Finite Group Databasebeta

← #24917421

Subgroups — #24917421

Order 10624.

418 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 133 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#19128265C211yesmaximal, characteristic
2#19128025C212yesmaximal, complemented
3#19128025C212yesmaximal, complemented
4#19128019C212yesmaximal, complemented
5#19128014C212yesmaximal, complemented
6#19127741C211yesmaximal, characteristic, complemented
7#19127737C211yesmaximal, characteristic, complemented
8#19127705C211yesmaximal, characteristic, complemented
9C2×#14581712C211yesmaximal, characteristic, complemented
10#19127668C211yesmaximal, characteristic, complemented
11#520×C83C211yesmaximal, characteristic, complemented
12C22×D332C2211yescharacteristic
13C2×#13050574C2211yescharacteristic
14C22×C83⋊C8C2211yescharacteristic
15#14581788—21no—
16#14581781—21no—
17#14581766C2212yes—
18#14581766C2212yes—
19#14581760—22no—
20#14581751C2211yescharacteristic
21#14581725C2211yescharacteristic, complemented
22#14581725C2211yescharacteristic, complemented
23#14581725C2212yescomplemented
24#14581723C2211yescharacteristic, complemented
25#14581723C2211yescharacteristic, complemented
26#14581723C2212yescomplemented
27#14581712C2212yes—
28#14581712C2212yes—
29#14581712C2212yes—
30#14581712C2212yes—
31C22⋊(C83⋊C8)C2212yescomplemented
32C22⋊(C83⋊C8)C2212yescomplemented
33C2×#41×C83C2211yescharacteristic
34C4×Q8×C83—21no—
35C4×D4×C83—21no—
36C4×D4×C83—22no—
37#117×C83C2211yescharacteristic
38#117×C83C2212yescomplemented
39C2×C4⋊C4×C83C2211yescharacteristic
40C2×C4⋊C4×C83C2212yescomplemented
41C23×D83—21no—
42C2×D332—22no—
43C2×D332—22no—
44C2×D332—21no—
45C2×D332C2312yes—
46C2×C22⋊D83—22no—
47C2×C22⋊D83—22no—
48#13050574—22no—
49C2×C83⋊C8—22no—
50C2×C83⋊C8C2311yescharacteristic
51C2×C83⋊C8C2311yescharacteristic
52C2×C83⋊C8C2312yes—
53C4⋊Dic83C2311yescharacteristic
54C4⋊Dic83C2311yescharacteristic
55C22⋊Dic83—21no—
56(C2×D83)⋊C4—22no—
57(C2×D83)⋊C4—22no—
58C4×Dic83—21no—
59#41×C83—22no—
60#41×C83—22no—
61C2×Q8×C83D411yescharacteristic
62C2×D4×C83D412yes—
63C2×D4×C83D411yescharacteristic
64C22×C4×C83—21no—
65C22×C4×C83—22no—
66C22×C4×C83C2312yes—
67C22×C4×C83C2311yescharacteristic
68C4⋊C4×C83—21no—
69C4⋊C4×C83—22no—
70C4⋊C4×C83C2312yes—
71C4⋊C4×C83C2311yescharacteristic
72C4⋊C4×C83C2311yescharacteristic
73C22⋊C4×C83—21no—
74C22⋊C4×C83—22no—
75C42×C83—21no—
76C42×C83—22no—
77C23×C83C2×D412yes—
78C23×C83C2×D411yescharacteristic
79C22×D83—22no—
80C22×D83—42no—
81C22×D83—81no—
82C22×D83—81no—
83Q8×C83—22no—
84D4×C83—24no—
85D4×C83—22no—
86C2×C4×C83—22no—
87C2×C4×C83—22no—
88C2×C4×C83—21no—
89C2×C4×C83—21no—
90C2×C4×C83—22no—
91C2×C4×C83—81no—
92C2×C4×C83—82no—
93C2×C4×C83C2×D412yes—
94C2×C4×C83C2×D411yescharacteristic
95C2×C4×C83#4112yes—
96C2×C4×C83#4112yes—
97C2×C4×C83C2411yescharacteristic
98C22⋊D83—82no—
99C22⋊D83—82no—
100C2×Dic83—21no—
101C2×Dic83—21no—
102D332—22no—
103D332—22no—
104D332—42no—
105C83⋊C8—21no—
106C83⋊C8—21no—
107C83⋊C8—22no—
108C22×C83—21no—
109C22×C83—21no—
110C22×C83—22no—
111C22×C83—22no—
112C22×C83—22no—
113C22×C83#12312yes—
114C22×C83C22×D411yescharacteristic
115C2×D83—42no—
116C2×D83—42no—
117C2×D83—42no—
118C2×D83—84no—
119C4×C83—22no—
120C4×C83—41no—
121C4×C83—42no—
122C4×C83—41no—
123C4×C83#12312yes—
124C4×C83C2×#4111yescharacteristic
125C4×C83C2×#4111yescharacteristic
126Dic83—41no—
127Dic83—41no—
128C2×C83—22no—
129C2×C83—42no—
130C2×C83C2×#12311yescharacteristic
131C2×C83#57611yescharacteristic
132C2×C83#57711yescharacteristic
133D83—82no—
134#3096—831nomaximal, Sylow
135C83#309611yescharacteristic, complemented, Sylow
136#535—831no—
137#520D8311yescharacteristic, complemented
138#482—832no—
139#482—832no—
140#481—832no—
141#480—832no—
142#428—831no—
143#427—831no—
144#418—831no—
145C2×#102—831no—
146#408—831no—
147C2×#41C2×D8311yescharacteristic
148C22×D4—831no—
149C2×M16—831no—
150C22×C8—831no—
151#127—1661no—
152#124—1661no—
153#121—832no—
154#121—832no—
155#120—1662no—
156C4×Q8—21no—
157C4×D4—21no—
158C4×D4—22no—
159#117—831no—
160#117C2×D8311yescharacteristic
161#117C2×D8312yescomplemented
162C2×C4⋊C4C2×D8311yescharacteristic
163C2×C4⋊C4C2×D8312yescomplemented
164#107—831no—
165#107—831no—
166#107—832no—
167#106—831no—
168#106—831no—
169#106—832no—
170#102—832no—
171#102—832no—
172#102—832no—
173#102—832no—
174C22⋊C8—832no—
175C22⋊C8—832no—
176C24—1661no—
177#41—22no—
178#41—22no—
179C2×Q8C22⋊D8311yescharacteristic
180C2×D4—832no—
181C2×D4—1662no—
182C2×D4—1662no—
183C2×D4—1661no—
184C2×D4—1662no—
185C2×D4—1662no—
186C2×D4C22⋊D8312yes—
187C2×D4C22⋊D8311yescharacteristic
188C22×C4—21no—
189C22×C4—22no—
190C22×C4C22×D8312yes—
191C22×C4C22×D8311yescharacteristic
192M16—1662no—
193C2×C8—831no—
194C2×C8—832no—
195C2×C8—831no—
196C2×C8—1662no—
197C4⋊C4—21no—
198C4⋊C4—22no—
199C4⋊C4—831no—
200C4⋊C4—831no—
201C4⋊C4C22×D8312yes—
202C4⋊C4C22×D8311yescharacteristic
203C4⋊C4C22×D8311yescharacteristic
204C22⋊C4—21no—
205C22⋊C4—22no—
206C22⋊C4—1661no—
207C22⋊C4—1662no—
208C22⋊C4—1662no—
209C42—21no—
210C42—22no—
211C42—1661no—
212C23—3322no—
213C23—6641no—
214C23—6641no—
215C23—1662no—
216C23C2×C22⋊D8312yes—
217C23C2×C22⋊D8311yescharacteristic
218Q8—22no—
219D4—24no—
220D4—22no—
221D4—3322no—
222D4—6642no—
223D4—6642no—
224D4—1662no—
225D4—1662no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—81no—
232C2×C4—82no—
233C2×C4—1661no—
234C2×C4—1661no—
235C2×C4C2×C22⋊D8312yes—
236C2×C4C2×C22⋊D8311yescharacteristic
237C2×C4#1305059012yes—
238C2×C4#1305059012yes—
239C2×C4C23×D8311yescharacteristic
240C8—1661no—
241C8—1661no—
242C8—1662no—
243C22—21no—
244C22—21no—
245C22—22no—
246C22—22no—
247C22—22no—
248C22—3322no—
249C22—3322no—
250C22—3322no—
251C22—6644no—
252C22#1458177512yes—
253C22C22×C22⋊D8311yescharacteristic
254C4—22no—
255C4—41no—
256C4—42no—
257C4—41no—
258C4—3321no—
259C4—3321no—
260C4#1458177512yes—
261C4C2×#1305059011yescharacteristic
262C4C2×#1305059011yescharacteristic
263C2—22no—
264C2—42no—
265C2—6642no—
266C2C2×#1458177511yescharacteristic
267C2#1912850411yescharacteristic
268C2#1912850911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#520×C83C2
socleC22×C83C22×D4
prime radical#19128265C2
hypercenter#520D83
hypocenterC83#3096
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×#41 ◁ #520

Complements

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

NH ≅ G/Ntype
2. #19128025C2semidirect
3. #19128025C2semidirect
4. #19128019C2semidirect
5. #19128014C2semidirect
6. #19127741C2semidirect
7. #19127737C2semidirect
8. #19127705C2semidirect
9. C2×#14581712C2semidirect
10. #19127668C2semidirect
11. #520×C83C2semidirect
21. #14581725C22semidirect
22. #14581725C22semidirect
23. #14581725C22semidirect
24. #14581723C22semidirect
25. #14581723C22semidirect
26. #14581723C22semidirect
31. C22⋊(C83⋊C8)C22semidirect
32. C22⋊(C83⋊C8)C22semidirect
38. #117×C83C22semidirect
40. C2×C4⋊C4×C83C22semidirect
135. C83#3096semidirect
137. #520D83semidirect
161. #117C2×D83semidirect
163. C2×C4⋊C4C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191282652
#191280252
#191280192
#191280142
#191277412
#191277372
#191277052
C2×#145817122
#191276682
#520×C832
#309683