Finite Group Databasebeta

← #46848157

Subgroups — #46848157

Order 34688.

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#31774703C214yesmaximal, complemented
2#31774157C214yesmaximal, complemented
3#31774079C212yesmaximal, complemented
4#31774064C212yesmaximal, complemented
5#31774015C212yesmaximal, complemented
6#492×C271C211yesmaximal, characteristic, complemented
7#22576208C2212yes—
8#22576205C2212yescomplemented
9#22576191C2214yescomplemented
10#22576189—24no—
11#22576187—24no—
12#22576186C2214yes—
13#22576182C2212yes—
14#22576181—24no—
15SD16×D271—24no—
16#22576166C2214yescomplemented
17#22576096C2212yescomplemented
18#22576088C2214yescomplemented
19#22576079C2212yescomplemented
20#22576077C2212yescomplemented
21C2×SD16×C271C2214yescomplemented
22C4⋊Q8×C271C2211yescharacteristic, complemented
23#127×C271C2211yescharacteristic
24C4×C8×C271C2211yescharacteristic, complemented
25#16864655—24no—
26#16864654—42no—
27#16864654C2314yes—
28#16864653—24no—
29Q8×D271—24no—
30C2×C22⋊D271—42no—
31D4×D271—24no—
32Dic1084D412yescomplemented
33#16864644—24no—
34#16864643D414yescomplemented
35#16864642—24no—
36#16864641—24no—
37D2168D412yescomplemented
38#16864639—24no—
39#16864638C2314yes—
40#16864637—24no—
41C8×D271—24no—
42C2×Q8×C271C2312yes—
43C2×D4×C271—41no—
44C2×D4×C271C2312yes—
45SD16×C271—24no—
46SD16×C271—24no—
47C2×C8×C271C2312yes—
48C4⋊C4×C271—41no—
49C42×C271C2311yescharacteristic
50C22×D271—42no—
51Dic542—42no—
52Dic542C2×D414yes—
53D1084—42no—
54D1084C2×D414yes—
55C22⋊D271—24no—
56C22⋊D271—44no—
57C2×Dic271—42no—
58C4×D271—24no—
59C4×D271—44no—
60C271⋊C8—24no—
61C23×C271—22no—
62Q8×C271—24no—
63D4×C271—24no—
64D4×C271—82no—
65C2×C4×C271—21no—
66C2×C4×C271—22no—
67C2×C4×C271C2411yescharacteristic
68C8×C271C2×D412yes—
69C8×C271C2×D412yes—
70C22×C271—42no—
71C22×C271—82no—
72C22×C271H(5,𝔽2)11yescharacteristic
73C2×D271—24no—
74C2×D271—162no—
75C4×C271—22no—
76C4×C271—42no—
77C4×C271C22×D412yes—
78Dic271—24no—
79C2×C271—21no—
80C2×C271—82no—
81C2×C271D4211yescharacteristic
82D271—44no—
83C271#329111yescharacteristic, complemented, Sylow
84#3291—2711nomaximal, Sylow
85#577—2714no—
86#492D27111yescharacteristic, complemented
87#471—2714no—
88#456—2712no—
89#453—2712no—
90#443—2712no—
912−1+4—2712no—
92H(5,𝔽2)—2712no—
93#137—2714no—
94#137—5424no—
95AGL(1,ℤ/8)—2714no—
96AGL(1,ℤ/8)—5424no—
97#135—2712no—
98#135—5424no—
99C2×SD16—5424no—
100C2×SD16C2×D27114yescomplemented
101#131—2714no—
102C4⋊Q8C2×D27111yescharacteristic, complemented
103#127C2×D27111yescharacteristic
104#108—2712no—
105#104—2714no—
106#101—2712no—
107#100—2712no—
108C4×C8C2×D27111yescharacteristic, complemented
109#41—2714no—
110#41—5424no—
111#41—5424no—
112#41—10842no—
113C2×Q8—5424no—
114C2×Q8C22×D27112yes—
115C2×D4—41no—
116C2×D4—5424no—
117C2×D4—10842no—
118C2×D4C22×D27112yes—
119Q16—2712no—
120Q16—5424no—
121SD16—24no—
122SD16—24no—
123SD16—2714no—
124SD16—5424no—
125SD16—5424no—
126D8—2712no—
127D8—5424no—
128M16—2714no—
129M16—5424no—
130C2×C8—5424no—
131C2×C8C22×D27112yes—
132C4⋊C4—41no—
133C42C22×D27111yescharacteristic
134C23—22no—
135C23—10842no—
136Q8—24no—
137Q8—2714no—
138Q8—10842no—
139D4—24no—
140D4—82no—
141D4—2714no—
142D4—5424no—
143D4—10842no—
144D4—10844no—
145C2×C4—21no—
146C2×C4—22no—
147C2×C4—5424no—
148C2×C4—10844no—
149C2×C4—10842no—
150C2×C4C23×D27111yescharacteristic
151C8—5424no—
152C8D4×D27112yes—
153C8D4×D27112yes—
154C22—42no—
155C22—82no—
156C22—5424no—
157C22—43362no—
158C22#2257620511yescharacteristic
159C4—22no—
160C4—42no—
161C4—5424no—
162C4C2×D4×D27112yes—
163C2—21no—
164C2—82no—
165C2—10844no—
166C2#3177453211yescharacteristic
subgroupG/H
centerC2#31774532
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#492×C271C2
socleC2×C271D42
hypercenter#492D271
hypocenterC271#3291
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C2 ◁ C2×C4 ◁ #492

Complements

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

NH ≅ G/Ntype
1. #31774703C2semidirect
2. #31774157C2semidirect
3. #31774079C2semidirect
4. #31774064C2semidirect
5. #31774015C2semidirect
6. #492×C271C2semidirect
8. #22576205C22semidirect
9. #22576191C22semidirect
16. #22576166C22semidirect
17. #22576096C22semidirect
18. #22576088C22semidirect
19. #22576079C22semidirect
20. #22576077C22semidirect
21. C2×SD16×C271C22semidirect
22. C4⋊Q8×C271C22semidirect
24. C4×C8×C271C22semidirect
32. Dic1084D4semidirect
34. #16864643D4semidirect
37. D2168D4semidirect
83. C271#3291semidirect
86. #492D271semidirect
100. C2×SD16C2×D271semidirect
102. C4⋊Q8C2×D271semidirect
108. C4×C8C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317747032
#317741572
#317740792
#317740642
#317740152
#492×C2712
#3291271