Finite Group Databasebeta

← #24917087

Subgroups — #24917087

Order 10624.

496 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 143 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×M16×D83C211yesmaximal, characteristic, complemented
2#19128208C211yesmaximal, characteristic
3C2×#14581768C211yesmaximal, characteristic, complemented
4#19127956C212yesmaximal, complemented
5#19127944C212yesmaximal, complemented
6#19127939C212yesmaximal, complemented
7#19127927C212yesmaximal, complemented
8#19127743C211yesmaximal, characteristic, complemented
9#19127710C211yesmaximal, characteristic, complemented
10#19127702C211yesmaximal, characteristic, complemented
11C2×#137×C83C211yesmaximal, characteristic, complemented
12C22×C4×D83C2211yescharacteristic
13C2×#13050574C2211yescharacteristic
14C2×#13050573C2212yes—
15M16×D83—24no—
16C2×C8×D83C2212yes—
17#14581773—21no—
18#14581768—21no—
19#14581768—21no—
20#14581768C2212yes—
21#14581768C2211yescharacteristic
22#14581768C2212yes—
23#14581768C2211yescharacteristic
24#14581763C2211yescharacteristic
25#14581763C2211yescharacteristic
26#14581760—21no—
27C4×C22⋊D83—21no—
28C4×C22⋊D83—21no—
29#14581751C2211yescharacteristic
30C2×C4⋊Dic83C2211yescharacteristic
31C2×Dic83⋊C4—21no—
32C2×(C2×D83)⋊C4—21no—
33#14581726C2212yescomplemented
34#14581724C2212yescomplemented
35#14581716C2212yescomplemented
36#14581716C2212yescomplemented
37#14581716C2212yescomplemented
38#14581713C2212yescomplemented
39C2×#41×C83C2211yescharacteristic
40C22×Q8×C83C2211yescharacteristic
41#137×C83—24no—
42C2×Q16×C83C2212yescomplemented
43C2×SD16×C83C2212yescomplemented
44C2×M16×C83C2211yescharacteristic
45C23×D83D411yescharacteristic
46C2×C22⋊D83—21no—
47C2×C22⋊D83—21no—
48C22×Dic83—21no—
49C22×Dic83D411yescharacteristic
50C2×C4×D83—22no—
51C2×C4×D83—22no—
52C2×C4×D83D411yescharacteristic
53C2×C4×D83D411yescharacteristic
54C2×C4×D83C2311yescharacteristic
55C2×C4×D83C2311yescharacteristic
56#13050574—22no—
57#13050573—24no—
58C2×C83⋊C8C2312yes—
59C8×D83—24no—
60C4⋊Dic83—21no—
61C4⋊Dic83C2311yescharacteristic
62C4⋊Dic83C2311yescharacteristic
63C4⋊Dic83C2312yes—
64Dic83⋊C4—22no—
65Dic83⋊C4—22no—
66Dic83⋊C4—21no—
67Dic83⋊C4—21no—
68C22⋊Dic83—21no—
69C22⋊Dic83—21no—
70C22⋊Dic83—21no—
71(C2×D83)⋊C4—21no—
72(C2×D83)⋊C4—21no—
73(C2×D83)⋊C4—22no—
74(C2×D83)⋊C4—22no—
75C4×Dic83—21no—
76#41×C83—22no—
77#41×C83—22no—
78C2×Q8×C83—21no—
79C2×Q8×C83—22no—
80C2×Q8×C83—22no—
81C2×Q8×C83C2312yes—
82C2×Q8×C83C2311yescharacteristic
83C2×D4×C83—21no—
84C2×D4×C83C2311yescharacteristic
85C22×C4×C83—21no—
86C22×C4×C83—21no—
87C22×C4×C83C2311yescharacteristic
88Q16×C83—24no—
89SD16×C83—24no—
90M16×C83D414yescomplemented
91C2×C8×C83C2312yes—
92C23×C83—21no—
93C23×C83C2×D411yescharacteristic
94C22×D83—22no—
95C22×D83—22no—
96C22×D83—41no—
97C22×D83C2×D411yescharacteristic
98C22×D83C2×D411yescharacteristic
99Q8×C83—22no—
100Q8×C83—21no—
101Q8×C83—22no—
102Q8×C83—21no—
103Q8×C83—42no—
104D4×C83—21no—
105D4×C83—21no—
106D4×C83—42no—
107C2×C4×C83—21no—
108C2×C4×C83—22no—
109C2×C4×C83—42no—
110C2×C4×C83—42no—
111C2×C4×C83C2×D411yescharacteristic
112C2×C4×C83C2×D412yes—
113C2×C4×C83#4112yes—
114C2×C4×C83C2411yescharacteristic
115C22⋊D83—42no—
116C22⋊D83—42no—
117C2×Dic83—22no—
118C2×Dic83—21no—
119C2×Dic83—21no—
120C2×Dic83—22no—
121C2×Dic83—42no—
122C2×Dic83C2×D411yescharacteristic
123C2×Dic83C2×D411yescharacteristic
124C4×D83—22no—
125C4×D83—22no—
126C4×D83—21no—
127C4×D83—21no—
128C4×D83—21no—
129C4×D83—21no—
130C8×C83C2×D414yes—
131C83⋊C8—22no—
132C22×C83—21no—
133C22×C83—21no—
134C22×C83—41no—
135C22×C83—41no—
136C22×C83—41no—
137C22×C83#12112yes—
138C22×C83C22×D411yescharacteristic
139C2×D83—21no—
140C2×D83—21no—
141C2×D83—21no—
142C2×D83—21no—
143C2×D83—21no—
144C2×D83—44no—
145C2×D83#13712yescomplemented
146C4×C83—41no—
147C4×C83—42no—
148C4×C83#12112yes—
149C4×C83C22×D411yescharacteristic
150C4×C83C2×#4111yescharacteristic
151Dic83—21no—
152Dic83—21no—
153Dic83—41no—
154Dic83—42no—
155Dic83—41no—
156C2×C83—22no—
157C2×C83—81no—
158C2×C83C2×#12111yescharacteristic
159C2×C83C2×#13711yescharacteristic
160C2×C83#57511yescharacteristic
161D83—22no—
162D83—41no—
163#3056—831nomaximal, Sylow
164C83#305611yescharacteristic, complemented, Sylow
165C2×#137D8311yescharacteristic, complemented
166C22×M16—831no—
167#525—831no—
168C2×C22⋊Q8—831no—
169#470—832no—
170#468—832no—
171#467—832no—
172#465—832no—
173#428—831no—
174#419—831no—
175#417—831no—
176C2×#41C2×D8311yescharacteristic
177C22×Q8C2×D8311yescharacteristic
178C23×C4—831no—
179#137—24no—
180C2×Q16C2×D8312yescomplemented
181C2×SD16C2×D8312yescomplemented
182C2×M16—831no—
183C2×M16—832no—
184C2×M16—1664no—
185C2×M16C2×D8311yescharacteristic
186C22×C8—832no—
187#123—1661no—
188C22⋊Q8—832no—
189C22⋊Q8—832no—
190C22⋊Q8—831no—
191C22⋊Q8—831no—
192C22⋊Q8—1661no—
193C22⋊Q8—1661no—
194#121—831no—
195#121—831no—
196#120—1661no—
197C4×D4—1661no—
198C4×D4—1661no—
199#117—831no—
200C2×C4⋊C4—831no—
201C2×C4⋊C4—1661no—
202C2×C22⋊C4—1661no—
203#107—832no—
204#106—832no—
205#103—832no—
206#103—832no—
207#103—832no—
208#102—832no—
209C24—831no—
210#41—22no—
211#41—22no—
212C2×Q8—21no—
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8C22×D8312yes—
216C2×Q8C22×D8311yescharacteristic
217C2×D4—21no—
218C2×D4—1661no—
219C2×D4—1661no—
220C2×D4C22×D8311yescharacteristic
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—831no—
224C22×C4—831no—
225C22×C4—831no—
226C22×C4—831no—
227C22×C4—831no—
228C22×C4—1661no—
229C22×C4—1662no—
230C22×C4—1662no—
231C22×C4C22×D8311yescharacteristic
232Q16—24no—
233SD16—24no—
234M16—1662no—
235M16—1664no—
236M16C22⋊D8314yescomplemented
237C2×C8—832no—
238C2×C8—1664no—
239C2×C8C22×D8312yes—
240C4⋊C4—832no—
241C4⋊C4—831no—
242C4⋊C4—831no—
243C4⋊C4—1662no—
244C4⋊C4—1662no—
245C4⋊C4—1661no—
246C4⋊C4—1661no—
247C4⋊C4—1661no—
248C22⋊C4—1662no—
249C22⋊C4—1662no—
250C22⋊C4—1661no—
251C22⋊C4—1661no—
252C22⋊C4—1661no—
253C22⋊C4—1661no—
254C22⋊C4—1661no—
255C42—1661no—
256C23—21no—
257C23—3321no—
258C23—831no—
259C23—831no—
260C23—1662no—
261C23—1662no—
262C23D4×D8311yescharacteristic
263Q8—22no—
264Q8—21no—
265Q8—22no—
266Q8—21no—
267Q8—42no—
268D4—21no—
269D4—21no—
270D4—42no—
271D4—3322no—
272D4—3322no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—3322no—
278C2×C4—831no—
279C2×C4—831no—
280C2×C4—1661no—
281C2×C4—1661no—
282C2×C4—1662no—
283C2×C4—1662no—
284C2×C4—1662no—
285C2×C4—1662no—
286C2×C4—1661no—
287C2×C4—1661no—
288C2×C4—1661no—
289C2×C4—1661no—
290C2×C4D4×D8311yescharacteristic
291C2×C4C2×C22⋊D8312yes—
292C2×C4#1305058912yes—
293C2×C4C23×D8311yescharacteristic
294C8—1662no—
295C8C2×C22⋊D8314yes—
296C22—21no—
297C22—21no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—3324no—
302C22—832no—
303C22—1661no—
304C22—1661no—
305C22—1661no—
306C22—1661no—
307C22—1661no—
308C22#1458176312yes—
309C22C2×D4×D8311yescharacteristic
310C4—41no—
311C4—42no—
312C4—3321no—
313C4—3322no—
314C4—3321no—
315C4—1661no—
316C4—1661no—
317C4#1458176312yes—
318C4C22×C22⋊D8311yescharacteristic
319C4C2×#1305058911yescharacteristic
320C2—22no—
321C2—81no—
322C2—3321no—
323C2—1662no—
324C2C2×#1458176311yescharacteristic
325C2#137×D8311yescharacteristic
326C2#1912850011yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
FittingC2×#137×C83C2
socleC22×C83C22×D4
prime radical#19128208C2
hypercenterC2×#137D83
hypocenterC83#3056
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×M16×D83C2semidirect
3. C2×#14581768C2semidirect
4. #19127956C2semidirect
5. #19127944C2semidirect
6. #19127939C2semidirect
7. #19127927C2semidirect
8. #19127743C2semidirect
9. #19127710C2semidirect
10. #19127702C2semidirect
11. C2×#137×C83C2semidirect
33. #14581726C22semidirect
34. #14581724C22semidirect
35. #14581716C22semidirect
36. #14581716C22semidirect
37. #14581716C22semidirect
38. #14581713C22semidirect
42. C2×Q16×C83C22semidirect
43. C2×SD16×C83C22semidirect
90. M16×C83D4semidirect
145. C2×D83#137semidirect
164. C83#3056semidirect
165. C2×#137D83semidirect
180. C2×Q16C2×D83semidirect
181. C2×SD16C2×D83semidirect
236. M16C22⋊D83semidirect

Maximal subgroups

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

subgroupindex
C2×M16×D832
#191282082
C2×#145817682
#191279562
#191279442
#191279392
#191279272
#191277432
#191277102
#191277022
C2×#137×C832
#305683