgroups — #36712575

Order 22784.

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

normal subgroups only

This group has 472 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#26033389C212yesmaximal, complemented
2#26022565C212yesmaximal, complemented
3#26022563C214yesmaximal, complemented
4#26022561C211yesmaximal, characteristic, complemented
5#26022558C212yesmaximal, complemented
6D32×D89C212yesmaximal, complemented
7#26022545C211yesmaximal, characteristic, complemented
8C2×D32×C89C211yesmaximal, characteristic, complemented
9#19152399—22no—
10#19152034—22no—
11#19152032—24no—
12#19152030C2212yescomplemented
13#19152027C2214yescomplemented
14D16×D89C2214yescomplemented
15#19152014C2211yescharacteristic
16Dic1424C2211yescharacteristic, complemented
17#19151437C2212yescomplemented
18#19151436C2214yescomplemented
19D2848C2211yescharacteristic, complemented
20#19151433C2214yescomplemented
21#19151432C2211yescharacteristic, complemented
22#19151431C2212yescomplemented
23C32×D89C2212yescomplemented
24C2×D16×C89C2212yescomplemented
25D32×C89C2212yescomplemented
26D32×C89C2212yescomplemented
27C2×C32×C89C2211yescharacteristic, complemented
28#15668057—42no—
29#15668038—42no—
30#15668036—44no—
31#15668034—22no—
32#15668031—24no—
33D8×D89—24no—
34#15668018D411yescharacteristic
35Dic712C2312yes—
36#15667957—22no—
37#15667956—24no—
38D1424C2312yes—
39#15667953—24no—
40#15667952D411yescharacteristic
41#15667951D412yes—
42C16×D89C2312yes—
43#15667918C2312yes—
44C2×D8×C89—22no—
45D16×C89—22no—
46D16×C89C2314yes—
47C2×C16×C89C2311yescharacteristic
48C32×C89C2311yescharacteristic
49C32×C89C2311yescharacteristic
50#13083574—42no—
51#13083574D811yescharacteristic
52#13083573—44no—
53#13083573—82no—
54C2×C22⋊D89—84no—
55D4×D89—44no—
56D4×D89—82no—
57Dic356—22no—
58#13083563—42no—
59#13083562—44no—
60D712—22no—
61#13083559—44no—
62#13083558D811yescharacteristic
63#13083557D812yes—
64C8×D89C2×D412yes—
65#13083549C2×D412yes—
66C2×D4×C89—42no—
67D8×C89—24no—
68D8×C89—42no—
69C2×C8×C89C2×D411yescharacteristic
70C16×C89C2×D411yescharacteristic
71C16×C89C2411yescharacteristic
72C23×C89—82no—
73C22×D89—84no—
74D4×C89—44no—
75D4×C89—82no—
76C2×C4×C89C2×D811yescharacteristic
77C22⋊D89—82no—
78C22⋊D89—84no—
79C22⋊D89—84no—
80C22⋊D89D1612yes—
81C2×Dic89—84no—
82D356—42no—
83D356D1611yescharacteristic
84C4×D89—82no—
85C4×D89C2×D812yes—
86Dic178—42no—
87Dic178D1611yescharacteristic
88C8×C89C2×D811yescharacteristic
89C8×C89C22×D411yescharacteristic
90#10623494C2×D812yes—
91C22×C89—84no—
92C22×C89—322no—
93C22×C89C2×D1611yescharacteristic
94C2×D89—82no—
95C2×D89—324no—
96C2×D89C2×D1612yes—
97C4×C89C2×D1611yescharacteristic
98C4×C89C22×D811yescharacteristic
99Dic89—82no—
100Dic89C2×D1612yes—
101#33982—891nomaximal, Sylow
102C2×C89—21no—
103C2×C89—164no—
104C2×C89C22×D1611yescharacteristic
105D89—22no—
106D89—162no—
107#3415—892no—
108#2263—892no—
109#2263—894no—
110#2262—892no—
111#2262—891no—
112C2×D32—892no—
113C2×D32D8911yescharacteristic, complemented
114#2258—891no—
115C89#3398211yescharacteristic, complemented, Sylow
116#576—1782no—
117#509—1784no—
118#509—1782no—
119#508—894no—
120#508—892no—
121C2×D16—894no—
122C2×D16C2×D8912yescomplemented
123#504—891no—
124Q64—891no—
125SD64—894no—
126SD64—892no—
127D32—894no—
128D32—891no—
129D32C2×D8912yescomplemented
130D32C2×D8912yescomplemented
131M64—891no—
132M64—892no—
133C2×C32—892no—
134C2×C32C2×D8911yescharacteristic, complemented
135H(5,𝔽2)—3562no—
136AGL(1,ℤ/8)—3564no—
137AGL(1,ℤ/8)—3562no—
138#135—1784no—
139#135—1782no—
140C2×D8—22no—
141C2×D8—1784no—
142#131—891no—
143Q32—892no—
144SD32—1784no—
145SD32—1782no—
146D16—22no—
147D16—892no—
148D16—1784no—
149D16C22×D8914yes—
150M32—891no—
151M32—892no—
152C2×C16—892no—
153C2×C16C22×D8911yescharacteristic
154C32—892no—
155C32C22×D8911yescharacteristic
156C32C22×D8911yescharacteristic
157#41—891no—
158#41—3564no—
159#41—3562no—
160#41—7122no—
161C2×D4—42no—
162C2×D4—3564no—
163C2×D4—7122no—
164C2×D4—7124no—
165Q16—1782no—
166SD16—3564no—
167SD16—3562no—
168D8—24no—
169D8—42no—
170D8—3564no—
171D8—1782no—
172M16—891no—
173M16—892no—
174C2×C8—892no—
175C2×C8D4×D8911yescharacteristic
176C16—892no—
177C16D4×D8911yescharacteristic
178C16C23×D8911yescharacteristic
179C23—82no—
180C23—7124no—
181Q8—891no—
182Q8—3562no—
183D4—44no—
184D4—82no—
185D4—892no—
186D4—891no—
187D4—3562no—
188D4—7124no—
189D4—7124no—
190D4—7122no—
191C2×C4—892no—
192C2×C4—7124no—
193C2×C4—7122no—
194C2×C4D8×D8911yescharacteristic
195C8—892no—
196C8D8×D8911yescharacteristic
197C8C2×D4×D8911yescharacteristic
198C22—84no—
199C22—322no—
200C22—28484no—
201C22—892no—
202C22—7122no—
203C22D16×D8911yescharacteristic
204C4—892no—
205C4—7122no—
206C4D16×D8911yescharacteristic
207C4C2×D8×D8911yescharacteristic
208C2—21no—
209C2—164no—
210C2—14242no—
211C2—1782no—
212C2C2×D16×D8911yescharacteristic
subgroupG/H
centerC2C2×D16×D89
commutator, cosocleC16×C89C24
FrattiniC16C23×D89
FittingC2×D32×C89C2
socleC2×C89C22×D16
hypercenterC2×D32D89
hypocenterC89#33982
LCSG ▷ C16×C89 ▷ C8×C89 ▷ C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ C2×D32

Complements

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

NH ≅ G/Ntype
1. #26033389C2semidirect
2. #26022565C2semidirect
3. #26022563C2semidirect
4. #26022561C2semidirect
5. #26022558C2semidirect
6. D32×D89C2semidirect
7. #26022545C2semidirect
8. C2×D32×C89C2semidirect
12. #19152030C22semidirect
13. #19152027C22semidirect
14. D16×D89C22semidirect
16. Dic1424C22semidirect
17. #19151437C22semidirect
18. #19151436C22semidirect
19. D2848C22semidirect
20. #19151433C22semidirect
21. #19151432C22semidirect
22. #19151431C22semidirect
23. C32×D89C22semidirect
24. C2×D16×C89C22semidirect
25. D32×C89C22semidirect
26. D32×C89C22semidirect
27. C2×C32×C89C22semidirect
113. C2×D32D89semidirect
115. C89#33982semidirect
122. C2×D16C2×D89semidirect
129. D32C2×D89semidirect
130. D32C2×D89semidirect
134. C2×C32C2×D89semidirect

Maximal subgroups

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

subgroupindex
#260333892
#260225652
#260225632
#260225612
#260225582
D32×D892
#260225452
C2×D32×C892
#3398289