Finite Group Databasebeta

← #46819344

Subgroups — #46819344

Order 34432.

800 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 209 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×#22575499C211yesmaximal, characteristic
2#31771079C212yesmaximal, complemented
3#31770579C211yesmaximal, characteristic, complemented
4#31770578C212yesmaximal, complemented
5#99×D269C218yesmaximal, complemented
6#410×C269C211yesmaximal, characteristic, complemented
7#22575499—24no—
8#22575499—24no—
9C2×#16864403C412yescomplemented
10C2×#16864402C412yescomplemented
11C2×#16864402C2211yescharacteristic
12C22×D1076C412yescomplemented
13C22×D1076C2211yescharacteristic
14C2×D4×D269C412yescomplemented
15C2×D4×D269C2214yes—
16C4×D1076—24no—
17#22575415—24no—
18#22575412C2212yescomplemented
19C22⋊C4×D269C2218yescomplemented
20#22575368C2214yescomplemented
21#22575367C2218yescomplemented
22C2×#41×C269C2211yescharacteristic
23#117×C269C2212yescomplemented
24#99×C269C2214yescomplemented
25C23×D269C2×C414yescomplemented
26C23×D269C2312yes—
27#16864403—28no—
28#16864402—24no—
29#16864402—24no—
30#16864402—22no—
31#16864402—22no—
32C2×Dic538C2×C411yescharacteristic, complemented
33C2×D1076—24no—
34C2×D1076—24no—
35C2×D1076C2×C412yes—
36C2×D1076C2×C412yes—
37C2×D1076C2×C414yescomplemented
38C2×D1076C2×C411yescharacteristic, complemented
39C2×D1076D414yes—
40C2×D1076D414yes—
41C2×C22⋊D269C2×C414yescomplemented
42C2×C22⋊D269C2312yes—
43D4×D269—216no—
44D4×D269—28no—
45C2×C4×D269—28no—
46C2×C4×D269C2×C412yescomplemented
47C2×C4×D269C2×C414yescomplemented
48C4⋊Dic269—22no—
49C22⋊Dic269C2314yes—
50#16864379—28no—
51C4×Dic269—22no—
52#41×C269—22no—
53#41×C269—22no—
54C2×Q8×C269C2×C411yescharacteristic, complemented
55C2×D4×C269C2×C411yescharacteristic, complemented
56C2×D4×C269C2312yes—
57C22×C4×C269C2×C412yescomplemented
58C22×C4×C269C2311yescharacteristic
59C4⋊C4×C269—22no—
60C22⋊C4×C269C2314yes—
61C42×C269—22no—
62C22×D269—28no—
63C22×D269—82no—
64C22×D269—84no—
65C22×D269C22×C412yes—
66C22×D269C22×C414yes—
67C22×D269C2×D418yes—
68Dic538—21no—
69Dic538—21no—
70D1076—28no—
71D1076—21no—
72D1076—21no—
73D1076C22⋊C418yes—
74D1076C22⋊C414yes—
75D1076C22⋊C414yes—
76C22⋊D269—24no—
77C22⋊D269—28no—
78C2×Dic269—24no—
79C2×Dic269C22×C412yes—
80C4×D269—28no—
81C4×D269—24no—
82C4×D269—88no—
83C23×C269C22×C412yes—
84C23×C269C2411yescharacteristic
85Q8×C269—22no—
86D4×C269—24no—
87D4×C269—22no—
88C2×C4×C269—22no—
89C2×C4×C269—22no—
90C2×C4×C269—24no—
91C2×C4×C269C22×C412yes—
92C2×C4×C269C22×C411yescharacteristic
93C2×C4×C269C22×C411yescharacteristic
94C2×C4×C269C2×D412yes—
95C2×C4×C269C2×D412yes—
96C22×C269—22no—
97C22×C269—41no—
98C22×C269—42no—
99C22×C269C23×C411yescharacteristic
100C22×C269C22×D412yes—
101C2×D269—22no—
102C2×D269—48no—
103C2×D269—44no—
104C2×D269—42no—
105C2×D269—88no—
106C2×D269C2×C22⋊C418yes—
107C4×C269—22no—
108C4×C269—44no—
109C4×C269C2×C22⋊C411yescharacteristic
110C4×C269C2×C22⋊C411yescharacteristic
111C4×C269C2×C22⋊C412yes—
112Dic269—22no—
113Dic269—44no—
114C2×C269—22no—
115C2×C269—21no—
116C2×C269—42no—
117C2×C269C22×C22⋊C411yescharacteristic
118D269—28no—
119D269—42no—
120C269#288311yescharacteristic, complemented, Sylow
121#2883—2691nomaximal, Sylow
122C2×H(5,𝔽2)—2691no—
123#518—2692no—
124#410—2691no—
125#410—2692no—
126#410D26911yescharacteristic, complemented
127C2×#99—2698no—
128H(5,𝔽2)—5384no—
129H(5,𝔽2)—5384no—
130C2×#41—2692no—
131C2×#41—2692no—
132C2×#41—2691no—
133C2×#41C2×D26911yescharacteristic
134C22×D4—2692no—
135C22×D4—2694no—
136C22×D4—2692no—
137C22×D4—2691no—
138C4×D4—5384no—
139C4×D4—5384no—
140#117—2692no—
141#117C2×D26912yescomplemented
142C2×C22⋊C4—2698no—
143#99—2694no—
144#99—2698no—
145#99C2×D26914yescomplemented
146C24—2692no—
147C24—2694no—
148#41—22no—
149#41—22no—
150#41—5384no—
151#41—5384no—
152#41—5388no—
153#41—5382no—
154#41—5382no—
155C2×Q8—2691no—
156C2×Q8C4×D26911yescharacteristic, complemented
157C2×D4—2694no—
158C2×D4—2692no—
159C2×D4—2692no—
160C2×D4—2692no—
161C2×D4—2694no—
162C2×D4—2691no—
163C2×D4—2694no—
164C2×D4—2694no—
165C2×D4—5384no—
166C2×D4—5384no—
167C2×D4—5388no—
168C2×D4—53816no—
169C2×D4C4×D26911yescharacteristic, complemented
170C2×D4C22×D26912yes—
171C22×C4—2694no—
172C22×C4—2692no—
173C22×C4—5388no—
174C22×C4C4×D26912yescomplemented
175C22×C4C22×D26911yescharacteristic
176C4⋊C4—22no—
177C4⋊C4—5382no—
178C22⋊C4—2694no—
179C22⋊C4—5388no—
180C22⋊C4C22×D26914yes—
181C42—22no—
182C42—5382no—
183C23—2694no—
184C23—2698no—
185C23—2692no—
186C23—5388no—
187C23—21522no—
188C23—21524no—
189C23C2×C4×D26912yes—
190C23C23×D26911yescharacteristic
191Q8—22no—
192Q8—5381no—
193Q8—5381no—
194D4—24no—
195D4—22no—
196D4—2694no—
197D4—2694no—
198D4—2698no—
199D4—5388no—
200D4—5388no—
201D4—5384no—
202D4—5381no—
203D4—5381no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—24no—
207C2×C4—2692no—
208C2×C4—5388no—
209C2×C4—5384no—
210C2×C4—5384no—
211C2×C4—21528no—
212C2×C4C2×C4×D26912yes—
213C2×C4C2×C4×D26911yescharacteristic
214C2×C4C2×C4×D26911yescharacteristic
215C2×C4D4×D26912yes—
216C2×C4D4×D26912yes—
217C22—22no—
218C22—41no—
219C22—42no—
220C22—2698no—
221C22—5382no—
222C22—10768no—
223C22—10764no—
224C22—10762no—
225C22—21528no—
226C22C22×C4×D26911yescharacteristic
227C22C2×D4×D26912yes—
228C4—22no—
229C4—44no—
230C4—5382no—
231C4—10764no—
232C4C22⋊C4×D26911yescharacteristic
233C4C22⋊C4×D26911yescharacteristic
234C4C22⋊C4×D26912yes—
235C2—22no—
236C2—21no—
237C2—42no—
238C2—5388no—
239C2—10762no—
240C2C2×C22⋊C4×D26911yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D269
commutatorC22×C269C23×C4
FrattiniC23C23×D269
Fitting#410×C269C2
socleC2×C269C22×C22⋊C4
prime radicalC2×#22575499C2
cosocleC23×C269C24
hypercenter#410D269
hypocenterC269#2883
LCSG ▷ C22×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #31771079C2semidirect
3. #31770579C2semidirect
4. #31770578C2semidirect
5. #99×D269C2semidirect
6. #410×C269C2semidirect
9. C2×#16864403C4semidirect
10. C2×#16864402C4semidirect
12. C22×D1076C4semidirect
14. C2×D4×D269C4semidirect
18. #22575412C22semidirect
19. C22⋊C4×D269C22semidirect
20. #22575368C22semidirect
21. #22575367C22semidirect
23. #117×C269C22semidirect
24. #99×C269C22semidirect
25. C23×D269C2×C4semidirect
32. C2×Dic538C2×C4semidirect
37. C2×D1076C2×C4semidirect
38. C2×D1076C2×C4semidirect
41. C2×C22⋊D269C2×C4semidirect
46. C2×C4×D269C2×C4semidirect
47. C2×C4×D269C2×C4semidirect
54. C2×Q8×C269C2×C4semidirect
55. C2×D4×C269C2×C4semidirect
57. C22×C4×C269C2×C4semidirect
120. C269#2883semidirect
126. #410D269semidirect
141. #117C2×D269semidirect
145. #99C2×D269semidirect
156. C2×Q8C4×D269semidirect
169. C2×D4C4×D269semidirect
174. C22×C4C4×D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754992
#317710792
#317705792
#317705782
#99×D2692
#410×C2692
#2883269