Finite Group Databasebeta

← #46819345

Subgroups — #46819345

Order 34432.

800 conjugacy classes of subgroups; only the 249 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 249 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
1C2×#22575498C211yesmaximal, characteristic
2#31771080C211yesmaximal, characteristic, complemented
3#31770578C214yesmaximal, complemented
4C2×#22575367C218yesmaximal, complemented
5#518×C269C211yesmaximal, characteristic, complemented
6#22575498—24no—
7#22575498—24no—
8C2×#16864402C2212yes—
9C2×#16864401C414yescomplemented
10C22×C22⋊D269C2214yes—
11C2×D4×D269C414yescomplemented
12D4×Dic269—24no—
13#22575412C2212yescomplemented
14C2×C22⋊Dic269C2214yescomplemented
15#22575367C22116yescomplemented
16C22×D4×C269C2211yescharacteristic
17C4×D4×C269—24no—
18#117×C269C2212yescomplemented
19C2×C22⋊C4×C269C2214yescomplemented
20C23×D269C2×C414yescomplemented
21#16864402—24no—
22#16864402—24no—
23#16864401—216no—
24C2×Dic538C2×C412yescomplemented
25C2×D1076C2×C412yescomplemented
26C2×C22⋊D269—28no—
27C2×C22⋊D269—28no—
28C2×C22⋊D269C2×C418yescomplemented
29C2×C22⋊D269C2314yes—
30D4×D269—216no—
31C22×Dic269—24no—
32C22×Dic269C2×C414yescomplemented
33C2×C4×D269C2×C414yescomplemented
34C4⋊Dic269—22no—
35C22⋊Dic269—24no—
36C22⋊Dic269C2314yes—
37C4×Dic269—22no—
38C24×C269C2312yes—
39C2×D4×C269C2×C414yes—
40C2×D4×C269D414yes—
41C2×D4×C269D414yes—
42C22×C4×C269—24no—
43C22×C4×C269C2311yescharacteristic
44C4⋊C4×C269—22no—
45C22⋊C4×C269—24no—
46C22⋊C4×C269C2314yes—
47C42×C269—22no—
48C22×D269—28no—
49C22×D269—84no—
50C22×D269C22×C414yes—
51Dic538—24no—
52D1076—24no—
53C22⋊D269—216no—
54C22⋊D269—24no—
55C22⋊D269—24no—
56C2×Dic269—28no—
57C2×Dic269—24no—
58C2×Dic269—84no—
59C2×Dic269C22×C414yes—
60C4×D269—28no—
61C23×C269—82no—
62C23×C269C22×C414yes—
63C23×C269C2×D414yes—
64C23×C269C2×D414yes—
65C23×C269C2411yescharacteristic
66D4×C269C22⋊C4116yes—
67C2×C4×C269—24no—
68C2×C4×C269—84no—
69C2×C4×C269C22×C412yes—
70C2×C4×C269C2×D412yes—
71C2×C4×C269C2×D412yes—
72C22×C269—44no—
73C22×C269—41no—
74C22×C269—44no—
75C22×C269—44no—
76C22×C269C2×C22⋊C418yes—
77C22×C269C23×C411yescharacteristic
78C22×C269C22×D411yescharacteristic
79C22×C269C22×D411yescharacteristic
80C2×D269—24no—
81C2×D269—44no—
82C2×D269—88no—
83C4×C269—44no—
84C4×C269C2×C22⋊C414yes—
85Dic269—24no—
86Dic269—44no—
87C2×C269—21no—
88C2×C269—21no—
89C2×C269—21no—
90C2×C269—28no—
91C2×C269C22×C22⋊C411yescharacteristic
92D269—44no—
93C269#288311yescharacteristic, complemented, Sylow
94#2883—2691nomaximal, Sylow
95C2×H(5,𝔽2)—2691no—
96#518—2691no—
97#518D26911yescharacteristic, complemented
98#410—2694no—
99C2×#99—2698no—
100H(5,𝔽2)—5384no—
101H(5,𝔽2)—5384no—
102C2×#41—2694no—
103C2×#41—2692no—
104C22×D4—2694no—
105C22×D4—2694no—
106C22×D4C2×D26911yescharacteristic
107C4×D4—24no—
108C4×D4—5384no—
109#117—2692no—
110#117C2×D26912yescomplemented
111C2×C22⋊C4—2694no—
112C2×C22⋊C4C2×D26914yescomplemented
113#99—26916no—
114C24—2694no—
115C24C22×D26912yes—
116#41—53816no—
117#41—5384no—
118#41—5384no—
119C2×Q8—2692no—
120C2×D4—2698no—
121C2×D4—2694no—
122C2×D4—2692no—
123C2×D4—5388no—
124C2×D4—5388no—
125C2×D4—53816no—
126C2×D4C4×D26914yes—
127C2×D4C22⋊D26914yes—
128C2×D4D107614yes—
129C22×C4—24no—
130C22×C4—2694no—
131C22×C4—2694no—
132C22×C4—5384no—
133C22×C4C22×D26911yescharacteristic
134C4⋊C4—22no—
135C4⋊C4—5382no—
136C22⋊C4—24no—
137C22⋊C4—2694no—
138C22⋊C4—5384no—
139C22⋊C4C22×D26914yes—
140C42—22no—
141C42—5382no—
142C23—82no—
143C23—2694no—
144C23—5388no—
145C23—21524no—
146C23C2×C4×D26914yes—
147C23C2×C22⋊D26914yes—
148C23C2×D107614yes—
149C23C23×D26911yescharacteristic
150Q8—5384no—
151D4—53816no—
152D4—5384no—
153D4—5384no—
154D4—5384no—
155D4#16864379116yes—
156C2×C4—24no—
157C2×C4—84no—
158C2×C4—2694no—
159C2×C4—5388no—
160C2×C4—5388no—
161C2×C4—5384no—
162C2×C4—21524no—
163C2×C4C2×C4×D26912yes—
164C2×C4C2×C22⋊D26912yes—
165C2×C4C2×D107612yes—
166C22—44no—
167C22—44no—
168C22—41no—
169C22—44no—
170C22—5384no—
171C22—10764no—
172C22—21528no—
173C22C2×#1686437918yes—
174C22C22×C4×D26911yescharacteristic
175C22C22×C22⋊D26911yescharacteristic
176C22C22×D107611yescharacteristic
177C4—44no—
178C4—5384no—
179C4—10764no—
180C4C2×#1686437914yes—
181C2—21no—
182C2—21no—
183C2—21no—
184C2—28no—
185C2—10764no—
186C2C22×#1686437911yescharacteristic
subgroupG/H
centerC2C22×#16864379
commutatorC22×C269C23×C4
FrattiniC23C23×D269
Fitting#518×C269C2
socleC2×C269C22×C22⋊C4
prime radicalC2×#22575498C2
cosocleC23×C269C24
hypercenter#518D269
hypocenterC269#2883
LCSG ▷ C22×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ C22×D4 ◁ #518

Complements

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

NH ≅ G/Ntype
2. #31771080C2semidirect
3. #31770578C2semidirect
4. C2×#22575367C2semidirect
5. #518×C269C2semidirect
9. C2×#16864401C4semidirect
11. C2×D4×D269C4semidirect
13. #22575412C22semidirect
14. C2×C22⋊Dic269C22semidirect
15. #22575367C22semidirect
18. #117×C269C22semidirect
19. C2×C22⋊C4×C269C22semidirect
20. C23×D269C2×C4semidirect
24. C2×Dic538C2×C4semidirect
25. C2×D1076C2×C4semidirect
28. C2×C22⋊D269C2×C4semidirect
32. C22×Dic269C2×C4semidirect
33. C2×C4×D269C2×C4semidirect
93. C269#2883semidirect
97. #518D269semidirect
110. #117C2×D269semidirect
112. C2×C22⋊C4C2×D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754982
#317710802
#317705782
C2×#225753672
#518×C2692
#2883269