Finite Group Databasebeta

← #24917409

Subgroups — #24917409

Order 10624.

410 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 127 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×#14581768C211yesmaximal, characteristic
2#117×D83C211yesmaximal, characteristic
3#19128041C212yesmaximal, complemented
4#19128040C212yesmaximal, complemented
5#19128035C212yesmaximal, complemented
6#19128030C212yesmaximal, complemented
7#19127744C211yesmaximal, characteristic, complemented
8#19127743C211yesmaximal, characteristic, complemented
9#19127710C211yesmaximal, characteristic, complemented
10#19127664C211yesmaximal, characteristic
11#419×C83C211yesmaximal, characteristic, complemented
12C22×C4×D83C2211yescharacteristic
13C2×#13050574C2211yescharacteristic
14#14581768—21no—
15#14581768—21no—
16#14581768C2212yes—
17#14581768C2212yes—
18#14581751C2211yescharacteristic
19#14581750C2211yescharacteristic
20#14581750C2211yescharacteristic
21#14581749—21no—
22#14581748—21no—
23C2×C4⋊Dic83C2211yescharacteristic
24C2×Dic83⋊C4—21no—
25C4⋊C4×D83C2211yescharacteristic
26C4⋊C4×D83C2211yescharacteristic
27C2×(C2×D83)⋊C4—21no—
28C22⋊C4×D83—21no—
29C42×D83—21no—
30#14581726C2212yescomplemented
31#14581725C2212yescomplemented
32#14581724C2212yescomplemented
33#14581723C2212yescomplemented
34#14581716C2212yescomplemented
35#14581716C2212yescomplemented
36(C2×D83)⋊C8C2212yes—
37(C2×D83)⋊C8C2212yes—
38C22×Q8×C83C2211yescharacteristic
39C2×M16×C83C2211yescharacteristic
40#117×C83C2211yescharacteristic
41#103×C83C2212yescomplemented
42#103×C83C2212yescomplemented
43C23×D83D411yescharacteristic
44C22×Dic83—21no—
45C22×Dic83D411yescharacteristic
46C2×C4×D83—22no—
47C2×C4×D83—22no—
48C2×C4×D83—21no—
49C2×C4×D83—21no—
50C2×C4×D83D411yescharacteristic
51C2×C4×D83D411yescharacteristic
52C2×C4×D83C2311yescharacteristic
53C2×C4×D83C2311yescharacteristic
54#13050574—22no—
55C2×C83⋊C8C2312yes—
56C4⋊Dic83—21no—
57C4⋊Dic83C2312yes—
58C4⋊Dic83C2311yescharacteristic
59C4⋊Dic83C2311yescharacteristic
60Dic83⋊C4—22no—
61Dic83⋊C4—22no—
62Dic83⋊C4—21no—
63Dic83⋊C4—21no—
64C22⋊Dic83—21no—
65(C2×D83)⋊C4—22no—
66(C2×D83)⋊C4—22no—
67(C2×D83)⋊C4—21no—
68(C2×D83)⋊C4—21no—
69C4×Dic83—21no—
70C4×Dic83—21no—
71C4×Dic83—21no—
72C2×Q8×C83—21no—
73C2×Q8×C83—22no—
74C2×Q8×C83—22no—
75C2×Q8×C83C2312yes—
76C22×C4×C83—21no—
77C22×C4×C83C2311yescharacteristic
78M16×C83—22no—
79C2×C8×C83C2312yes—
80C4⋊C4×C83C2311yescharacteristic
81C4⋊C4×C83C2311yescharacteristic
82C22⋊C4×C83—21no—
83C42×C83—21no—
84C23×C83C2×D411yescharacteristic
85C22×D83—22no—
86C22×D83—22no—
87C22×D83—41no—
88C22×D83C2×D411yescharacteristic
89C22×D83C2×D411yescharacteristic
90Q8×C83—22no—
91Q8×C83—22no—
92Q8×C83—42no—
93C2×C4×C83—21no—
94C2×C4×C83—21no—
95C2×C4×C83—22no—
96C2×C4×C83—42no—
97C2×C4×C83C2×D411yescharacteristic
98C2×C4×C83#4112yes—
99C2×C4×C83#4112yes—
100C2×C4×C83C2411yescharacteristic
101C2×Dic83—22no—
102C2×Dic83—21no—
103C2×Dic83—21no—
104C2×Dic83—22no—
105C2×Dic83—42no—
106C2×Dic83C2×D411yescharacteristic
107C2×Dic83C2×D411yescharacteristic
108C4×D83—22no—
109C4×D83—22no—
110C4×D83—21no—
111C4×D83—21no—
112C4×D83—21no—
113C4×D83—21no—
114C4×D83—42no—
115C4×D83—42no—
116C8×C83—22no—
117C83⋊C8—22no—
118C22×C83—21no—
119C22×C83—21no—
120C22×C83#12312yes—
121C22×C83C22×D411yescharacteristic
122C2×D83—21no—
123C2×D83—21no—
124C2×D83—21no—
125C2×D83—21no—
126C2×D83—21no—
127C2×D83—44no—
128C2×D83#13712yes—
129C4×C83—42no—
130C4×C83—41no—
131C4×C83—41no—
132C4×C83#12312yes—
133C4×C83C2×#4111yescharacteristic
134C4×C83C2×#4111yescharacteristic
135Dic83—21no—
136Dic83—21no—
137Dic83—41no—
138Dic83—41no—
139Dic83—42no—
140C2×C83—22no—
141C2×C83C2×#12311yescharacteristic
142C2×C83C2×#13711yescharacteristic
143C2×C83#57511yescharacteristic
144D83—22no—
145D83—41no—
146#3095—831nomaximal, Sylow
147C83#309511yescharacteristic, complemented, Sylow
148C2×C22⋊Q8—831no—
149C2×#117—831no—
150#485—832no—
151#485—832no—
152#484—832no—
153#483—832no—
154#428—831no—
155#428—831no—
156#419—831no—
157#419D8311yescharacteristic, complemented
158#407—831no—
159C22×Q8C2×D8311yescharacteristic
160C23×C4—831no—
161C2×M16—831no—
162C2×M16C2×D8311yescharacteristic
163C22⋊Q8—832no—
164C22⋊Q8—832no—
165C22⋊Q8—1661no—
166C22⋊Q8—1661no—
167#117—831no—
168#117—831no—
169#117—831no—
170#117—1661no—
171#117—1661no—
172#117C2×D8311yescharacteristic
173C2×C4⋊C4—831no—
174C2×C4⋊C4—831no—
175C2×C4⋊C4—831no—
176C2×C4⋊C4—1661no—
177C2×C22⋊C4—1661no—
178C2×C22⋊C4—1661no—
179C2×C42—1661no—
180#107—832no—
181#107—832no—
182#106—832no—
183#106—832no—
184#103—832no—
185#103—832no—
186#103C2×D8312yescomplemented
187#103C2×D8312yescomplemented
188C22⋊C8—832no—
189C22⋊C8—832no—
190C24—831no—
191C2×Q8—21no—
192C2×Q8—22no—
193C2×Q8—22no—
194C2×Q8C22×D8312yes—
195C22×C4—21no—
196C22×C4—831no—
197C22×C4—831no—
198C22×C4—831no—
199C22×C4—831no—
200C22×C4—831no—
201C22×C4—1661no—
202C22×C4—1661no—
203C22×C4—1661no—
204C22×C4—1662no—
205C22×C4—1662no—
206C22×C4C22×D8311yescharacteristic
207M16—22no—
208M16—1662no—
209C2×C8—832no—
210C2×C8C22×D8312yes—
211C4⋊C4—831no—
212C4⋊C4—831no—
213C4⋊C4—832no—
214C4⋊C4—1661no—
215C4⋊C4—1661no—
216C4⋊C4—1662no—
217C4⋊C4—1662no—
218C4⋊C4—1661no—
219C4⋊C4C22×D8311yescharacteristic
220C4⋊C4C22×D8311yescharacteristic
221C22⋊C4—21no—
222C22⋊C4—1661no—
223C22⋊C4—1661no—
224C22⋊C4—1662no—
225C22⋊C4—1662no—
226C22⋊C4—1661no—
227C42—21no—
228C42—1661no—
229C42—1661no—
230C42—1661no—
231C23—3321no—
232C23—831no—
233C23—831no—
234C23—1662no—
235C23—1662no—
236C23D4×D8311yescharacteristic
237Q8—22no—
238Q8—22no—
239Q8—42no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—3322no—
245C2×C4—3322no—
246C2×C4—3322no—
247C2×C4—831no—
248C2×C4—831no—
249C2×C4—1661no—
250C2×C4—1661no—
251C2×C4—1662no—
252C2×C4—1662no—
253C2×C4—1662no—
254C2×C4—1662no—
255C2×C4—1661no—
256C2×C4—1661no—
257C2×C4—1661no—
258C2×C4—1661no—
259C2×C4D4×D8311yescharacteristic
260C2×C4#1305058912yes—
261C2×C4#1305059012yes—
262C2×C4C23×D8311yescharacteristic
263C8—22no—
264C8—1662no—
265C22—21no—
266C22—21no—
267C22—3324no—
268C22—832no—
269C22—1661no—
270C22—1661no—
271C22—1661no—
272C22—1661no—
273C22—1661no—
274C22#1458177412yes—
275C22C2×D4×D8311yescharacteristic
276C4—42no—
277C4—41no—
278C4—41no—
279C4—3321no—
280C4—3321no—
281C4—3322no—
282C4—1661no—
283C4—1661no—
284C4#1458177412yes—
285C4C2×#1305058911yescharacteristic
286C4C2×#1305059011yescharacteristic
287C2—22no—
288C2—3321no—
289C2—1662no—
290C2C2×#1458177411yescharacteristic
291C2#137×D8311yescharacteristic
292C2#1912850011yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#419×C83C2
socleC22×C83C22×D4
prime radicalC23×D83D4
hypercenter#419D83
hypocenterC83#3095
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
3. #19128041C2semidirect
4. #19128040C2semidirect
5. #19128035C2semidirect
6. #19128030C2semidirect
7. #19127744C2semidirect
8. #19127743C2semidirect
9. #19127710C2semidirect
11. #419×C83C2semidirect
30. #14581726C22semidirect
31. #14581725C22semidirect
32. #14581724C22semidirect
33. #14581723C22semidirect
34. #14581716C22semidirect
35. #14581716C22semidirect
41. #103×C83C22semidirect
42. #103×C83C22semidirect
147. C83#3095semidirect
157. #419D83semidirect
186. #103C2×D83semidirect
187. #103C2×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817682
#117×D832
#191280412
#191280402
#191280352
#191280302
#191277442
#191277432
#191277102
#191276642
#419×C832
#309583