Finite Group Databasebeta

← #44562165

Subgroups — #44562165

Order 32128.

476 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 125 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#30632492C212yesmaximal, complemented
2#30632489C212yesmaximal, complemented
3#30632064C211yesmaximal, characteristic, complemented
4#30631958C212yesmaximal, complemented
5#30631957C212yesmaximal, complemented
6#30631856C214yesmaximal, complemented
7#30631805C211yesmaximal, characteristic, complemented
8#443×C251C211yesmaximal, characteristic, complemented
9#22548761C2214yescomplemented
10#22548743—24no—
11#22548743C2214yescomplemented
12#22548742—24no—
13#22548740C2214yescomplemented
14#22548738—22no—
15#22548737—22no—
16#22548737C2211yescharacteristic
17C2×D2008—22no—
18C2×D2008C2212yescomplemented
19C2×D2008C2212yescomplemented
20D8×D251—22no—
21#22548721C2212yescomplemented
22#22548712C2212yescomplemented
23#22548651C2211yescharacteristic, complemented
24#22548643C2212yescomplemented
25#22548631C2214yescomplemented
26#135×C251C2211yescharacteristic
27#131×C251C2212yescomplemented
28#108×C251C2211yescharacteristic, complemented
29#104×C251C2212yescomplemented
30C4×C8×C251C2211yescharacteristic, complemented
31#16860811—24no—
32#16860810—24no—
33#16860810C2312yes—
34C2×D1004—24no—
35C2×D1004—42no—
36C2×D1004C2314yes—
37D4×D251—24no—
38D4×D251—44no—
39Dic1004D411yescharacteristic, complemented
40#16860799—24no—
41#16860799D412yescomplemented
42#16860797—24no—
43D2008—24no—
44D2008—22no—
45D2008—22no—
46D2008—22no—
47D2008—22no—
48D2008D411yescharacteristic, complemented
49#16860795—24no—
50#16860794C2312yes—
51#16860793—22no—
52C8×D251—22no—
53#41×C251C2312yes—
54Q16×C251D411yescharacteristic, complemented
55SD16×C251D412yescomplemented
56D8×C251D411yescharacteristic, complemented
57M16×C251—22no—
58M16×C251C2312yes—
59C2×C8×C251—22no—
60C2×C8×C251C2311yescharacteristic
61C2×C8×C251C2311yescharacteristic
62C42×C251C2311yescharacteristic
63C22×D251—24no—
64C22×D251—44no—
65Dic502C2×D412yes—
66D1004—24no—
67D1004—24no—
68D1004—44no—
69D1004—84no—
70D1004C2×D412yes—
71C22⋊D251—22no—
72C22⋊D251—44no—
73C4×D251—22no—
74C4×D251—44no—
75C251⋊C8—22no—
76Q8×C251C2×D412yes—
77D4×C251—22no—
78D4×C251C2×D412yes—
79C2×C4×C251—21no—
80C2×C4×C251—22no—
81C2×C4×C251C2411yescharacteristic
82C8×C251—22no—
83C8×C251C2×D411yescharacteristic
84C8×C251C2×D411yescharacteristic
85C8×C251C2×D411yescharacteristic
86C8×C251C2×D411yescharacteristic
87C22×C251—22no—
88C22×C251H(5,𝔽2)11yescharacteristic
89C2×D251—22no—
90C2×D251—44no—
91C2×D251—84no—
92C2×D251—164no—
93C4×C251—22no—
94C4×C251—22no—
95C4×C251C22×D411yescharacteristic
96C4×C251C22×D411yescharacteristic
97Dic251—22no—
98C2×C251—21no—
99C2×C251—42no—
100C2×C251D4211yescharacteristic
101D251—42no—
102D251—84no—
103C251#329211yescharacteristic, complemented, Sylow
104#3292—2511nomaximal, Sylow
105#576—2512no—
106#576—2512no—
107#493—2511no—
108#472—2512no—
109#472—2512no—
110#453—2514no—
111#443—2511no—
112#443D25111yescharacteristic, complemented
113H(5,𝔽2)—2514no—
114AGL(1,ℤ/8)—5024no—
115AGL(1,ℤ/8)—5024no—
116AGL(1,ℤ/8)—2514no—
117AGL(1,ℤ/8)—2514no—
118#135—5022no—
119#135—5022no—
120#135—2511no—
121#135C2×D25111yescharacteristic
122C2×D8—5022no—
123C2×D8—5022no—
124C2×D8—2512no—
125C2×D8—2512no—
126#131—2512no—
127#131C2×D25112yescomplemented
128#127—2512no—
129#108—2511no—
130#108C2×D25111yescharacteristic, complemented
131#104—2512no—
132#104C2×D25112yescomplemented
133#100—2514no—
134C4×C8C2×D25111yescharacteristic, complemented
135#41—5024no—
136#41—5024no—
137#41—2512no—
138#41C22×D25112yes—
139C2×D4—10042no—
140C2×D4—10044no—
141C2×D4—5024no—
142C2×D4—5024no—
143C2×D4—2514no—
144Q16—2511no—
145Q16D100411yescharacteristic, complemented
146SD16—5024no—
147SD16—5024no—
148SD16—2512no—
149SD16D100412yescomplemented
150D8—5024no—
151D8—5024no—
152D8—5022no—
153D8—5022no—
154D8—5022no—
155D8—5022no—
156D8—2511no—
157D8D100411yescharacteristic, complemented
158M16—22no—
159M16—5022no—
160M16—2512no—
161M16C22×D25112yes—
162C2×C8—22no—
163C2×C8—5022no—
164C2×C8C22×D25111yescharacteristic
165C2×C8C22×D25111yescharacteristic
166C42C22×D25111yescharacteristic
167C23—10044no—
168C23—5024no—
169Q8—2512no—
170Q8C2×D100412yes—
171D4—22no—
172D4—20084no—
173D4—10044no—
174D4—10044no—
175D4—5022no—
176D4—5024no—
177D4—5024no—
178D4—2512no—
179D4C2×D100412yes—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—10044no—
183C2×C4—5022no—
184C2×C4C23×D25111yescharacteristic
185C8—22no—
186C8—5022no—
187C8D4×D25111yescharacteristic
188C8D4×D25111yescharacteristic
189C8C2×D100411yescharacteristic
190C8C2×D100411yescharacteristic
191C22—22no—
192C22—40164no—
193C22—20084no—
194C22—10044no—
195C22—5022no—
196C22#2254876111yescharacteristic
197C4—22no—
198C4—22no—
199C4—5022no—
200C4C2×D4×D25111yescharacteristic
201C4C22×D100411yescharacteristic
202C2—21no—
203C2—42no—
204C2—20084no—
205C2—10042no—
206C2D4×D100411yescharacteristic
subgroupG/H
centerC2D4×D1004
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#443×C251C2
socleC2×C251D42
hypercenter#443D251
hypocenterC251#3292
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C2×C4 ◁ #443

Complements

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

NH ≅ G/Ntype
1. #30632492C2semidirect
2. #30632489C2semidirect
3. #30632064C2semidirect
4. #30631958C2semidirect
5. #30631957C2semidirect
6. #30631856C2semidirect
7. #30631805C2semidirect
8. #443×C251C2semidirect
9. #22548761C22semidirect
11. #22548743C22semidirect
13. #22548740C22semidirect
18. C2×D2008C22semidirect
19. C2×D2008C22semidirect
21. #22548721C22semidirect
22. #22548712C22semidirect
23. #22548651C22semidirect
24. #22548643C22semidirect
25. #22548631C22semidirect
27. #131×C251C22semidirect
28. #108×C251C22semidirect
29. #104×C251C22semidirect
30. C4×C8×C251C22semidirect
39. Dic1004D4semidirect
41. #16860799D4semidirect
48. D2008D4semidirect
54. Q16×C251D4semidirect
55. SD16×C251D4semidirect
56. D8×C251D4semidirect
103. C251#3292semidirect
112. #443D251semidirect
127. #131C2×D251semidirect
130. #108C2×D251semidirect
132. #104C2×D251semidirect
134. C4×C8C2×D251semidirect
145. Q16D1004semidirect
149. SD16D1004semidirect
157. D8D1004semidirect

Maximal subgroups

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

subgroupindex
#306324922
#306324892
#306320642
#306319582
#306319572
#306318562
#306318052
#443×C2512
#3292251