Finite Group Databasebeta

← #24919252

Subgroups — #24919252

Order 10624.

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

normal subgroups only

This group has 454 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#19128514C211yesmaximal, characteristic, complemented
2#19128510C212yesmaximal, complemented
3#19128506C211yesmaximal, characteristic, complemented
4#19128086C211yesmaximal, characteristic, complemented
5#19127972C211yesmaximal, characteristic, complemented
6#19127967C212yesmaximal, complemented
7#19127877C212yesmaximal, complemented
8#19127876C212yesmaximal, complemented
9#19127817C211yesmaximal, characteristic, complemented
10#19127816C211yesmaximal, characteristic, complemented
11#473×C83C211yesmaximal, characteristic, complemented
12#14581840C2212yescomplemented
13#14581838C2212yes—
14#14581824—22no—
15#14581824—22no—
16#14581823C2212yescomplemented
17#14581820—22no—
18#14581820C2212yes—
19#14581820C2212yescomplemented
20#14581819—22no—
21#14581815—22no—
22#14581815—21no—
23#14581814—21no—
24#14581814C2211yescharacteristic
25#14581812C2211yescharacteristic
26Q16×D83—21no—
27C2×#13050577C2212yescomplemented
28SD16×D83—22no—
29C2×D664—21no—
30C2×D664C2211yescharacteristic
31#14581800C2212yescomplemented
32#14581798C2211yescharacteristic, complemented
33#14581781C2212yescomplemented
34#14581727C2211yescharacteristic, complemented
35#14581719C2211yescharacteristic
36#14581718C2211yescharacteristic, complemented
37#14581718C2212yescomplemented
38#14581708C2212yescomplemented
39C8×Dic83C2211yescharacteristic, complemented
40#137×C83C2212yes—
41C2×Q16×C83C2211yescharacteristic, complemented
42#131×C83C2211yescharacteristic
43#108×C83C2211yescharacteristic, complemented
44#101×C83C2212yescomplemented
45#13050591—22no—
46#13050591—22no—
47#13050591—22no—
48#13050590—22no—
49#13050590—42no—
50#13050590C2311yescharacteristic
51#13050590C2312yes—
52Q8×D83—22no—
53Q8×D83—22no—
54C2×D332—22no—
55C2×D332C2312yes—
56D4×D83—22no—
57D4×D83—42no—
58Dic332D411yescharacteristic
59#13050580—22no—
60#13050580D411yescharacteristic
61#13050579—22no—
62#13050579—22no—
63#13050579D412yes—
64#13050578D411yescharacteristic
65#13050577—22no—
66#13050577—22no—
67#13050577—22no—
68#13050577D411yescharacteristic
69D664—22no—
70D664—22no—
71D664—21no—
72D664—21no—
73D664D411yescharacteristic
74#13050575D411yescharacteristic
75#13050574C2311yescharacteristic
76#13050573—21no—
77#13050573—22no—
78C2×C83⋊C8C2311yescharacteristic
79C8×D83—21no—
80C8×D83—22no—
81(C2×D83)⋊C4—42no—
82C4×Dic83C2311yescharacteristic
83#41×C83C2311yescharacteristic
84C2×Q8×C83C2312yes—
85Q16×C83—22no—
86Q16×C83—21no—
87Q16×C83—21no—
88SD16×C83—22no—
89M16×C83—21no—
90M16×C83C2312yes—
91M16×C83C2311yescharacteristic
92C2×C8×C83—21no—
93C2×C8×C83C2311yescharacteristic
94C22×D83—22no—
95C22×D83—42no—
96Q8×C83—22no—
97Q8×C83—22no—
98Q8×C83C2×D411yescharacteristic
99D4×C83—21no—
100D4×C83C2×D411yescharacteristic
101C2×C4×C83—21no—
102C2×C4×C83—22no—
103C2×C4×C83C2411yescharacteristic
104C22⋊D83—22no—
105C22⋊D83—21no—
106C22⋊D83—42no—
107C2×Dic83—21no—
108D332—22no—
109D332—22no—
110D332—42no—
111D332—42no—
112D332C2×D411yescharacteristic
113D332C2×D412yes—
114C4×D83—22no—
115C4×D83—21no—
116C4×D83—42no—
117C4×D83—42no—
118C4×D83—42no—
119Dic166—42no—
120Dic166C2×D411yescharacteristic
121Dic166C2×D412yes—
122C8×C83—21no—
123C8×C83—22no—
124C8×C83C2×D411yescharacteristic
125C8×C83C2×D411yescharacteristic
126C83⋊C8—21no—
127C83⋊C8C2×D411yescharacteristic
128C83⋊C8C2×D411yescharacteristic
129C22×C83—21no—
130C22×C83H(5,𝔽2)11yescharacteristic
131C2×D83—21no—
132C2×D83—22no—
133C2×D83—42no—
134C2×D83—82no—
135C2×D83—162no—
136C4×C83—21no—
137C4×C83—42no—
138C4×C83C22×D411yescharacteristic
139C4×C83C22×D411yescharacteristic
140Dic83—21no—
141Dic83—22no—
142Dic83—41no—
143C2×C83—21no—
144C2×C83—41no—
145C2×C83D4211yescharacteristic
146D83—41no—
147D83—42no—
148D83—82no—
149#3290—831nomaximal, Sylow
150C83#329011yescharacteristic, complemented, Sylow
151#578—831no—
152#577—832no—
153#576—831no—
154#495—831no—
155#473D8311yescharacteristic, complemented
156#472—831no—
157#471—832no—
158#455—832no—
159#454—832no—
160#443—831no—
161#443—831no—
1622−1+4—832no—
163H(5,𝔽2)—832no—
164#137—832no—
165#137—1662no—
166#137—1662no—
167#137C2×D8312yes—
168AGL(1,ℤ/8)—832no—
169AGL(1,ℤ/8)—832no—
170AGL(1,ℤ/8)—1662no—
171AGL(1,ℤ/8)—1662no—
172#135—831no—
173#135—831no—
174#135—1661no—
175#135—1661no—
176#135—1662no—
177C2×Q16—1661no—
178C2×Q16C2×D8311yescharacteristic, complemented
179C2×SD16—832no—
180C2×SD16—1662no—
181C2×D8—831no—
182C2×D8—1661no—
183#131—832no—
184#131—831no—
185#131C2×D8311yescharacteristic
186#124—832no—
187#108—831no—
188#108C2×D8311yescharacteristic, complemented
189#104—831no—
190#104—831no—
191#104—832no—
192#101C2×D8312yescomplemented
193#100—832no—
194C4×C8—831no—
195#41—3322no—
196#41—831no—
197#41—832no—
198#41—1662no—
199#41—1662no—
200#41—1662no—
201#41—1662no—
202#41C22×D8311yescharacteristic
203C2×Q8—1662no—
204C2×Q8—1662no—
205C2×Q8C22×D8312yes—
206C2×D4—3322no—
207C2×D4—832no—
208C2×D4—1662no—
209C2×D4—1662no—
210Q16—22no—
211Q16—21no—
212Q16—21no—
213Q16—831no—
214Q16—831no—
215Q16—1662no—
216SD16—22no—
217SD16—831no—
218SD16—831no—
219SD16—832no—
220SD16—1662no—
221SD16—1662no—
222SD16—1662no—
223SD16—1662no—
224SD16—1662no—
225D8—831no—
226D8—831no—
227D8—1662no—
228D8—1662no—
229D8—1661no—
230D8—1661no—
231M16—21no—
232M16—831no—
233M16—1662no—
234M16—1661no—
235M16C22×D8312yes—
236M16C22×D8311yescharacteristic
237C2×C8—21no—
238C2×C8—831no—
239C2×C8—1662no—
240C2×C8—1661no—
241C2×C8C22×D8311yescharacteristic
242C22⋊C4—3322no—
243C42—831no—
244C23—3322no—
245C23—1662no—
246Q8—22no—
247Q8—22no—
248Q8—3322no—
249Q8—831no—
250Q8—832no—
251Q8D4×D8311yescharacteristic
252D4—21no—
253D4—3322no—
254D4—3322no—
255D4—3322no—
256D4—831no—
257D4—832no—
258D4—1662no—
259D4—1661no—
260D4—1662no—
261D4—1662no—
262D4D4×D8311yescharacteristic
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—3322no—
266C2×C4—3322no—
267C2×C4—3322no—
268C2×C4—1661no—
269C2×C4—1662no—
270C2×C4—1661no—
271C2×C4C23×D8311yescharacteristic
272C8—21no—
273C8—22no—
274C8—831no—
275C8—831no—
276C8—1661no—
277C8D4×D8311yescharacteristic
278C8D4×D8311yescharacteristic
279C22—21no—
280C22—13282no—
281C22—3322no—
282C22—6642no—
283C22—1661no—
284C22—1662no—
285C22#1458183811yescharacteristic
286C4—21no—
287C4—42no—
288C4—3321no—
289C4—1662no—
290C4—1661no—
291C4C2×D4×D8311yescharacteristic
292C4C2×D4×D8311yescharacteristic
293C2—21no—
294C2—41no—
295C2—3322no—
296C2—3321no—
297C2—6642no—
298C2#1912833511yescharacteristic
subgroupG/H
centerC2#19128335
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#473×C83C2
socleC2×C83D42
hypercenter#473D83
hypocenterC83#3290
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C2×C4 ◁ #473

Complements

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

NH ≅ G/Ntype
1. #19128514C2semidirect
2. #19128510C2semidirect
3. #19128506C2semidirect
4. #19128086C2semidirect
5. #19127972C2semidirect
6. #19127967C2semidirect
7. #19127877C2semidirect
8. #19127876C2semidirect
9. #19127817C2semidirect
10. #19127816C2semidirect
11. #473×C83C2semidirect
12. #14581840C22semidirect
16. #14581823C22semidirect
19. #14581820C22semidirect
27. C2×#13050577C22semidirect
31. #14581800C22semidirect
32. #14581798C22semidirect
33. #14581781C22semidirect
34. #14581727C22semidirect
36. #14581718C22semidirect
37. #14581718C22semidirect
38. #14581708C22semidirect
39. C8×Dic83C22semidirect
41. C2×Q16×C83C22semidirect
43. #108×C83C22semidirect
44. #101×C83C22semidirect
150. C83#3290semidirect
155. #473D83semidirect
178. C2×Q16C2×D83semidirect
188. #108C2×D83semidirect
192. #101C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191285142
#191285102
#191285062
#191280862
#191279722
#191279672
#191278772
#191278762
#191278172
#191278162
#473×C832
#329083