Finite Group Databasebeta

← #49133057

Subgroups — #49133057

Order 36224.

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#31827269C212yesmaximal, complemented
2#31827268C212yesmaximal, complemented
3#31826841C211yesmaximal, characteristic, complemented
4#31826734C212yesmaximal, complemented
5#31826733C212yesmaximal, complemented
6#31826634C212yesmaximal, complemented
7#31826632C211yesmaximal, characteristic, complemented
8#31826579C212yesmaximal, complemented
9#453×C283C211yesmaximal, characteristic, complemented
10#22583656C2212yescomplemented
11#22583655C2211yescharacteristic, complemented
12#22583638C2212yescomplemented
13#22583637—22no—
14#22583636—22no—
15#22583636C2212yescomplemented
16#22583635—22no—
17#22583635C2212yescomplemented
18#22583634—22no—
19#22583631—22no—
20#22583630—22no—
21#22583630C2212yes—
22C2×#17985377—22no—
23C2×#17985377C2212yescomplemented
24C2×#17985377C2212yescomplemented
25D8×D283—22no—
26#22583618C2212yescomplemented
27#22583617C2212yescomplemented
28#22583607C2211yescharacteristic, complemented
29#22583545C2212yescomplemented
30#22583538C2212yescomplemented
31#22583527C2211yescharacteristic, complemented
32#22583526C2212yescomplemented
33C4×C283⋊C8C2211yescharacteristic, complemented
34H(5,𝔽2)×C283C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C283C2212yescomplemented
36#127×C283C2211yescharacteristic, complemented
37#104×C283C2212yescomplemented
38#100×C283C2211yescharacteristic, complemented
39#17985393—22no—
40#17985392—22no—
41#17985392C2312yes—
42#17985391—22no—
43C2×D1132—22no—
44C2×D1132—41no—
45C2×D1132C2312yes—
46C2×C22⋊D283—41no—
47D4×D283—22no—
48D4×D283—22no—
49D4×D283—42no—
50#17985382D412yescomplemented
51#17985381—22no—
52#17985380—22no—
53#17985380D412yescomplemented
54#17985379—22no—
55#17985379D412yescomplemented
56D2264—22no—
57#17985377—22no—
58#17985377—22no—
59#17985377—22no—
60#17985377—22no—
61#17985377—22no—
62#17985377—22no—
63#17985377D412yescomplemented
64#17985376—22no—
65#17985376C2312yes—
66#17985375—22no—
67C2×C283⋊C8—22no—
68C2×C283⋊C8C2312yes—
69C8×D283—22no—
70#41×C283—22no—
71#41×C283C2312yes—
72C2×D4×C283—22no—
73C2×D4×C283—41no—
74C2×D4×C283—41no—
75C2×D4×C283C2311yescharacteristic
76C2×D4×C283C2311yescharacteristic
77SD16×C283—22no—
78D8×C283—22no—
79M16×C283C2312yes—
80C42×C283C2311yescharacteristic
81C22×D283—22no—
82C22×D283—41no—
83C22×D283—42no—
84Dic566C2×D412yes—
85D1132—22no—
86D1132—22no—
87D1132—42no—
88D1132—82no—
89D1132C2×D412yes—
90C22⋊D283—22no—
91C22⋊D283—42no—
92C22⋊D283—42no—
93C2×Dic283—41no—
94C4×D283—22no—
95C4×D283—42no—
96C283⋊C8—22no—
97C283⋊C8C2×D412yes—
98C283⋊C8C2×D412yes—
99C23×C283—21no—
100C23×C283—21no—
101C23×C283—41no—
102Q8×C283C2×D412yes—
103D4×C283—22no—
104D4×C283—22no—
105D4×C283—22no—
106D4×C283—42no—
107D4×C283—82no—
108D4×C283C2×D412yes—
109C2×C4×C283—21no—
110C2×C4×C283—22no—
111C2×C4×C283—41no—
112C2×C4×C283C2411yescharacteristic
113C8×C283—22no—
114C22×C283—22no—
115C22×C283—41no—
116C22×C283—41no—
117C22×C283—81no—
118C22×C283—81no—
119C22×C283—161no—
120C22×C283H(5,𝔽2)11yescharacteristic
121C2×D283—22no—
122C2×D283—42no—
123C2×D283—82no—
124C2×D283—161no—
125C2×D283—162no—
126C4×C283—22no—
127C4×C283—22no—
128C4×C283C22×D412yes—
129Dic283—22no—
130C2×C283—21no—
131C2×C283—42no—
132C2×C283—81no—
133C2×C283—81no—
134C2×C283D4211yescharacteristic
135D283—42no—
136D283—82no—
137C283#329211yescharacteristic, complemented, Sylow
138#3292—2831nomaximal, Sylow
139#576—2832no—
140#576—2832no—
141#493—2831no—
142#472—2832no—
143#472—2832no—
144#453—2831no—
145#453—2832no—
146#453D28311yescharacteristic, complemented
147#443—2832no—
148H(5,𝔽2)—2832no—
149H(5,𝔽2)—2831no—
150H(5,𝔽2)C2×D28311yescharacteristic, complemented
151AGL(1,ℤ/8)—2832no—
152AGL(1,ℤ/8)—2832no—
153AGL(1,ℤ/8)—2832no—
154AGL(1,ℤ/8)—5662no—
155AGL(1,ℤ/8)—5662no—
156AGL(1,ℤ/8)—5662no—
157AGL(1,ℤ/8)—5662no—
158AGL(1,ℤ/8)C2×D28312yescomplemented
159#135—2832no—
160#135—5662no—
161#135—5662no—
162C2×D8—2832no—
163C2×D8—2832no—
164C2×D8—5662no—
165C2×D8—5662no—
166#131—2832no—
167#131—2832no—
168#127—2831no—
169#127C2×D28311yescharacteristic, complemented
170#108—2832no—
171#104—2832no—
172#104C2×D28312yescomplemented
173#100—2831no—
174#100—2832no—
175#100C2×D28311yescharacteristic, complemented
176C4×C8—2831no—
177#41—22no—
178#41—2832no—
179#41—5662no—
180#41—5662no—
181#41—5662no—
182#41C22×D28312yes—
183C2×D4—22no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—2832no—
187C2×D4—5662no—
188C2×D4—5662no—
189C2×D4—5662no—
190C2×D4—11321no—
191C2×D4—11322no—
192C2×D4—11321no—
193C2×D4C22×D28311yescharacteristic
194C2×D4C22×D28311yescharacteristic
195Q16—2832no—
196SD16—22no—
197SD16—2832no—
198SD16—2832no—
199SD16—5662no—
200SD16—5662no—
201SD16—5662no—
202D8—22no—
203D8—2832no—
204D8—5662no—
205D8—5662no—
206D8—5662no—
207D8—5662no—
208D8—5662no—
209D8—5662no—
210D8—5662no—
211M16—2832no—
212M16—5662no—
213M16—5662no—
214M16C22×D28312yes—
215C2×C8—2832no—
216C2×C8—5662no—
217C2×C8—5662no—
218C42C22×D28311yescharacteristic
219C23—21no—
220C23—21no—
221C23—41no—
222C23—5662no—
223C23—11322no—
224C23—11321no—
225Q8—2832no—
226Q8D4×D28312yes—
227D4—22no—
228D4—22no—
229D4—22no—
230D4—42no—
231D4—82no—
232D4—2832no—
233D4—5662no—
234D4—5662no—
235D4—5662no—
236D4—11322no—
237D4—11322no—
238D4—11322no—
239D4—22642no—
240D4D4×D28312yes—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—5662no—
245C2×C4—11322no—
246C2×C4—11321no—
247C2×C4C23×D28311yescharacteristic
248C8—22no—
249C8—2832no—
250C8—2832no—
251C8—5662no—
252C22—22no—
253C22—41no—
254C22—41no—
255C22—81no—
256C22—81no—
257C22—161no—
258C22—5662no—
259C22—11322no—
260C22—45282no—
261C22—45281no—
262C22—22642no—
263C22#2258365511yescharacteristic
264C4—22no—
265C4—22no—
266C4—5662no—
267C4C2×D4×D28312yes—
268C2—21no—
269C2—42no—
270C2—81no—
271C2—81no—
272C2—11322no—
273C2—22642no—
274C2#3182709711yescharacteristic
subgroupG/H
centerC2#31827097
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#453×C283C2
socleC2×C283D42
hypercenter#453D283
hypocenterC283#3292
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #31827269C2semidirect
2. #31827268C2semidirect
3. #31826841C2semidirect
4. #31826734C2semidirect
5. #31826733C2semidirect
6. #31826634C2semidirect
7. #31826632C2semidirect
8. #31826579C2semidirect
9. #453×C283C2semidirect
10. #22583656C22semidirect
11. #22583655C22semidirect
12. #22583638C22semidirect
15. #22583636C22semidirect
17. #22583635C22semidirect
23. C2×#17985377C22semidirect
24. C2×#17985377C22semidirect
26. #22583618C22semidirect
27. #22583617C22semidirect
28. #22583607C22semidirect
29. #22583545C22semidirect
30. #22583538C22semidirect
31. #22583527C22semidirect
32. #22583526C22semidirect
33. C4×C283⋊C8C22semidirect
34. H(5,𝔽2)×C283C22semidirect
35. AGL(1,ℤ/8)×C283C22semidirect
36. #127×C283C22semidirect
37. #104×C283C22semidirect
38. #100×C283C22semidirect
50. #17985382D4semidirect
53. #17985380D4semidirect
55. #17985379D4semidirect
63. #17985377D4semidirect
137. C283#3292semidirect
146. #453D283semidirect
150. H(5,𝔽2)C2×D283semidirect
158. AGL(1,ℤ/8)C2×D283semidirect
169. #127C2×D283semidirect
172. #104C2×D283semidirect
175. #100C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318272692
#318272682
#318268412
#318267342
#318267332
#318266342
#318266322
#318265792
#453×C2832
#3292283