Finite Group Databasebeta

← #24919259

Subgroups — #24919259

Order 10624.

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#19128505C212yesmaximal, complemented
2#19128504C212yesmaximal, complemented
3#19128077C211yesmaximal, characteristic, complemented
4#19127970C212yesmaximal, complemented
5#19127969C212yesmaximal, complemented
6#19127870C212yesmaximal, complemented
7#19127868C211yesmaximal, characteristic, complemented
8#19127815C212yesmaximal, complemented
9#453×C83C211yesmaximal, characteristic, complemented
10#14581838C2212yescomplemented
11#14581837C2211yescharacteristic, complemented
12#14581820C2212yescomplemented
13#14581819—22no—
14#14581818—22no—
15#14581818C2212yescomplemented
16#14581817—22no—
17#14581817C2212yescomplemented
18#14581816—22no—
19#14581813—22no—
20#14581812—22no—
21#14581812C2212yes—
22C2×#13050575—22no—
23C2×#13050575C2212yescomplemented
24C2×#13050575C2212yescomplemented
25D8×D83—22no—
26#14581800C2212yescomplemented
27#14581799C2212yescomplemented
28#14581789C2211yescharacteristic, complemented
29#14581727C2212yescomplemented
30#14581720C2212yescomplemented
31#14581709C2211yescharacteristic, complemented
32#14581708C2212yescomplemented
33C4×C83⋊C8C2211yescharacteristic, complemented
34H(5,𝔽2)×C83C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C83C2212yescomplemented
36#127×C83C2211yescharacteristic, complemented
37#104×C83C2212yescomplemented
38#100×C83C2211yescharacteristic, complemented
39#13050591—22no—
40#13050590—22no—
41#13050590C2312yes—
42#13050589—22no—
43C2×D332—22no—
44C2×D332—41no—
45C2×D332C2312yes—
46C2×C22⋊D83—41no—
47D4×D83—22no—
48D4×D83—22no—
49D4×D83—42no—
50#13050580D412yescomplemented
51#13050579—22no—
52#13050578—22no—
53#13050578D412yescomplemented
54#13050577—22no—
55#13050577D412yescomplemented
56D664—22no—
57#13050575—22no—
58#13050575—22no—
59#13050575—22no—
60#13050575—22no—
61#13050575—22no—
62#13050575—22no—
63#13050575D412yescomplemented
64#13050574—22no—
65#13050574C2312yes—
66#13050573—22no—
67C2×C83⋊C8—22no—
68C2×C83⋊C8C2312yes—
69C8×D83—22no—
70#41×C83—22no—
71#41×C83C2312yes—
72C2×D4×C83—22no—
73C2×D4×C83—41no—
74C2×D4×C83—41no—
75C2×D4×C83C2311yescharacteristic
76C2×D4×C83C2311yescharacteristic
77SD16×C83—22no—
78D8×C83—22no—
79M16×C83C2312yes—
80C42×C83C2311yescharacteristic
81C23×C83—21no—
82C23×C83—21no—
83C23×C83—41no—
84C22×D83—22no—
85C22×D83—41no—
86C22×D83—42no—
87Q8×C83C2×D412yes—
88D4×C83—22no—
89D4×C83—22no—
90D4×C83—22no—
91D4×C83—42no—
92D4×C83—82no—
93D4×C83C2×D412yes—
94C2×C4×C83—21no—
95C2×C4×C83—22no—
96C2×C4×C83—41no—
97C2×C4×C83C2411yescharacteristic
98C22⋊D83—22no—
99C22⋊D83—42no—
100C22⋊D83—42no—
101C2×Dic83—41no—
102D332—22no—
103D332—22no—
104D332—42no—
105D332—82no—
106D332C2×D412yes—
107C4×D83—22no—
108C4×D83—42no—
109Dic166C2×D412yes—
110C8×C83—22no—
111C83⋊C8—22no—
112C83⋊C8C2×D412yes—
113C83⋊C8C2×D412yes—
114C22×C83—22no—
115C22×C83—41no—
116C22×C83—41no—
117C22×C83—81no—
118C22×C83—81no—
119C22×C83—161no—
120C22×C83H(5,𝔽2)11yescharacteristic
121C2×D83—22no—
122C2×D83—42no—
123C2×D83—82no—
124C2×D83—161no—
125C2×D83—162no—
126C4×C83—22no—
127C4×C83—22no—
128C4×C83C22×D412yes—
129Dic83—22no—
130C2×C83—21no—
131C2×C83—42no—
132C2×C83—81no—
133C2×C83—81no—
134C2×C83D4211yescharacteristic
135D83—42no—
136D83—82no—
137#3292—831nomaximal, Sylow
138C83#329211yescharacteristic, complemented, Sylow
139#576—832no—
140#576—832no—
141#493—831no—
142#472—832no—
143#472—832no—
144#453—831no—
145#453—832no—
146#453D8311yescharacteristic, complemented
147#443—832no—
148H(5,𝔽2)—832no—
149H(5,𝔽2)—831no—
150H(5,𝔽2)C2×D8311yescharacteristic, complemented
151AGL(1,ℤ/8)—832no—
152AGL(1,ℤ/8)—832no—
153AGL(1,ℤ/8)—832no—
154AGL(1,ℤ/8)—1662no—
155AGL(1,ℤ/8)—1662no—
156AGL(1,ℤ/8)—1662no—
157AGL(1,ℤ/8)—1662no—
158AGL(1,ℤ/8)C2×D8312yescomplemented
159#135—832no—
160#135—1662no—
161#135—1662no—
162C2×D8—832no—
163C2×D8—832no—
164C2×D8—1662no—
165C2×D8—1662no—
166#131—832no—
167#131—832no—
168#127—831no—
169#127C2×D8311yescharacteristic, complemented
170#108—832no—
171#104—832no—
172#104C2×D8312yescomplemented
173#100—831no—
174#100—832no—
175#100C2×D8311yescharacteristic, complemented
176C4×C8—831no—
177#41—22no—
178#41—832no—
179#41—1662no—
180#41—1662no—
181#41—1662no—
182#41C22×D8312yes—
183C2×D4—22no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—3321no—
187C2×D4—3322no—
188C2×D4—3321no—
189C2×D4—832no—
190C2×D4—1662no—
191C2×D4—1662no—
192C2×D4—1662no—
193C2×D4C22×D8311yescharacteristic
194C2×D4C22×D8311yescharacteristic
195Q16—832no—
196SD16—22no—
197SD16—832no—
198SD16—832no—
199SD16—1662no—
200SD16—1662no—
201SD16—1662no—
202D8—22no—
203D8—832no—
204D8—1662no—
205D8—1662no—
206D8—1662no—
207D8—1662no—
208D8—1662no—
209D8—1662no—
210D8—1662no—
211M16—832no—
212M16—1662no—
213M16—1662no—
214M16C22×D8312yes—
215C2×C8—832no—
216C2×C8—1662no—
217C2×C8—1662no—
218C42C22×D8311yescharacteristic
219C23—21no—
220C23—21no—
221C23—41no—
222C23—3322no—
223C23—3321no—
224C23—1662no—
225Q8—832no—
226Q8D4×D8312yes—
227D4—22no—
228D4—22no—
229D4—22no—
230D4—42no—
231D4—82no—
232D4—3322no—
233D4—3322no—
234D4—3322no—
235D4—832no—
236D4—6642no—
237D4—1662no—
238D4—1662no—
239D4—1662no—
240D4D4×D8312yes—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—3322no—
245C2×C4—3321no—
246C2×C4—1662no—
247C2×C4C23×D8311yescharacteristic
248C8—22no—
249C8—832no—
250C8—832no—
251C8—1662no—
252C22—22no—
253C22—41no—
254C22—41no—
255C22—81no—
256C22—81no—
257C22—161no—
258C22—13282no—
259C22—13281no—
260C22—3322no—
261C22—6642no—
262C22—1662no—
263C22#1458183711yescharacteristic
264C4—22no—
265C4—22no—
266C4—1662no—
267C4C2×D4×D8312yes—
268C2—21no—
269C2—42no—
270C2—81no—
271C2—81no—
272C2—3322no—
273C2—6642no—
274C2#1912833311yescharacteristic
subgroupG/H
centerC2#19128333
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#453×C83C2
socleC2×C83D42
hypercenter#453D83
hypocenterC83#3292
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #19128505C2semidirect
2. #19128504C2semidirect
3. #19128077C2semidirect
4. #19127970C2semidirect
5. #19127969C2semidirect
6. #19127870C2semidirect
7. #19127868C2semidirect
8. #19127815C2semidirect
9. #453×C83C2semidirect
10. #14581838C22semidirect
11. #14581837C22semidirect
12. #14581820C22semidirect
15. #14581818C22semidirect
17. #14581817C22semidirect
23. C2×#13050575C22semidirect
24. C2×#13050575C22semidirect
26. #14581800C22semidirect
27. #14581799C22semidirect
28. #14581789C22semidirect
29. #14581727C22semidirect
30. #14581720C22semidirect
31. #14581709C22semidirect
32. #14581708C22semidirect
33. C4×C83⋊C8C22semidirect
34. H(5,𝔽2)×C83C22semidirect
35. AGL(1,ℤ/8)×C83C22semidirect
36. #127×C83C22semidirect
37. #104×C83C22semidirect
38. #100×C83C22semidirect
50. #13050580D4semidirect
53. #13050578D4semidirect
55. #13050577D4semidirect
63. #13050575D4semidirect
138. C83#3292semidirect
146. #453D83semidirect
150. H(5,𝔽2)C2×D83semidirect
158. AGL(1,ℤ/8)C2×D83semidirect
169. #127C2×D83semidirect
172. #104C2×D83semidirect
175. #100C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191285052
#191285042
#191280772
#191279702
#191279692
#191278702
#191278682
#191278152
#453×C832
#329283