Finite Group Databasebeta

← #31792589

Subgroups — #31792589

Order 17536.

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#22578208C212yesmaximal, complemented
2#22578205C212yesmaximal, complemented
3#22577780C211yesmaximal, characteristic, complemented
4#22577674C212yesmaximal, complemented
5#22577673C212yesmaximal, complemented
6#22577572C214yesmaximal, complemented
7#22577521C211yesmaximal, characteristic, complemented
8#443×C137C211yesmaximal, characteristic, complemented
9#17984134C2214yescomplemented
10#17984116—24no—
11#17984116C2214yescomplemented
12#17984115—24no—
13#17984113C2214yescomplemented
14#17984111—22no—
15#17984110—22no—
16#17984110C2211yescharacteristic
17C2×D1096—22no—
18C2×D1096C2212yescomplemented
19C2×D1096C2212yescomplemented
20D8×D137—22no—
21#17984094C2212yescomplemented
22#17984085C2212yescomplemented
23#17984024C2211yescharacteristic, complemented
24#17984016C2212yescomplemented
25#17984004C2214yescomplemented
26#135×C137C2211yescharacteristic
27#131×C137C2212yescomplemented
28#108×C137C2211yescharacteristic, complemented
29#104×C137C2212yescomplemented
30C4×C8×C137C2211yescharacteristic, complemented
31#14555511—24no—
32#14555510—24no—
33#14555510C2312yes—
34C2×D548—24no—
35C2×D548—42no—
36C2×D548C2314yes—
37D4×D137—24no—
38D4×D137—44no—
39Dic548D411yescharacteristic, complemented
40#14555499—24no—
41#14555499D412yescomplemented
42#14555497—24no—
43D1096—24no—
44D1096—22no—
45D1096—22no—
46D1096—22no—
47D1096—22no—
48D1096D411yescharacteristic, complemented
49#14555495—24no—
50#14555494C2312yes—
51#14555493—22no—
52C8×D137—22no—
53#41×C137C2312yes—
54Q16×C137D411yescharacteristic, complemented
55SD16×C137D412yescomplemented
56D8×C137D411yescharacteristic, complemented
57M16×C137—22no—
58M16×C137C2312yes—
59C2×C8×C137—22no—
60C2×C8×C137C2311yescharacteristic
61C2×C8×C137C2311yescharacteristic
62C42×C137C2311yescharacteristic
63C22×D137—24no—
64C22×D137—44no—
65Dic274C2×D412yes—
66D548—24no—
67D548—24no—
68D548—44no—
69D548—84no—
70D548C2×D412yes—
71C22⋊D137—22no—
72C22⋊D137—44no—
73C4×D137—22no—
74C4×D137—44no—
75#11764040—22no—
76Q8×C137C2×D412yes—
77D4×C137—22no—
78D4×C137C2×D412yes—
79C2×C4×C137—21no—
80C2×C4×C137—22no—
81C2×C4×C137C2411yescharacteristic
82C8×C137—22no—
83C8×C137C2×D411yescharacteristic
84C8×C137C2×D411yescharacteristic
85C8×C137C2×D411yescharacteristic
86C8×C137C2×D411yescharacteristic
87C22×C137—22no—
88C22×C137H(5,𝔽2)11yescharacteristic
89C2×D137—22no—
90C2×D137—44no—
91C2×D137—84no—
92C2×D137—164no—
93C4×C137—22no—
94C4×C137—22no—
95C4×C137C22×D411yescharacteristic
96C4×C137C22×D411yescharacteristic
97Dic137—22no—
98C2×C137—21no—
99C2×C137—42no—
100C2×C137D4211yescharacteristic
101D137—42no—
102D137—84no—
103C137#329211yescharacteristic, complemented, Sylow
104#3292—1371nomaximal, Sylow
105#576—1372no—
106#576—1372no—
107#493—1371no—
108#472—1372no—
109#472—1372no—
110#453—1374no—
111#443—1371no—
112#443D13711yescharacteristic, complemented
113H(5,𝔽2)—1374no—
114AGL(1,ℤ/8)—2744no—
115AGL(1,ℤ/8)—2744no—
116AGL(1,ℤ/8)—1374no—
117AGL(1,ℤ/8)—1374no—
118#135—2742no—
119#135—2742no—
120#135—1371no—
121#135C2×D13711yescharacteristic
122C2×D8—2742no—
123C2×D8—2742no—
124C2×D8—1372no—
125C2×D8—1372no—
126#131—1372no—
127#131C2×D13712yescomplemented
128#127—1372no—
129#108—1371no—
130#108C2×D13711yescharacteristic, complemented
131#104—1372no—
132#104C2×D13712yescomplemented
133#100—1374no—
134C4×C8C2×D13711yescharacteristic, complemented
135#41—2744no—
136#41—2744no—
137#41—1372no—
138#41C22×D13712yes—
139C2×D4—2744no—
140C2×D4—2744no—
141C2×D4—5482no—
142C2×D4—5484no—
143C2×D4—1374no—
144Q16—1371no—
145Q16D54811yescharacteristic, complemented
146SD16—2744no—
147SD16—2744no—
148SD16—1372no—
149SD16D54812yescomplemented
150D8—2744no—
151D8—2744no—
152D8—2742no—
153D8—2742no—
154D8—2742no—
155D8—2742no—
156D8—1371no—
157D8D54811yescharacteristic, complemented
158M16—22no—
159M16—2742no—
160M16—1372no—
161M16C22×D13712yes—
162C2×C8—22no—
163C2×C8—2742no—
164C2×C8C22×D13711yescharacteristic
165C2×C8C22×D13711yescharacteristic
166C42C22×D13711yescharacteristic
167C23—2744no—
168C23—5484no—
169Q8—1372no—
170Q8C2×D54812yes—
171D4—22no—
172D4—2742no—
173D4—2744no—
174D4—2744no—
175D4—5484no—
176D4—5484no—
177D4—10964no—
178D4—1372no—
179D4C2×D54812yes—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—2742no—
183C2×C4—5484no—
184C2×C4C23×D13711yescharacteristic
185C8—22no—
186C8—2742no—
187C8D4×D13711yescharacteristic
188C8D4×D13711yescharacteristic
189C8C2×D54811yescharacteristic
190C8C2×D54811yescharacteristic
191C22—22no—
192C22—2742no—
193C22—5484no—
194C22—10964no—
195C22—21924no—
196C22#1798413411yescharacteristic
197C4—22no—
198C4—22no—
199C4—2742no—
200C4C2×D4×D13711yescharacteristic
201C4C22×D54811yescharacteristic
202C2—21no—
203C2—42no—
204C2—5482no—
205C2—10964no—
206C2D4×D54811yescharacteristic
subgroupG/H
centerC2D4×D548
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#443×C137C2
socleC2×C137D42
hypercenter#443D137
hypocenterC137#3292
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ #443

Complements

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

NH ≅ G/Ntype
1. #22578208C2semidirect
2. #22578205C2semidirect
3. #22577780C2semidirect
4. #22577674C2semidirect
5. #22577673C2semidirect
6. #22577572C2semidirect
7. #22577521C2semidirect
8. #443×C137C2semidirect
9. #17984134C22semidirect
11. #17984116C22semidirect
13. #17984113C22semidirect
18. C2×D1096C22semidirect
19. C2×D1096C22semidirect
21. #17984094C22semidirect
22. #17984085C22semidirect
23. #17984024C22semidirect
24. #17984016C22semidirect
25. #17984004C22semidirect
27. #131×C137C22semidirect
28. #108×C137C22semidirect
29. #104×C137C22semidirect
30. C4×C8×C137C22semidirect
39. Dic548D4semidirect
41. #14555499D4semidirect
48. D1096D4semidirect
54. Q16×C137D4semidirect
55. SD16×C137D4semidirect
56. D8×C137D4semidirect
103. C137#3292semidirect
112. #443D137semidirect
127. #131C2×D137semidirect
130. #108C2×D137semidirect
132. #104C2×D137semidirect
134. C4×C8C2×D137semidirect
145. Q16D548semidirect
149. SD16D548semidirect
157. D8D548semidirect

Maximal subgroups

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

subgroupindex
#225782082
#225782052
#225777802
#225776742
#225776732
#225775722
#225775212
#443×C1372
#3292137