Finite Group Databasebeta

← #24919249

Subgroups — #24919249

Order 10624.

454 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 125 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#19128509C212yesmaximal, complemented
2#19128503C211yesmaximal, characteristic, complemented
3#19128086C211yesmaximal, characteristic, complemented
4#19127973C211yesmaximal, characteristic, complemented
5#19127971C211yesmaximal, characteristic, complemented
6#19127966C212yesmaximal, complemented
7#19127877C212yesmaximal, complemented
8#19127874C212yesmaximal, complemented
9#19127818C211yesmaximal, characteristic, complemented
10#19127816C211yesmaximal, characteristic, complemented
11#578×C83C211yesmaximal, characteristic, complemented
12#14581838C2212yes—
13#14581823—22no—
14#14581823C2212yes—
15#14581819—22no—
16#14581817—22no—
17#14581817C2212yes—
18#14581817C2212yescomplemented
19#14581815—21no—
20#14581812—22no—
21#14581812C2211yescharacteristic
22C2×#13050577—22no—
23C2×#13050577C2212yescomplemented
24C2×D664C2211yescharacteristic
25D8×D83—21no—
26#14581799C2212yescomplemented
27#14581798C2211yescharacteristic
28#14581781C2212yescomplemented
29#14581728C2211yescharacteristic, complemented
30#14581727C2211yescharacteristic, complemented
31#14581719C2211yescharacteristic, complemented
32#14581719C2212yescomplemented
33#14581718C2211yescharacteristic
34#14581711C2212yescomplemented
35#14581708C2212yescomplemented
36C8×Dic83C2211yescharacteristic, complemented
372−1+4×C83C2212yescomplemented
38#137×C83—22no—
39#137×C83C2212yescomplemented
40#135×C83—21no—
41#135×C83C2211yescharacteristic
42C2×Q16×C83—21no—
43C2×Q16×C83C2211yescharacteristic, complemented
44#131×C83C2211yescharacteristic, complemented
45#13050591—22no—
46#13050590—22no—
47#13050590C2311yescharacteristic
48C2×D332—22no—
49C2×D332C2312yes—
50D4×D83—22no—
51D4×D83—42no—
52#13050580—22no—
53#13050580D411yescharacteristic
54#13050578D411yescharacteristic
55#13050577—22no—
56#13050577—22no—
57#13050577—22no—
58#13050577—22no—
59#13050577—22no—
60#13050577D411yescharacteristic
61D664—21no—
62D664—21no—
63#13050575—22no—
64#13050575—22no—
65#13050575D411yescharacteristic
66#13050574—22no—
67#13050574C2311yescharacteristic
68#13050574C2312yes—
69#13050573—21no—
70C2×C83⋊C8—22no—
71C2×C83⋊C8C2311yescharacteristic
72C8×D83—21no—
73(C2×D83)⋊C4—42no—
74C4×Dic83C2311yescharacteristic
75#41×C83—22no—
76#41×C83—22no—
77#41×C83—42no—
78#41×C83C2312yes—
79#41×C83C2311yescharacteristic
80C2×Q8×C83—22no—
81C2×Q8×C83—22no—
82C2×Q8×C83C2312yes—
83Q16×C83—22no—
84Q16×C83—21no—
85Q16×C83—21no—
86Q16×C83D411yescharacteristic
87SD16×C83—22no—
88SD16×C83D412yes—
89D8×C83D411yescharacteristic
90M16×C83—21no—
91M16×C83C2311yescharacteristic
92C2×C8×C83—21no—
93C2×C8×C83C2311yescharacteristic
94C22×D83—22no—
95C22×D83—42no—
96Q8×C83—22no—
97Q8×C83—22no—
98Q8×C83—42no—
99Q8×C83C2×D411yescharacteristic
100Q8×C83C2×D412yes—
101D4×C83—21no—
102D4×C83—22no—
103D4×C83—42no—
104D4×C83C2×D411yescharacteristic
105D4×C83C2×D412yes—
106C2×C4×C83—21no—
107C2×C4×C83—22no—
108C2×C4×C83—22no—
109C2×C4×C83—42no—
110C2×C4×C83—42no—
111C2×C4×C83C2411yescharacteristic
112C22⋊D83—21no—
113C22⋊D83—42no—
114C2×Dic83—21no—
115D332—22no—
116D332—22no—
117D332—42no—
118D332C2×D411yescharacteristic
119C4×D83—21no—
120C4×D83—42no—
121Dic166C2×D411yescharacteristic
122C8×C83—21no—
123C8×C83C2×D411yescharacteristic
124C8×C83C2×D411yescharacteristic
125C83⋊C8—21no—
126C83⋊C8—22no—
127C83⋊C8C2×D411yescharacteristic
128C83⋊C8C2×D411yescharacteristic
129C22×C83—22no—
130C22×C83—21no—
131C22×C83H(5,𝔽2)11yescharacteristic
132C2×D83—21no—
133C2×D83—42no—
134C2×D83—82no—
135C2×D83—162no—
136C4×C83—22no—
137C4×C83—21no—
138C4×C83—42no—
139C4×C83C22×D411yescharacteristic
140C4×C83C22×D411yescharacteristic
141Dic83—21no—
142Dic83—41no—
143C2×C83—21no—
144C2×C83—42no—
145C2×C83—41no—
146C2×C83D4211yescharacteristic
147D83—41no—
148D83—82no—
149#3290—831nomaximal, Sylow
150C83#329011yescharacteristic, complemented, Sylow
151#578D8311yescharacteristic, complemented
152#577—832no—
153#576—831no—
154#495—831no—
155#473—831no—
156#472—831no—
157#471—832no—
158#455—832no—
159#454—832no—
160#443—831no—
161#443—831no—
1622−1+4C2×D8312yescomplemented
163H(5,𝔽2)—832no—
164#137—22no—
165#137—832no—
166#137—1662no—
167#137C2×D8312yescomplemented
168AGL(1,ℤ/8)—832no—
169AGL(1,ℤ/8)—832no—
170AGL(1,ℤ/8)—1662no—
171AGL(1,ℤ/8)—1662no—
172#135—21no—
173#135—831no—
174#135—1662no—
175#135—1661no—
176#135C2×D8311yescharacteristic
177C2×Q16—21no—
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, complemented
186#124—832no—
187#108—831no—
188#108—831no—
189#104—831no—
190#104—832no—
191#104—831no—
192#101—832no—
193#100—832no—
194C4×C8—831no—
195#41—22no—
196#41—22no—
197#41—42no—
198#41—831no—
199#41—1662no—
200#41—1662no—
201#41C22×D8311yescharacteristic
202#41C22×D8312yes—
203C2×Q8—22no—
204C2×Q8—22no—
205C2×Q8C22×D8312yes—
206C2×D4—3322no—
207C2×D4—832no—
208C2×D4—1662no—
209C2×D4—1662no—
210Q16—22no—
211Q16—21no—
212Q16—21no—
213Q16—831no—
214Q16—1662no—
215Q16C22⋊D8311yescharacteristic
216SD16—22no—
217SD16—831no—
218SD16—831no—
219SD16—1662no—
220SD16—1662no—
221SD16—1662no—
222SD16—1662no—
223SD16—1662no—
224SD16C22⋊D8312yes—
225D8—831no—
226D8—1662no—
227D8—1662no—
228D8—1661no—
229D8—1661no—
230D8C22⋊D8311yescharacteristic
231M16—21no—
232M16—832no—
233M16—831no—
234M16—1662no—
235M16—1661no—
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—42no—
249Q8—831no—
250Q8D4×D8311yescharacteristic
251Q8C2×C22⋊D8312yes—
252D4—22no—
253D4—21no—
254D4—42no—
255D4—3322no—
256D4—3322no—
257D4—831no—
258D4—1661no—
259D4—1662no—
260D4—1662no—
261D4D4×D8311yescharacteristic
262D4C2×C22⋊D8312yes—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—3322no—
269C2×C4—1661no—
270C2×C4—1661no—
271C2×C4C23×D8311yescharacteristic
272C8—21no—
273C8—831no—
274C8—831no—
275C8—1662no—
276C8—1661no—
277C8C2×C22⋊D8311yescharacteristic
278C8C2×C22⋊D8311yescharacteristic
279C22—21no—
280C22—22no—
281C22—13282no—
282C22—3322no—
283C22—6642no—
284C22—1661no—
285C22#1458183711yescharacteristic
286C4—21no—
287C4—22no—
288C4—42no—
289C4—3321no—
290C4—1661no—
291C4C2×D4×D8311yescharacteristic
292C4C22×C22⋊D8311yescharacteristic
293C2—21no—
294C2—42no—
295C2—41no—
296C2—3321no—
297C2—6642no—
298C2D4×C22⋊D8311yescharacteristic
subgroupG/H
centerC2D4×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#578×C83C2
socleC2×C83D42
hypercenter#578D83
hypocenterC83#3290
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C2×C4 ◁ #578

Complements

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

NH ≅ G/Ntype
1. #19128509C2semidirect
2. #19128503C2semidirect
3. #19128086C2semidirect
4. #19127973C2semidirect
5. #19127971C2semidirect
6. #19127966C2semidirect
7. #19127877C2semidirect
8. #19127874C2semidirect
9. #19127818C2semidirect
10. #19127816C2semidirect
11. #578×C83C2semidirect
18. #14581817C22semidirect
23. C2×#13050577C22semidirect
26. #14581799C22semidirect
28. #14581781C22semidirect
29. #14581728C22semidirect
30. #14581727C22semidirect
31. #14581719C22semidirect
32. #14581719C22semidirect
34. #14581711C22semidirect
35. #14581708C22semidirect
36. C8×Dic83C22semidirect
37. 2−1+4×C83C22semidirect
39. #137×C83C22semidirect
43. C2×Q16×C83C22semidirect
44. #131×C83C22semidirect
150. C83#3290semidirect
151. #578D83semidirect
162. 2−1+4C2×D83semidirect
167. #137C2×D83semidirect
178. C2×Q16C2×D83semidirect
185. #131C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191285092
#191285032
#191280862
#191279732
#191279712
#191279662
#191278772
#191278742
#191278182
#191278162
#578×C832
#329083