Finite Group Databasebeta

← #46822681

Subgroups — #46822681

Order 34432.

476 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 476 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#31771409C212yesmaximal, complemented
2#31771408C212yesmaximal, complemented
3#31770981C211yesmaximal, characteristic, complemented
4#31770874C212yesmaximal, complemented
5#31770873C212yesmaximal, complemented
6#31770774C212yesmaximal, complemented
7#31770772C211yesmaximal, characteristic, complemented
8#31770719C212yesmaximal, complemented
9#453×C269C211yesmaximal, characteristic, complemented
10#22575499C2212yescomplemented
11#22575498C2211yescharacteristic, complemented
12#22575481C2212yescomplemented
13#22575480—22no—
14#22575479—22no—
15#22575479C2212yescomplemented
16#22575478—22no—
17#22575478C2212yescomplemented
18#22575477—22no—
19#22575474—22no—
20#22575473—22no—
21#22575473C2212yes—
22C2×#16864387—22no—
23C2×#16864387C2212yescomplemented
24C2×#16864387C2212yescomplemented
25D8×D269—22no—
26#22575461C2212yescomplemented
27#22575460C2212yescomplemented
28#22575450C2211yescharacteristic, complemented
29#22575388C2212yescomplemented
30#22575381C2212yescomplemented
31#22575370C2211yescharacteristic, complemented
32#22575369C2212yescomplemented
33C4×#14524130C2211yescharacteristic, complemented
34H(5,𝔽2)×C269C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C269C2212yescomplemented
36#127×C269C2211yescharacteristic, complemented
37#104×C269C2212yescomplemented
38#100×C269C2211yescharacteristic, complemented
39#16864403—22no—
40#16864402—22no—
41#16864402C2312yes—
42#16864401—22no—
43C2×D1076—22no—
44C2×D1076—41no—
45C2×D1076C2312yes—
46C2×C22⋊D269—41no—
47D4×D269—22no—
48D4×D269—22no—
49D4×D269—42no—
50#16864392D412yescomplemented
51#16864391—22no—
52#16864390—22no—
53#16864390D412yescomplemented
54#16864389—22no—
55#16864389D412yescomplemented
56D2152—22no—
57#16864387—22no—
58#16864387—22no—
59#16864387—22no—
60#16864387—22no—
61#16864387—22no—
62#16864387—22no—
63#16864387D412yescomplemented
64#16864386—22no—
65#16864386C2312yes—
66#16864385—22no—
67C2×#14524130—22no—
68C2×#14524130C2312yes—
69C8×D269—22no—
70#41×C269—22no—
71#41×C269C2312yes—
72C2×D4×C269—22no—
73C2×D4×C269—41no—
74C2×D4×C269—41no—
75C2×D4×C269C2311yescharacteristic
76C2×D4×C269C2311yescharacteristic
77SD16×C269—22no—
78D8×C269—22no—
79M16×C269C2312yes—
80C42×C269C2311yescharacteristic
81C22×D269—22no—
82C22×D269—41no—
83C22×D269—42no—
84Dic538C2×D412yes—
85D1076—22no—
86D1076—22no—
87D1076—42no—
88D1076—82no—
89D1076C2×D412yes—
90C22⋊D269—22no—
91C22⋊D269—42no—
92C22⋊D269—42no—
93C2×Dic269—41no—
94C4×D269—22no—
95C4×D269—42no—
96#14524130—22no—
97#14524130C2×D412yes—
98#14524130C2×D412yes—
99C23×C269—21no—
100C23×C269—21no—
101C23×C269—41no—
102Q8×C269C2×D412yes—
103D4×C269—22no—
104D4×C269—22no—
105D4×C269—22no—
106D4×C269—42no—
107D4×C269—82no—
108D4×C269C2×D412yes—
109C2×C4×C269—21no—
110C2×C4×C269—22no—
111C2×C4×C269—41no—
112C2×C4×C269C2411yescharacteristic
113C8×C269—22no—
114C22×C269—22no—
115C22×C269—41no—
116C22×C269—41no—
117C22×C269—81no—
118C22×C269—81no—
119C22×C269—161no—
120C22×C269H(5,𝔽2)11yescharacteristic
121C2×D269—22no—
122C2×D269—42no—
123C2×D269—82no—
124C2×D269—161no—
125C2×D269—162no—
126C4×C269—22no—
127C4×C269—22no—
128C4×C269C22×D412yes—
129Dic269—22no—
130C2×C269—21no—
131C2×C269—42no—
132C2×C269—81no—
133C2×C269—81no—
134C2×C269D4211yescharacteristic
135D269—42no—
136D269—82no—
137C269#329211yescharacteristic, complemented, Sylow
138#3292—2691nomaximal, Sylow
139#576—2692no—
140#576—2692no—
141#493—2691no—
142#472—2692no—
143#472—2692no—
144#453—2691no—
145#453—2692no—
146#453D26911yescharacteristic, complemented
147#443—2692no—
148H(5,𝔽2)—2692no—
149H(5,𝔽2)—2691no—
150H(5,𝔽2)C2×D26911yescharacteristic, complemented
151AGL(1,ℤ/8)—2692no—
152AGL(1,ℤ/8)—2692no—
153AGL(1,ℤ/8)—2692no—
154AGL(1,ℤ/8)—5382no—
155AGL(1,ℤ/8)—5382no—
156AGL(1,ℤ/8)—5382no—
157AGL(1,ℤ/8)—5382no—
158AGL(1,ℤ/8)C2×D26912yescomplemented
159#135—2692no—
160#135—5382no—
161#135—5382no—
162C2×D8—2692no—
163C2×D8—2692no—
164C2×D8—5382no—
165C2×D8—5382no—
166#131—2692no—
167#131—2692no—
168#127—2691no—
169#127C2×D26911yescharacteristic, complemented
170#108—2692no—
171#104—2692no—
172#104C2×D26912yescomplemented
173#100—2691no—
174#100—2692no—
175#100C2×D26911yescharacteristic, complemented
176C4×C8—2691no—
177#41—22no—
178#41—2692no—
179#41—5382no—
180#41—5382no—
181#41—5382no—
182#41C22×D26912yes—
183C2×D4—22no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—2692no—
187C2×D4—5382no—
188C2×D4—5382no—
189C2×D4—5382no—
190C2×D4—10761no—
191C2×D4—10762no—
192C2×D4—10761no—
193C2×D4C22×D26911yescharacteristic
194C2×D4C22×D26911yescharacteristic
195Q16—2692no—
196SD16—22no—
197SD16—2692no—
198SD16—2692no—
199SD16—5382no—
200SD16—5382no—
201SD16—5382no—
202D8—22no—
203D8—2692no—
204D8—5382no—
205D8—5382no—
206D8—5382no—
207D8—5382no—
208D8—5382no—
209D8—5382no—
210D8—5382no—
211M16—2692no—
212M16—5382no—
213M16—5382no—
214M16C22×D26912yes—
215C2×C8—2692no—
216C2×C8—5382no—
217C2×C8—5382no—
218C42C22×D26911yescharacteristic
219C23—21no—
220C23—21no—
221C23—41no—
222C23—5382no—
223C23—10762no—
224C23—10761no—
225Q8—2692no—
226Q8D4×D26912yes—
227D4—22no—
228D4—22no—
229D4—22no—
230D4—42no—
231D4—82no—
232D4—2692no—
233D4—5382no—
234D4—5382no—
235D4—5382no—
236D4—10762no—
237D4—10762no—
238D4—10762no—
239D4—21522no—
240D4D4×D26912yes—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—5382no—
245C2×C4—10762no—
246C2×C4—10761no—
247C2×C4C23×D26911yescharacteristic
248C8—22no—
249C8—2692no—
250C8—2692no—
251C8—5382no—
252C22—22no—
253C22—41no—
254C22—41no—
255C22—81no—
256C22—81no—
257C22—161no—
258C22—5382no—
259C22—10762no—
260C22—21522no—
261C22—43042no—
262C22—43041no—
263C22#2257549811yescharacteristic
264C4—22no—
265C4—22no—
266C4—5382no—
267C4C2×D4×D26912yes—
268C2—21no—
269C2—42no—
270C2—81no—
271C2—81no—
272C2—10762no—
273C2—21522no—
274C2#3177123711yescharacteristic
subgroupG/H
centerC2#31771237
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#453×C269C2
socleC2×C269D42
hypercenter#453D269
hypocenterC269#3292
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #31771409C2semidirect
2. #31771408C2semidirect
3. #31770981C2semidirect
4. #31770874C2semidirect
5. #31770873C2semidirect
6. #31770774C2semidirect
7. #31770772C2semidirect
8. #31770719C2semidirect
9. #453×C269C2semidirect
10. #22575499C22semidirect
11. #22575498C22semidirect
12. #22575481C22semidirect
15. #22575479C22semidirect
17. #22575478C22semidirect
23. C2×#16864387C22semidirect
24. C2×#16864387C22semidirect
26. #22575461C22semidirect
27. #22575460C22semidirect
28. #22575450C22semidirect
29. #22575388C22semidirect
30. #22575381C22semidirect
31. #22575370C22semidirect
32. #22575369C22semidirect
33. C4×#14524130C22semidirect
34. H(5,𝔽2)×C269C22semidirect
35. AGL(1,ℤ/8)×C269C22semidirect
36. #127×C269C22semidirect
37. #104×C269C22semidirect
38. #100×C269C22semidirect
50. #16864392D4semidirect
53. #16864390D4semidirect
55. #16864389D4semidirect
63. #16864387D4semidirect
137. C269#3292semidirect
146. #453D269semidirect
150. H(5,𝔽2)C2×D269semidirect
158. AGL(1,ℤ/8)C2×D269semidirect
169. #127C2×D269semidirect
172. #104C2×D269semidirect
175. #100C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317714092
#317714082
#317709812
#317708742
#317708732
#317707742
#317707722
#317707192
#453×C2692
#3292269