Finite Group Databasebeta

← #46820277

Subgroups — #46820277

Order 34432.

766 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 766 conjugacy classes of subgroups — too many to draw. Only its 161 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#31771112C213yesmaximal, complemented
2#31770795C214yesmaximal, complemented
3#31770792C214yesmaximal, complemented
4#31770579C213yesmaximal, complemented
5#585×C269C211yesmaximal, characteristic, complemented
6C22×C4×D269C2211yescharacteristic, complemented
7#22575434C2216yescomplemented
8#22575429—23no—
9#22575424—23no—
10#22575421C2216yescomplemented
11C4×C22⋊D269—26no—
12#22575412C2213yescomplemented
13#22575368C22112yescomplemented
142−1+4×C269—23no—
15H(5,𝔽2)×C269—23no—
16H(5,𝔽2)×C269C2214yes—
17C2×#41×C269—26no—
18C2×#41×C269C2213yes—
19C23×D269C2311yescharacteristic
20C2×C22⋊D269—26no—
21C22×Dic269C2311yescharacteristic
22C2×C4×D269—23no—
23C2×C4×D269—23no—
24C4⋊Dic269—23no—
25#16864381—26no—
26C22⋊Dic269—26no—
27C22⋊Dic269C2316yes—
28#16864379—26no—
29C4×Dic269—23no—
30#41×C269—24no—
31#41×C269—212no—
32#41×C269—26no—
33#41×C269—212no—
34#41×C269—26no—
35C2×Q8×C269—26no—
36C2×Q8×C269D413yescomplemented
37C2×D4×C269—26no—
38C2×D4×C269—212no—
39C2×D4×C269D413yescomplemented
40C2×D4×C269C2316yes—
41C22×C4×C269—24no—
42C22×C4×C269D416yescomplemented
43C22×C4×C269C2311yescharacteristic
44C22×D269—23no—
45C22×D269—42no—
46C22⋊D269—412no—
47C2×Dic269—26no—
48C2×Dic269—23no—
49C4×D269—43no—
50C4×D269—41no—
51C23×C269—24no—
52C23×C269C2×D416yes—
53C23×C269C2411yescharacteristic
54Q8×C269—24no—
55Q8×C269—26no—
56D4×C269—212no—
57D4×C269—212no—
58D4×C269—26no—
59C2×C4×C269—212no—
60C2×C4×C269—26no—
61C2×C4×C269—26no—
62C2×C4×C269C2×D416yes—
63C2×C4×C269C2×D413yes—
64C2×C4×C269C2×D413yes—
65C22×C269—26no—
66C22×C269—41no—
67C22×C269—46no—
68C22×C269—84no—
69C22×C269C22×D413yes—
70C2×D269—26no—
71C2×D269—43no—
72C2×D269—41no—
73C4×C269—26no—
74C4×C269#12013yes—
75C4×C269#12011yescharacteristic
76Dic269—46no—
77Dic269—41no—
78C2×C269—23no—
79C2×C269—46no—
80C2×C269C2×#12011yescharacteristic
81D269—42no—
82C269#302511yescharacteristic, complemented, Sylow
83#3025—2691nomaximal, Sylow
84#585D26911yescharacteristic, complemented
85#525—2693no—
86#458—2694no—
87B(4,𝔽2)—2694no—
88#410—2693no—
892−1+4—23no—
90H(5,𝔽2)—23no—
91H(5,𝔽2)C2×D26914yes—
92C2×#41—26no—
93C2×#41C2×D26913yes—
94C23×C4—2691no—
95#123—2696no—
96C22⋊Q8—5383no—
97#121—5383no—
98#120—2696no—
99C4×D4—5386no—
100#117—2693no—
101#99—26912no—
102C24—2691no—
103#41—24no—
104#41—212no—
105#41—26no—
106#41—212no—
107#41—26no—
108C2×Q8—26no—
109C2×Q8C22⋊D26913yescomplemented
110C2×D4—26no—
111C2×D4—212no—
112C2×D4—5386no—
113C2×D4C22⋊D26913yescomplemented
114C2×D4C22×D26916yes—
115C22×C4—24no—
116C22×C4—2691no—
117C22×C4—5383no—
118C22×C4—5383no—
119C22×C4C22⋊D26916yescomplemented
120C22×C4C22×D26911yescharacteristic
121C4⋊C4—5383no—
122C4⋊C4—5386no—
123C22⋊C4—2696no—
124C22⋊C4—5386no—
125C22⋊C4—5386no—
126C42—5383no—
127C23—24no—
128C23—5383no—
129C23—10762no—
130C23C2×C22⋊D26916yes—
131C23C23×D26911yescharacteristic
132Q8—24no—
133Q8—26no—
134D4—212no—
135D4—212no—
136D4—26no—
137D4—107612no—
138C2×C4—212no—
139C2×C4—26no—
140C2×C4—26no—
141C2×C4—5386no—
142C2×C4—5383no—
143C2×C4—10763no—
144C2×C4—10761no—
145C2×C4C2×C22⋊D26916yes—
146C2×C4C2×C22⋊D26913yes—
147C2×C4C2×C22⋊D26913yes—
148C22—26no—
149C22—41no—
150C22—46no—
151C22—84no—
152C22—5386no—
153C22—10763no—
154C22—10761no—
155C22C22×C22⋊D26913yes—
156C4—26no—
157C4—10766no—
158C4—10761no—
159C4#2257542213yes—
160C4#2257542211yescharacteristic
161C2—23no—
162C2—46no—
163C2—10762no—
164C2C2×#2257542211yescharacteristic
subgroupG/H
centerC4#22575422
commutator, cosocleC23×C269C24
FrattiniC23C23×D269
Fitting#585×C269C2
socleC2×C269C2×#120
hypercenter#585D269
hypocenterC269#3025
LCSG ▷ C23×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C4 ◁ C22×C4 ◁ #585

Complements

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

NH ≅ G/Ntype
1. #31771112C2semidirect
2. #31770795C2semidirect
3. #31770792C2semidirect
4. #31770579C2semidirect
5. #585×C269C2semidirect
6. C22×C4×D269C22semidirect
7. #22575434C22semidirect
10. #22575421C22semidirect
12. #22575412C22semidirect
13. #22575368C22semidirect
36. C2×Q8×C269D4semidirect
39. C2×D4×C269D4semidirect
42. C22×C4×C269D4semidirect
82. C269#3025semidirect
84. #585D269semidirect
109. C2×Q8C22⋊D269semidirect
113. C2×D4C22⋊D269semidirect
119. C22×C4C22⋊D269semidirect

Maximal subgroups

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

subgroupindex
#317711122
#317707952
#317707922
#317705792
#585×C2692
#3025269