Finite Group Databasebeta

← #30677761

Subgroups — #30677761

Order 16768.

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#22554117C212yesmaximal, complemented
2#22554114C212yesmaximal, complemented
3#22553689C211yesmaximal, characteristic, complemented
4#22553583C212yesmaximal, complemented
5#22553582C212yesmaximal, complemented
6#22553481C214yesmaximal, complemented
7#22553430C211yesmaximal, characteristic, complemented
8#443×C131C211yesmaximal, characteristic, complemented
9#16862292C2214yescomplemented
10#16862274—24no—
11#16862274C2214yescomplemented
12#16862273—24no—
13#16862271C2214yescomplemented
14#16862269—22no—
15#16862268—22no—
16#16862268C2211yescharacteristic
17C2×D1048—22no—
18C2×D1048C2212yescomplemented
19C2×D1048C2212yescomplemented
20D8×D131—22no—
21#16862252C2212yescomplemented
22#16862243C2212yescomplemented
23#16862182C2211yescharacteristic, complemented
24#16862174C2212yescomplemented
25#16862162C2214yescomplemented
26#135×C131C2211yescharacteristic
27#131×C131C2212yescomplemented
28#108×C131C2211yescharacteristic, complemented
29#104×C131C2212yescomplemented
30C4×C8×C131C2211yescharacteristic, complemented
31#14523514—24no—
32#14523513—24no—
33#14523513C2312yes—
34C2×D524—24no—
35C2×D524—42no—
36C2×D524C2314yes—
37D4×D131—24no—
38D4×D131—44no—
39Dic524D411yescharacteristic, complemented
40#14523502—24no—
41#14523502D412yescomplemented
42#14523500—24no—
43D1048—24no—
44D1048—22no—
45D1048—22no—
46D1048—22no—
47D1048—22no—
48D1048D411yescharacteristic, complemented
49#14523498—24no—
50#14523497C2312yes—
51#14523496—22no—
52C8×D131—22no—
53#41×C131C2312yes—
54Q16×C131D411yescharacteristic, complemented
55SD16×C131D412yescomplemented
56D8×C131D411yescharacteristic, complemented
57M16×C131—22no—
58M16×C131C2312yes—
59C2×C8×C131—22no—
60C2×C8×C131C2311yescharacteristic
61C2×C8×C131C2311yescharacteristic
62C42×C131C2311yescharacteristic
63C22×D131—24no—
64C22×D131—44no—
65Dic262C2×D412yes—
66D524—24no—
67D524—24no—
68D524—44no—
69D524—84no—
70D524C2×D412yes—
71C22⋊D131—22no—
72C22⋊D131—44no—
73C4×D131—22no—
74C4×D131—44no—
75C131⋊C8—22no—
76Q8×C131C2×D412yes—
77D4×C131—22no—
78D4×C131C2×D412yes—
79C2×C4×C131—21no—
80C2×C4×C131—22no—
81C2×C4×C131C2411yescharacteristic
82C8×C131—22no—
83C8×C131C2×D411yescharacteristic
84C8×C131C2×D411yescharacteristic
85C8×C131C2×D411yescharacteristic
86C8×C131C2×D411yescharacteristic
87C22×C131—22no—
88C22×C131H(5,𝔽2)11yescharacteristic
89C2×D131—22no—
90C2×D131—44no—
91C2×D131—84no—
92C2×D131—164no—
93C4×C131—22no—
94C4×C131—22no—
95C4×C131C22×D411yescharacteristic
96C4×C131C22×D411yescharacteristic
97Dic131—22no—
98C2×C131—21no—
99C2×C131—42no—
100C2×C131D4211yescharacteristic
101D131—42no—
102D131—84no—
103C131#329211yescharacteristic, complemented, Sylow
104#3292—1311nomaximal, Sylow
105#576—1312no—
106#576—1312no—
107#493—1311no—
108#472—1312no—
109#472—1312no—
110#453—1314no—
111#443—1311no—
112#443D13111yescharacteristic, complemented
113H(5,𝔽2)—1314no—
114AGL(1,ℤ/8)—2624no—
115AGL(1,ℤ/8)—2624no—
116AGL(1,ℤ/8)—1314no—
117AGL(1,ℤ/8)—1314no—
118#135—2622no—
119#135—2622no—
120#135—1311no—
121#135C2×D13111yescharacteristic
122C2×D8—2622no—
123C2×D8—2622no—
124C2×D8—1312no—
125C2×D8—1312no—
126#131—1312no—
127#131C2×D13112yescomplemented
128#127—1312no—
129#108—1311no—
130#108C2×D13111yescharacteristic, complemented
131#104—1312no—
132#104C2×D13112yescomplemented
133#100—1314no—
134C4×C8C2×D13111yescharacteristic, complemented
135#41—2624no—
136#41—2624no—
137#41—1312no—
138#41C22×D13112yes—
139C2×D4—2624no—
140C2×D4—2624no—
141C2×D4—5242no—
142C2×D4—5244no—
143C2×D4—1314no—
144Q16—1311no—
145Q16D52411yescharacteristic, complemented
146SD16—2624no—
147SD16—2624no—
148SD16—1312no—
149SD16D52412yescomplemented
150D8—2624no—
151D8—2624no—
152D8—2622no—
153D8—2622no—
154D8—2622no—
155D8—2622no—
156D8—1311no—
157D8D52411yescharacteristic, complemented
158M16—22no—
159M16—2622no—
160M16—1312no—
161M16C22×D13112yes—
162C2×C8—22no—
163C2×C8—2622no—
164C2×C8C22×D13111yescharacteristic
165C2×C8C22×D13111yescharacteristic
166C42C22×D13111yescharacteristic
167C23—2624no—
168C23—5244no—
169Q8—1312no—
170Q8C2×D52412yes—
171D4—22no—
172D4—2622no—
173D4—2624no—
174D4—2624no—
175D4—5244no—
176D4—5244no—
177D4—10484no—
178D4—1312no—
179D4C2×D52412yes—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—2622no—
183C2×C4—5244no—
184C2×C4C23×D13111yescharacteristic
185C8—22no—
186C8—2622no—
187C8D4×D13111yescharacteristic
188C8D4×D13111yescharacteristic
189C8C2×D52411yescharacteristic
190C8C2×D52411yescharacteristic
191C22—22no—
192C22—2622no—
193C22—5244no—
194C22—10484no—
195C22—20964no—
196C22#1686229211yescharacteristic
197C4—22no—
198C4—22no—
199C4—2622no—
200C4C2×D4×D13111yescharacteristic
201C4C22×D52411yescharacteristic
202C2—21no—
203C2—42no—
204C2—5242no—
205C2—10484no—
206C2D4×D52411yescharacteristic
subgroupG/H
centerC2D4×D524
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#443×C131C2
socleC2×C131D42
hypercenter#443D131
hypocenterC131#3292
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C2 ◁ C2×C4 ◁ #443

Complements

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

NH ≅ G/Ntype
1. #22554117C2semidirect
2. #22554114C2semidirect
3. #22553689C2semidirect
4. #22553583C2semidirect
5. #22553582C2semidirect
6. #22553481C2semidirect
7. #22553430C2semidirect
8. #443×C131C2semidirect
9. #16862292C22semidirect
11. #16862274C22semidirect
13. #16862271C22semidirect
18. C2×D1048C22semidirect
19. C2×D1048C22semidirect
21. #16862252C22semidirect
22. #16862243C22semidirect
23. #16862182C22semidirect
24. #16862174C22semidirect
25. #16862162C22semidirect
27. #131×C131C22semidirect
28. #108×C131C22semidirect
29. #104×C131C22semidirect
30. C4×C8×C131C22semidirect
39. Dic524D4semidirect
41. #14523502D4semidirect
48. D1048D4semidirect
54. Q16×C131D4semidirect
55. SD16×C131D4semidirect
56. D8×C131D4semidirect
103. C131#3292semidirect
112. #443D131semidirect
127. #131C2×D131semidirect
130. #108C2×D131semidirect
132. #104C2×D131semidirect
134. C4×C8C2×D131semidirect
145. Q16D524semidirect
149. SD16D524semidirect
157. D8D524semidirect

Maximal subgroups

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

subgroupindex
#225541172
#225541142
#225536892
#225535832
#225535822
#225534812
#225534302
#443×C1312
#3292131