Finite Group Databasebeta

← #31792582

Subgroups — #31792582

Order 17536.

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#22578209C214yesmaximal, complemented
2#22577663C214yesmaximal, complemented
3#22577585C212yesmaximal, complemented
4#22577570C212yesmaximal, complemented
5#22577521C212yesmaximal, complemented
6#492×C137C211yesmaximal, characteristic, complemented
7#17984136C2212yes—
8#17984133C2212yescomplemented
9#17984119C2214yescomplemented
10#17984117—24no—
11#17984115—24no—
12#17984114C2214yes—
13#17984110C2212yes—
14#17984109—24no—
15SD16×D137—24no—
16#17984094C2214yescomplemented
17#17984024C2212yescomplemented
18#17984016C2214yescomplemented
19#17984007C2212yescomplemented
20#17984005C2212yescomplemented
21C2×SD16×C137C2214yescomplemented
22C4⋊Q8×C137C2211yescharacteristic, complemented
23#127×C137C2211yescharacteristic
24C4×C8×C137C2211yescharacteristic, complemented
25#14555511—24no—
26#14555510—42no—
27#14555510C2314yes—
28#14555509—24no—
29Q8×D137—24no—
30C2×C22⋊D137—42no—
31D4×D137—24no—
32Dic548D412yescomplemented
33#14555500—24no—
34#14555499D414yescomplemented
35#14555498—24no—
36#14555497—24no—
37D1096D412yescomplemented
38#14555495—24no—
39#14555494C2314yes—
40#14555493—24no—
41C8×D137—24no—
42C2×Q8×C137C2312yes—
43C2×D4×C137—41no—
44C2×D4×C137C2312yes—
45SD16×C137—24no—
46SD16×C137—24no—
47C2×C8×C137C2312yes—
48C4⋊C4×C137—41no—
49C42×C137C2311yescharacteristic
50C22×D137—42no—
51Dic274—42no—
52Dic274C2×D414yes—
53D548—42no—
54D548C2×D414yes—
55C22⋊D137—24no—
56C22⋊D137—44no—
57C2×Dic137—42no—
58C4×D137—24no—
59C4×D137—44no—
60#11764040—24no—
61C23×C137—22no—
62Q8×C137—24no—
63D4×C137—24no—
64D4×C137—82no—
65C2×C4×C137—21no—
66C2×C4×C137—22no—
67C2×C4×C137C2411yescharacteristic
68C8×C137C2×D412yes—
69C8×C137C2×D412yes—
70C22×C137—42no—
71C22×C137—82no—
72C22×C137H(5,𝔽2)11yescharacteristic
73C2×D137—24no—
74C2×D137—162no—
75C4×C137—22no—
76C4×C137—42no—
77C4×C137C22×D412yes—
78Dic137—24no—
79C2×C137—21no—
80C2×C137—82no—
81C2×C137D4211yescharacteristic
82D137—44no—
83C137#329111yescharacteristic, complemented, Sylow
84#3291—1371nomaximal, Sylow
85#577—1374no—
86#492D13711yescharacteristic, complemented
87#471—1374no—
88#456—1372no—
89#453—1372no—
90#443—1372no—
912−1+4—1372no—
92H(5,𝔽2)—1372no—
93#137—2744no—
94#137—1374no—
95AGL(1,ℤ/8)—2744no—
96AGL(1,ℤ/8)—1374no—
97#135—2744no—
98#135—1372no—
99C2×SD16—2744no—
100C2×SD16C2×D13714yescomplemented
101#131—1374no—
102C4⋊Q8C2×D13711yescharacteristic, complemented
103#127C2×D13711yescharacteristic
104#108—1372no—
105#104—1374no—
106#101—1372no—
107#100—1372no—
108C4×C8C2×D13711yescharacteristic, complemented
109#41—2744no—
110#41—2744no—
111#41—5482no—
112#41—1374no—
113C2×Q8—2744no—
114C2×Q8C22×D13712yes—
115C2×D4—41no—
116C2×D4—2744no—
117C2×D4—5482no—
118C2×D4C22×D13712yes—
119Q16—2744no—
120Q16—1372no—
121SD16—24no—
122SD16—24no—
123SD16—2744no—
124SD16—2744no—
125SD16—1374no—
126D8—2744no—
127D8—1372no—
128M16—2744no—
129M16—1374no—
130C2×C8—2744no—
131C2×C8C22×D13712yes—
132C4⋊C4—41no—
133C42C22×D13711yescharacteristic
134C23—22no—
135C23—5482no—
136Q8—24no—
137Q8—5482no—
138Q8—1374no—
139D4—24no—
140D4—82no—
141D4—2744no—
142D4—5482no—
143D4—5484no—
144D4—1374no—
145C2×C4—21no—
146C2×C4—22no—
147C2×C4—2744no—
148C2×C4—5484no—
149C2×C4—5482no—
150C2×C4C23×D13711yescharacteristic
151C8—2744no—
152C8D4×D13712yes—
153C8D4×D13712yes—
154C22—42no—
155C22—82no—
156C22—2744no—
157C22—21922no—
158C22#1798413311yescharacteristic
159C4—22no—
160C4—42no—
161C4—2744no—
162C4C2×D4×D13712yes—
163C2—21no—
164C2—82no—
165C2—5484no—
166C2#2257803811yescharacteristic
subgroupG/H
centerC2#22578038
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#492×C137C2
socleC2×C137D42
hypercenter#492D137
hypocenterC137#3291
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ #492

Complements

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

NH ≅ G/Ntype
1. #22578209C2semidirect
2. #22577663C2semidirect
3. #22577585C2semidirect
4. #22577570C2semidirect
5. #22577521C2semidirect
6. #492×C137C2semidirect
8. #17984133C22semidirect
9. #17984119C22semidirect
16. #17984094C22semidirect
17. #17984024C22semidirect
18. #17984016C22semidirect
19. #17984007C22semidirect
20. #17984005C22semidirect
21. C2×SD16×C137C22semidirect
22. C4⋊Q8×C137C22semidirect
24. C4×C8×C137C22semidirect
32. Dic548D4semidirect
34. #14555499D4semidirect
37. D1096D4semidirect
83. C137#3291semidirect
86. #492D137semidirect
100. C2×SD16C2×D137semidirect
102. C4⋊Q8C2×D137semidirect
108. C4×C8C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225782092
#225776632
#225775852
#225775702
#225775212
#492×C1372
#3291137