Finite Group Databasebeta

← #30677754

Subgroups — #30677754

Order 16768.

460 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 117 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#22554118C214yesmaximal, complemented
2#22553572C214yesmaximal, complemented
3#22553494C212yesmaximal, complemented
4#22553479C212yesmaximal, complemented
5#22553430C212yesmaximal, complemented
6#492×C131C211yesmaximal, characteristic, complemented
7#16862294C2212yes—
8#16862291C2212yescomplemented
9#16862277C2214yescomplemented
10#16862275—24no—
11#16862273—24no—
12#16862272C2214yes—
13#16862268C2212yes—
14#16862267—24no—
15SD16×D131—24no—
16#16862252C2214yescomplemented
17#16862182C2212yescomplemented
18#16862174C2214yescomplemented
19#16862165C2212yescomplemented
20#16862163C2212yescomplemented
21C2×SD16×C131C2214yescomplemented
22C4⋊Q8×C131C2211yescharacteristic, complemented
23#127×C131C2211yescharacteristic
24C4×C8×C131C2211yescharacteristic, complemented
25#14523514—24no—
26#14523513—42no—
27#14523513C2314yes—
28#14523512—24no—
29Q8×D131—24no—
30C2×C22⋊D131—42no—
31D4×D131—24no—
32Dic524D412yescomplemented
33#14523503—24no—
34#14523502D414yescomplemented
35#14523501—24no—
36#14523500—24no—
37D1048D412yescomplemented
38#14523498—24no—
39#14523497C2314yes—
40#14523496—24no—
41C8×D131—24no—
42C2×Q8×C131C2312yes—
43C2×D4×C131—41no—
44C2×D4×C131C2312yes—
45SD16×C131—24no—
46SD16×C131—24no—
47C2×C8×C131C2312yes—
48C4⋊C4×C131—41no—
49C42×C131C2311yescharacteristic
50C22×D131—42no—
51Dic262—42no—
52Dic262C2×D414yes—
53D524—42no—
54D524C2×D414yes—
55C22⋊D131—24no—
56C22⋊D131—44no—
57C2×Dic131—42no—
58C4×D131—24no—
59C4×D131—44no—
60C131⋊C8—24no—
61C23×C131—22no—
62Q8×C131—24no—
63D4×C131—24no—
64D4×C131—82no—
65C2×C4×C131—21no—
66C2×C4×C131—22no—
67C2×C4×C131C2411yescharacteristic
68C8×C131C2×D412yes—
69C8×C131C2×D412yes—
70C22×C131—42no—
71C22×C131—82no—
72C22×C131H(5,𝔽2)11yescharacteristic
73C2×D131—24no—
74C2×D131—162no—
75C4×C131—22no—
76C4×C131—42no—
77C4×C131C22×D412yes—
78Dic131—24no—
79C2×C131—21no—
80C2×C131—82no—
81C2×C131D4211yescharacteristic
82D131—44no—
83C131#329111yescharacteristic, complemented, Sylow
84#3291—1311nomaximal, Sylow
85#577—1314no—
86#492D13111yescharacteristic, complemented
87#471—1314no—
88#456—1312no—
89#453—1312no—
90#443—1312no—
912−1+4—1312no—
92H(5,𝔽2)—1312no—
93#137—2624no—
94#137—1314no—
95AGL(1,ℤ/8)—2624no—
96AGL(1,ℤ/8)—1314no—
97#135—2624no—
98#135—1312no—
99C2×SD16—2624no—
100C2×SD16C2×D13114yescomplemented
101#131—1314no—
102C4⋊Q8C2×D13111yescharacteristic, complemented
103#127C2×D13111yescharacteristic
104#108—1312no—
105#104—1314no—
106#101—1312no—
107#100—1312no—
108C4×C8C2×D13111yescharacteristic, complemented
109#41—2624no—
110#41—2624no—
111#41—5242no—
112#41—1314no—
113C2×Q8—2624no—
114C2×Q8C22×D13112yes—
115C2×D4—41no—
116C2×D4—2624no—
117C2×D4—5242no—
118C2×D4C22×D13112yes—
119Q16—2624no—
120Q16—1312no—
121SD16—24no—
122SD16—24no—
123SD16—2624no—
124SD16—2624no—
125SD16—1314no—
126D8—2624no—
127D8—1312no—
128M16—2624no—
129M16—1314no—
130C2×C8—2624no—
131C2×C8C22×D13112yes—
132C4⋊C4—41no—
133C42C22×D13111yescharacteristic
134C23—22no—
135C23—5242no—
136Q8—24no—
137Q8—5242no—
138Q8—1314no—
139D4—24no—
140D4—2624no—
141D4—82no—
142D4—5242no—
143D4—5244no—
144D4—1314no—
145C2×C4—21no—
146C2×C4—22no—
147C2×C4—2624no—
148C2×C4—5244no—
149C2×C4—5242no—
150C2×C4C23×D13111yescharacteristic
151C8—2624no—
152C8D4×D13112yes—
153C8D4×D13112yes—
154C22—42no—
155C22—2624no—
156C22—82no—
157C22—20962no—
158C22#1686229111yescharacteristic
159C4—22no—
160C4—42no—
161C4—2624no—
162C4C2×D4×D13112yes—
163C2—21no—
164C2—82no—
165C2—5244no—
166C2#2255394711yescharacteristic
subgroupG/H
centerC2#22553947
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#492×C131C2
socleC2×C131D42
hypercenter#492D131
hypocenterC131#3291
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C2 ◁ C2×C4 ◁ #492

Complements

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

NH ≅ G/Ntype
1. #22554118C2semidirect
2. #22553572C2semidirect
3. #22553494C2semidirect
4. #22553479C2semidirect
5. #22553430C2semidirect
6. #492×C131C2semidirect
8. #16862291C22semidirect
9. #16862277C22semidirect
16. #16862252C22semidirect
17. #16862182C22semidirect
18. #16862174C22semidirect
19. #16862165C22semidirect
20. #16862163C22semidirect
21. C2×SD16×C131C22semidirect
22. C4⋊Q8×C131C22semidirect
24. C4×C8×C131C22semidirect
32. Dic524D4semidirect
34. #14523502D4semidirect
37. D1048D4semidirect
83. C131#3291semidirect
86. #492D131semidirect
100. C2×SD16C2×D131semidirect
102. C4⋊Q8C2×D131semidirect
108. C4×C8C2×D131semidirect

Maximal subgroups

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

subgroupindex
#225541182
#225535722
#225534942
#225534792
#225534302
#492×C1312
#3291131