Finite Group Databasebeta

← #24919232

Subgroups — #24919232

Order 10624.

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

normal subgroups only

This group has 464 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#19128510C212yesmaximal, complemented
2#19128504C212yesmaximal, complemented
3#19128091C211yesmaximal, characteristic, complemented
4#19127970C212yesmaximal, complemented
5#19127966C212yesmaximal, complemented
6#19127873C211yesmaximal, characteristic, complemented
7#19127870C212yesmaximal, complemented
8#19127820C212yesmaximal, complemented
9#455×C83C211yesmaximal, characteristic, complemented
10#14581840C2211yescharacteristic, complemented
11#14581838C2212yescomplemented
12#14581824—22no—
13#14581820C2212yescomplemented
14#14581819—22no—
15#14581818—22no—
16#14581818C2212yescomplemented
17#14581817—22no—
18#14581817C2212yes—
19#14581815—22no—
20#14581812—22no—
21#14581812C2212yes—
22C2×#13050577C2212yescomplemented
23SD16×D83—22no—
24C2×#13050575—22no—
25C2×#13050575C2212yescomplemented
26#14581800C2212yescomplemented
27#14581799C2212yescomplemented
28#14581789C2211yescharacteristic, complemented
29#14581727C2212yescomplemented
30#14581720C2212yescomplemented
31#14581711C2211yescharacteristic, complemented
32#14581708C2212yescomplemented
33#14581703C2211yescharacteristic, complemented
34H(5,𝔽2)×C83C2211yescharacteristic
35#137×C83C2212yescomplemented
36#124×C83C2211yescharacteristic, complemented
37#104×C83C2212yescomplemented
38#100×C83C2211yescharacteristic, complemented
39#13050591—22no—
40#13050591—22no—
41#13050590—22no—
42#13050590—41no—
43#13050590C2312yes—
44Q8×D83—22no—
45C2×D332—22no—
46C2×D332—41no—
47C2×D332C2312yes—
48D4×D83—22no—
49D4×D83—42no—
50#13050580D412yescomplemented
51#13050579—22no—
52#13050578—22no—
53#13050578D412yescomplemented
54#13050577—22no—
55#13050577—22no—
56#13050577—22no—
57#13050577D412yescomplemented
58D664—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×Q8×C83C2311yescharacteristic
73C2×D4×C83—22no—
74C2×D4×C83—41no—
75C2×D4×C83C2311yescharacteristic
76Q16×C83—22no—
77SD16×C83—22no—
78M16×C83C2312yes—
79C22⋊C4×C83—41no—
80C42×C83C2311yescharacteristic
81C23×C83—21no—
82C23×C83—41no—
83C22×D83—22no—
84C22×D83—42no—
85Q8×C83—22no—
86Q8×C83C2×D412yes—
87D4×C83—22no—
88D4×C83—22no—
89D4×C83—42no—
90D4×C83C2×D412yes—
91C2×C4×C83—22no—
92C2×C4×C83—21no—
93C2×C4×C83—21no—
94C2×C4×C83—41no—
95C2×C4×C83C2411yescharacteristic
96C22⋊D83—22no—
97C22⋊D83—42no—
98D332—22no—
99D332—22no—
100D332—42no—
101D332—41no—
102D332—82no—
103D332C2×D412yes—
104C4×D83—22no—
105C4×D83—42no—
106C4×D83—42no—
107Dic166—41no—
108Dic166C2×D412yes—
109C8×C83—22no—
110C83⋊C8—22no—
111C83⋊C8C2×D412yes—
112C83⋊C8C2×D412yes—
113C22×C83—22no—
114C22×C83—41no—
115C22×C83—81no—
116C22×C83—161no—
117C22×C83H(5,𝔽2)11yescharacteristic
118C2×D83—22no—
119C2×D83—42no—
120C2×D83—82no—
121C2×D83—162no—
122C4×C83—22no—
123C4×C83—41no—
124C4×C83—41no—
125C4×C83C22×D412yes—
126Dic83—22no—
127C2×C83—21no—
128C2×C83—42no—
129C2×C83—81no—
130C2×C83D4211yescharacteristic
131D83—42no—
132D83—82no—
133#3288—831nomaximal, Sylow
134C83#328811yescharacteristic, complemented, Sylow
135#577—832no—
136#576—832no—
137#496—831no—
138#472—832no—
139#471—832no—
140#455D8311yescharacteristic, complemented
141#454—831no—
142#453—832no—
143#444—832no—
1442−1+4—831no—
145H(5,𝔽2)—832no—
146H(5,𝔽2)C2×D8311yescharacteristic
147#137—1662no—
148#137C2×D8312yescomplemented
149AGL(1,ℤ/8)—832no—
150AGL(1,ℤ/8)—832no—
151AGL(1,ℤ/8)—832no—
152AGL(1,ℤ/8)—1662no—
153AGL(1,ℤ/8)—1662no—
154AGL(1,ℤ/8)—1662no—
155#135—832no—
156#135—1662no—
157#135—1662no—
158C2×SD16—832no—
159C2×SD16—1662no—
160C2×D8—832no—
161C2×D8—1662no—
162#131—832no—
163#131—832no—
164#127—831no—
165#124C2×D8311yescharacteristic, complemented
166#108—832no—
167#104—832no—
168#104C2×D8312yescomplemented
169#101—831no—
170#100—832no—
171#100C2×D8311yescharacteristic, complemented
172#97—831no—
173#41—22no—
174#41—3321no—
175#41—832no—
176#41—1662no—
177#41—1662no—
178#41—1662no—
179#41C22×D8312yes—
180C2×Q8—1662no—
181C2×Q8C22×D8311yescharacteristic
182C2×D4—22no—
183C2×D4—41no—
184C2×D4—3321no—
185C2×D4—3322no—
186C2×D4—832no—
187C2×D4—1662no—
188C2×D4—1662no—
189C2×D4C22×D8311yescharacteristic
190Q16—22no—
191Q16—832no—
192SD16—22no—
193SD16—832no—
194SD16—832no—
195SD16—1662no—
196SD16—1662no—
197SD16—1662no—
198SD16—1662no—
199SD16—1662no—
200D8—832no—
201D8—1662no—
202D8—1662no—
203D8—1662no—
204D8—1662no—
205D8—1662no—
206M16—832no—
207M16—1662no—
208M16—1662no—
209M16C22×D8312yes—
210C2×C8—832no—
211C2×C8—1662no—
212C2×C8—1662no—
213C22⋊C4—41no—
214C42C22×D8311yescharacteristic
215C23—21no—
216C23—41no—
217C23—3322no—
218C23—1662no—
219Q8—22no—
220Q8—3321no—
221Q8—832no—
222Q8D4×D8312yes—
223D4—22no—
224D4—22no—
225D4—42no—
226D4—3322no—
227D4—3321no—
228D4—3322no—
229D4—832no—
230D4—6642no—
231D4—1662no—
232D4—1662no—
233D4—1662no—
234D4D4×D8312yes—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—41no—
239C2×C4—3322no—
240C2×C4—3322no—
241C2×C4—1662no—
242C2×C4C23×D8311yescharacteristic
243C8—22no—
244C8—832no—
245C8—832no—
246C8—1662no—
247C22—22no—
248C22—41no—
249C22—81no—
250C22—161no—
251C22—13282no—
252C22—3322no—
253C22—6642no—
254C22—1662no—
255C22#1458183711yescharacteristic
256C4—22no—
257C4—41no—
258C4—41no—
259C4—1662no—
260C4C2×D4×D8312yes—
261C2—21no—
262C2—42no—
263C2—81no—
264C2—3322no—
265C2—6642no—
266C2#1912833311yescharacteristic
subgroupG/H
centerC2#19128333
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#455×C83C2
socleC2×C83D42
hypercenter#455D83
hypocenterC83#3288
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C2×C4 ◁ #455

Complements

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

NH ≅ G/Ntype
1. #19128510C2semidirect
2. #19128504C2semidirect
3. #19128091C2semidirect
4. #19127970C2semidirect
5. #19127966C2semidirect
6. #19127873C2semidirect
7. #19127870C2semidirect
8. #19127820C2semidirect
9. #455×C83C2semidirect
10. #14581840C22semidirect
11. #14581838C22semidirect
13. #14581820C22semidirect
16. #14581818C22semidirect
22. C2×#13050577C22semidirect
25. C2×#13050575C22semidirect
26. #14581800C22semidirect
27. #14581799C22semidirect
28. #14581789C22semidirect
29. #14581727C22semidirect
30. #14581720C22semidirect
31. #14581711C22semidirect
32. #14581708C22semidirect
33. #14581703C22semidirect
35. #137×C83C22semidirect
36. #124×C83C22semidirect
37. #104×C83C22semidirect
38. #100×C83C22semidirect
50. #13050580D4semidirect
53. #13050578D4semidirect
57. #13050577D4semidirect
63. #13050575D4semidirect
134. C83#3288semidirect
140. #455D83semidirect
148. #137C2×D83semidirect
165. #124C2×D83semidirect
168. #104C2×D83semidirect
171. #100C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191285102
#191285042
#191280912
#191279702
#191279662
#191278732
#191278702
#191278202
#455×C832
#328883