Finite Group Databasebeta

← #24916729

Subgroups — #24916729

Order 10624.

784 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#14581838C211yesmaximal, characteristic, complemented
2C2×#14581818C211yesmaximal, characteristic, complemented
3C22×#13050575C211yesmaximal, characteristic, complemented
4#19127844C214yesmaximal, complemented
5#19127834C214yesmaximal, complemented
6#19127705C211yesmaximal, characteristic, complemented
7#19127695C211yesmaximal, characteristic, complemented
8#19127668C211yesmaximal, characteristic, complemented
9#535×C83C211yesmaximal, characteristic, complemented
10#14581838—24no—
11#14581838—24no—
12C2×#13050591—21no—
13C2×#13050590C2211yescharacteristic
14C2×#13050590C2212yescomplemented
15C22×D332C2211yescharacteristic
16C22×D332C2212yescomplemented
17C2×D4×D83—22no—
18C2×D4×D83—21no—
19#14581818—24no—
20C2×#13050578C2212yescomplemented
21C2×#13050575—24no—
22C2×#13050575C2212yescomplemented
23C2×#13050575C2214yescomplemented
24C2×#13050574C2211yescharacteristic, complemented
25C22×C83⋊C8C2211yescharacteristic, complemented
26#14581715C2212yescomplemented
27#14581712C2212yescomplemented
28#14581712C2214yescomplemented
29C22⋊(C83⋊C8)C2214yescomplemented
30C2×#41×C83C2211yescharacteristic
31C22×D4×C83C2211yescharacteristic, complemented
32#127×C83—21no—
33#124×C83—21no—
34#121×C83C2214yescomplemented
35#120×C83—22no—
36#117×C83C2211yescharacteristic, complemented
37C23×D83—21no—
38C23×D83—22no—
39#13050591—28no—
40#13050590—22no—
41#13050590—24no—
42#13050590—24no—
43#13050590D414yescomplemented
44C2×Dic166C2311yescharacteristic
45C2×D332—21no—
46C2×D332—22no—
47C2×D332—22no—
48C2×D332—24no—
49C2×D332—24no—
50C2×D332D414yescomplemented
51C2×D332C2312yes—
52C2×D332C2311yescharacteristic
53C2×C22⋊D83—21no—
54C2×C22⋊D83—22no—
55D4×D83—216no—
56D4×D83—28no—
57C2×C4×D83—21no—
58C2×C4×D83—22no—
59#13050578—24no—
60#13050575—28no—
61#13050575—24no—
62#13050574—22no—
63C2×C83⋊C8—22no—
64C2×C83⋊C8C2312yes—
65C2×C83⋊C8C2312yes—
66C24×C83—21no—
67#41×C83—22no—
68#41×C83—22no—
69C2×Q8×C83D411yescharacteristic
70C2×D4×C83—24no—
71C2×D4×C83—22no—
72C2×D4×C83—22no—
73C2×D4×C83—21no—
74C2×D4×C83D412yes—
75C2×D4×C83D411yescharacteristic
76C2×D4×C83C2312yes—
77C22×C4×C83C2312yes—
78C22×C4×C83C2311yescharacteristic
79C4⋊C4×C83C2312yes—
80C22⋊C4×C83—21no—
81C22⋊C4×C83—24no—
82C42×C83—21no—
83C23×C83—22no—
84C23×C83—42no—
85C23×C83—82no—
86C23×C83C2×D412yes—
87C23×C83C2×D411yescharacteristic
88C22×D83—21no—
89C22×D83—22no—
90C22×D83—28no—
91C22×D83—24no—
92C22×D83—82no—
93C22×D83—84no—
94Q8×C83—22no—
95D4×C83—24no—
96D4×C83—22no—
97D4×C83—22no—
98D4×C83—22no—
99D4×C83—42no—
100D4×C83—84no—
101C2×C4×C83—22no—
102C2×C4×C83—22no—
103C2×C4×C83—22no—
104C2×C4×C83C2×D412yes—
105C2×C4×C83C2×D412yes—
106C2×C4×C83C2×D412yes—
107C2×C4×C83C2×D411yescharacteristic
108C2×C4×C83C2411yescharacteristic
109C22⋊D83—28no—
110C22⋊D83—24no—
111C2×Dic83—21no—
112D332—28no—
113D332—24no—
114D332—21no—
115D332—22no—
116D332C2×D412yes—
117D332C2×D414yes—
118C4×D83—28no—
119C4×D83—24no—
120Dic166—21no—
121Dic166C2×D412yes—
122C83⋊C8—22no—
123C83⋊C8—22no—
124C22×C83—22no—
125C22×C83—22no—
126C22×C83—24no—
127C22×C83—42no—
128C22×C83—44no—
129C22×C83—84no—
130C22×C83#12012yes—
131C22×C83C22×D411yescharacteristic
132C2×D83—24no—
133C2×D83—22no—
134C2×D83—44no—
135C2×D83—42no—
136C2×D83—84no—
137C2×D83—88no—
138C4×C83—22no—
139C4×C83—42no—
140C4×C83#12012yes—
141C4×C83C22×D411yescharacteristic
142C4×C83C22×D411yescharacteristic
143Dic83—22no—
144C2×C83—22no—
145C2×C83—42no—
146C2×C83—82no—
147C2×C83C2×#12011yescharacteristic
148C2×C83#57612yes—
149D83—44no—
150D83—42no—
151#3008—831nomaximal, Sylow
152C83#300811yescharacteristic, complemented, Sylow
153C2×H(5,𝔽2)—831no—
154C2×AGL(1,ℤ/8)—831no—
155C22×D8—831no—
156#535D8311yescharacteristic, complemented
157#449—834no—
158#447—834no—
159#418—831no—
160#416—831no—
161#408—831no—
162H(5,𝔽2)—1664no—
163H(5,𝔽2)—1664no—
164C2×#41—831no—
165C2×#41—832no—
166C2×#41—1661no—
167C2×#41C2×D8311yescharacteristic
168C22×D4—831no—
169C22×D4—832no—
170C22×D4—1662no—
171C22×D4—1661no—
172C22×D4C2×D8311yescharacteristic, complemented
173AGL(1,ℤ/8)—1664no—
174C2×SD16—832no—
175C2×D8—834no—
176C2×D8—832no—
177C2×D8—1664no—
178C2×M16—831no—
179C22×C8—831no—
180#127—21no—
181#124—21no—
182#121C2×D8314yescomplemented
183#120—22no—
184#117C2×D8311yescharacteristic, complemented
185#103—832no—
186#102—832no—
187#102—834no—
188C22⋊C8—834no—
189C24—21no—
190C24—1662no—
191C24—1661no—
192#41—22no—
193#41—22no—
194#41—834no—
195#41—1664no—
196#41—1664no—
197#41—1668no—
198#41—1662no—
199C2×Q8—831no—
200C2×Q8C22⋊D8311yescharacteristic
201C2×D4—24no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—21no—
205C2×D4—834no—
206C2×D4—832no—
207C2×D4—831no—
208C2×D4—1664no—
209C2×D4—1664no—
210C2×D4—16616no—
211C2×D4—1668no—
212C2×D4—1662no—
213C2×D4—1661no—
214C2×D4—1662no—
215C2×D4—1662no—
216C2×D4—1661no—
217C2×D4C22⋊D8312yes—
218C2×D4C22⋊D8311yescharacteristic
219C2×D4C22×D8312yes—
220C22×C4—1662no—
221C22×C4—1661no—
222C22×C4C22×D8312yes—
223C22×C4C22×D8311yescharacteristic
224SD16—1664no—
225D8—1668no—
226D8—1664no—
227M16—1662no—
228C2×C8—832no—
229C2×C8—832no—
230C2×C8—1662no—
231C4⋊C4C22×D8312yes—
232C22⋊C4—21no—
233C22⋊C4—24no—
234C42—21no—
235C23—22no—
236C23—42no—
237C23—82no—
238C23—6642no—
239C23—6644no—
240C23—1668no—
241C23—1662no—
242C23—1661no—
243C23—1664no—
244C23C2×C22⋊D8312yes—
245C23C2×C22⋊D8311yescharacteristic
246Q8—22no—
247Q8—832no—
248Q8—1661no—
249D4—24no—
250D4—22no—
251D4—22no—
252D4—22no—
253D4—42no—
254D4—84no—
255D4—834no—
256D4—832no—
257D4—1668no—
258D4—1668no—
259D4—1664no—
260D4—1664no—
261D4—1661no—
262D4—1662no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—1668no—
267C2×C4—1661no—
268C2×C4—1664no—
269C2×C4D4×D8312yes—
270C2×C4D4×D8312yes—
271C2×C4C2×C22⋊D8312yes—
272C2×C4C2×C22⋊D8311yescharacteristic
273C2×C4C23×D8311yescharacteristic
274C8—1662no—
275C8—1662no—
276C22—22no—
277C22—22no—
278C22—24no—
279C22—42no—
280C22—44no—
281C22—84no—
282C22—3324no—
283C22—3322no—
284C22—6644no—
285C22—6648no—
286C22—1664no—
287C22—1662no—
288C22#1458176012yes—
289C22C22×C22⋊D8311yescharacteristic
290C4—22no—
291C4—42no—
292C4—1662no—
293C4#1458176012yes—
294C4C2×D4×D8311yescharacteristic
295C4C2×D4×D8311yescharacteristic
296C2—22no—
297C2—42no—
298C2—82no—
299C2—3324no—
300C2—3322no—
301C2C2×#1458176011yescharacteristic
302C2#1912850412yes—
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#535×C83C2
socleC22×C83C22×D4
hypercenter#535D83
hypocenterC83#3008
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#14581838C2semidirect
2. C2×#14581818C2semidirect
3. C22×#13050575C2semidirect
4. #19127844C2semidirect
5. #19127834C2semidirect
6. #19127705C2semidirect
7. #19127695C2semidirect
8. #19127668C2semidirect
9. #535×C83C2semidirect
14. C2×#13050590C22semidirect
16. C22×D332C22semidirect
20. C2×#13050578C22semidirect
22. C2×#13050575C22semidirect
23. C2×#13050575C22semidirect
24. C2×#13050574C22semidirect
25. C22×C83⋊C8C22semidirect
26. #14581715C22semidirect
27. #14581712C22semidirect
28. #14581712C22semidirect
29. C22⋊(C83⋊C8)C22semidirect
31. C22×D4×C83C22semidirect
34. #121×C83C22semidirect
36. #117×C83C22semidirect
43. #13050590D4semidirect
50. C2×D332D4semidirect
152. C83#3008semidirect
156. #535D83semidirect
172. C22×D4C2×D83semidirect
182. #121C2×D83semidirect
184. #117C2×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145818382
C2×#145818182
C22×#130505752
#191278442
#191278342
#191277052
#191276952
#191276682
#535×C832
#300883