Finite Group Databasebeta

← #36712576

Subgroups — #36712576

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#26033390C212yesmaximal, complemented
2#26022566C214yesmaximal, complemented
3#26022564C212yesmaximal, complemented
4#26022562C212yesmaximal, complemented
5C2×D2848C211yesmaximal, characteristic, complemented
6D32×D89C212yesmaximal, complemented
7#26022545C211yesmaximal, characteristic, complemented
8C89×#2262C211yesmaximal, characteristic, complemented
9#19152400—22no—
10#19152035—24no—
11#19152033—22no—
12#19152031C2214yescomplemented
13C2×D1424C2212yescomplemented
14D16×D89C2214yescomplemented
15#19152014C2211yescharacteristic
16#19151435C2214yescomplemented
17D2848C2212yescomplemented
18D2848C2212yescomplemented
19#19151433C2214yescomplemented
20#19151432C2211yescharacteristic, complemented
21#19151431C2212yescomplemented
22C32×D89C2212yescomplemented
23#508×C89C2212yescomplemented
24Q64×C89C2211yescharacteristic, complemented
25SD64×C89C2212yescomplemented
26D32×C89C2211yescharacteristic, complemented
27C2×C32×C89C2211yescharacteristic, complemented
28#15668058—42no—
29#15668039—44no—
30#15668037—42no—
31#15668035—24no—
32C2×D712—22no—
33D8×D89—24no—
34#15668018D411yescharacteristic
35#15667955—24no—
36D1424—22no—
37D1424C2314yes—
38#15667953—24no—
39#15667952D411yescharacteristic
40#15667951D412yes—
41C16×D89C2312yes—
42#15667918C2312yes—
43#135×C89—22no—
44Q32×C89C2312yes—
45SD32×C89—22no—
46D16×C89C2312yes—
47C2×C16×C89C2311yescharacteristic
48C32×C89C2311yescharacteristic
49C32×C89C2311yescharacteristic
50#13083575—44no—
51#13083574—82no—
52#13083574D811yescharacteristic
53C2×D356—42no—
54C2×D356—82no—
55D4×D89—44no—
56D4×D89—84no—
57#13083561—44no—
58D712—24no—
59D712—42no—
60#13083559—44no—
61#13083558D811yescharacteristic
62#13083557D812yes—
63C8×D89C2×D412yes—
64#13083549C2×D412yes—
65#41×C89—42no—
66Q16×C89—22no—
67SD16×C89—42no—
68D8×C89—22no—
69C2×C8×C89C2×D411yescharacteristic
70C16×C89C2×D411yescharacteristic
71C16×C89C2411yescharacteristic
72C22×D89—82no—
73C22×D89—84no—
74Q8×C89—42no—
75D4×C89—42no—
76D4×C89—82no—
77C2×C4×C89—82no—
78C2×C4×C89C2×D811yescharacteristic
79C22⋊D89—84no—
80C22⋊D89D1612yes—
81D356—44no—
82D356—84no—
83D356—82no—
84D356D1611yescharacteristic
85C4×D89—84no—
86C4×D89C2×D812yes—
87Dic178D1611yescharacteristic
88C8×C89C2×D811yescharacteristic
89C8×C89C22×D411yescharacteristic
90#10623494C2×D812yes—
91C22×C89—82no—
92C22×C89C2×D1611yescharacteristic
93C2×D89—84no—
94C2×D89—324no—
95C2×D89—322no—
96C2×D89C2×D1612yes—
97C4×C89—82no—
98C4×C89C2×D1611yescharacteristic
99C4×C89C22×D811yescharacteristic
100Dic89C2×D1612yes—
101#33982—891nomaximal, Sylow
102C2×C89—21no—
103C2×C89—162no—
104C2×C89C22×D1611yescharacteristic
105D89—22no—
106D89—164no—
107#3415—892no—
108#2263—892no—
109#2263—894no—
110#2262—892no—
111#2262D8911yescharacteristic, complemented
112C2×D32—892no—
113C2×D32—891no—
114#2258—891no—
115C89#3398211yescharacteristic, complemented, Sylow
116#576—1782no—
117#509—1784no—
118#509—1782no—
119#508—894no—
120#508C2×D8912yescomplemented
121C2×D16—894no—
122C2×D16—892no—
123#504—891no—
124Q64C2×D8911yescharacteristic, complemented
125SD64—894no—
126SD64C2×D8912yescomplemented
127D32—894no—
128D32—892no—
129D32—892no—
130D32C2×D8911yescharacteristic, complemented
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—22no—
139#135—1784no—
140C2×D8—1784no—
141C2×D8—1782no—
142#131—891no—
143Q32C22×D8912yes—
144SD32—22no—
145SD32—1784no—
146D16—894no—
147D16—1784no—
148D16—1782no—
149D16C22×D8912yes—
150M32—891no—
151M32—892no—
152C2×C16—892no—
153C2×C16C22×D8911yescharacteristic
154C32—892no—
155C32C22×D8911yescharacteristic
156C32C22×D8911yescharacteristic
157#41—42no—
158#41—891no—
159#41—3564no—
160#41—7122no—
161C2×D4—3564no—
162C2×D4—3562no—
163C2×D4—7122no—
164C2×D4—7124no—
165Q16—22no—
166SD16—42no—
167SD16—3564no—
168D8—22no—
169D8—3564no—
170D8—3562no—
171D8—1784no—
172M16—891no—
173M16—892no—
174C2×C8—892no—
175C2×C8D4×D8911yescharacteristic
176C16—892no—
177C16D4×D8911yescharacteristic
178C16C23×D8911yescharacteristic
179C23—7124no—
180C23—7122no—
181Q8—42no—
182Q8—891no—
183D4—42no—
184D4—82no—
185D4—892no—
186D4—891no—
187D4—3564no—
188D4—7124no—
189D4—7124no—
190D4—7122no—
191C2×C4—82no—
192C2×C4—892no—
193C2×C4—7124no—
194C2×C4D8×D8911yescharacteristic
195C8—892no—
196C8D8×D8911yescharacteristic
197C8C2×D4×D8911yescharacteristic
198C22—82no—
199C22—28482no—
200C22—28484no—
201C22—892no—
202C22—7124no—
203C22D16×D8911yescharacteristic
204C4—82no—
205C4—892no—
206C4D16×D8911yescharacteristic
207C4C2×D8×D8911yescharacteristic
208C2—21no—
209C2—162no—
210C2—14244no—
211C2—1782no—
212C2C2×D16×D8911yescharacteristic
subgroupG/H
centerC2C2×D16×D89
commutator, cosocleC16×C89C24
FrattiniC16C23×D89
FittingC89×#2262C2
socleC2×C89C22×D16
hypercenter#2262D89
hypocenterC89#33982
LCSG ▷ C16×C89 ▷ C8×C89 ▷ C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #2262

Complements

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

NH ≅ G/Ntype
1. #26033390C2semidirect
2. #26022566C2semidirect
3. #26022564C2semidirect
4. #26022562C2semidirect
5. C2×D2848C2semidirect
6. D32×D89C2semidirect
7. #26022545C2semidirect
8. C89×#2262C2semidirect
12. #19152031C22semidirect
13. C2×D1424C22semidirect
14. D16×D89C22semidirect
16. #19151435C22semidirect
17. D2848C22semidirect
18. D2848C22semidirect
19. #19151433C22semidirect
20. #19151432C22semidirect
21. #19151431C22semidirect
22. C32×D89C22semidirect
23. #508×C89C22semidirect
24. Q64×C89C22semidirect
25. SD64×C89C22semidirect
26. D32×C89C22semidirect
27. C2×C32×C89C22semidirect
111. #2262D89semidirect
115. C89#33982semidirect
120. #508C2×D89semidirect
124. Q64C2×D89semidirect
126. SD64C2×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
#260333902
#260225662
#260225642
#260225622
C2×D28482
D32×D892
#260225452
C89×#22622
#3398289