Finite Group Databasebeta

← #24910727

Subgroups — #24910727

Order 10624.

560 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 147 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×#14581754C211yesmaximal, characteristic
2C2×#14581750C211yesmaximal, characteristic, complemented
3#19127640C211yesmaximal, characteristic, complemented
4#19127638C211yesmaximal, characteristic, complemented
5#19127628C212yesmaximal, complemented
6#19127627C211yesmaximal, characteristic, complemented
7#19127618C212yesmaximal
8C4⋊C4×Dic83C212yesmaximal, complemented
9#19127548C212yesmaximal, complemented
10#19127548C211yesmaximal, characteristic, complemented
11#398×C83C211yesmaximal, characteristic, complemented
12C22×D332C2211yescharacteristic
13C22×C4×D83C2211yescharacteristic
14C22×C4×D83C2211yescharacteristic
15#14581754—24no—
16#14581750—24no—
17C2×C4⋊Dic83C2212yes—
18C2×C4⋊Dic83C2211yescharacteristic
19C2×(C2×D83)⋊C4C2212yes—
20C2×(C2×D83)⋊C4C2212yes—
21C2×(C2×D83)⋊C4C2212yes—
22C2×(C2×D83)⋊C4C2212yes—
23C2×(C2×D83)⋊C4C2211yescharacteristic
24C2×(C2×D83)⋊C4C2211yescharacteristic
25C2×C4×Dic83C2211yescharacteristic
26C2×C4×Dic83C2212yes—
27C2×C4×Dic83C2212yes—
28C83⋊#95C2212yes—
29C83⋊#95C2212yes—
30C83⋊#95C2211yescharacteristic
31C83⋊#95C2212yes—
32C2×C4⋊C4×C83C2212yes—
33C2×C4⋊C4×C83C2211yescharacteristic
34C2×C4⋊C4×C83C2211yescharacteristic
35#95×C83C2212yes—
36#95×C83C2211yescharacteristic
37C23×D83C2311yescharacteristic
38C23×D83C2311yescharacteristic
39C2×D332—22no—
40C2×D332—22no—
41C2×D332—22no—
42C22×Dic83C2312yes—
43C22×Dic83C2311yescharacteristic
44C22×Dic83C2312yes—
45C22×Dic83C2311yescharacteristic
46C2×C4×D83—24no—
47C2×C4×D83—22no—
48C2×C4×D83—24no—
49C2×C4×D83D414yes—
50C4⋊Dic83—22no—
51C4⋊Dic83—24no—
52(C2×D83)⋊C4—24no—
53(C2×D83)⋊C4—22no—
54(C2×D83)⋊C4—24no—
55(C2×D83)⋊C4—22no—
56(C2×D83)⋊C4—24no—
57(C2×D83)⋊C4—24no—
58C4×Dic83—24no—
59C4×Dic83—22no—
60C4×Dic83—24no—
61C22×C4×C83C2311yescharacteristic
62C22×C4×C83C2312yes—
63C22×C4×C83C2311yescharacteristic
64C22×C4×C83C2311yescharacteristic
65C22×C4×C83C2312yes—
66C4⋊C4×C83—22no—
67C4⋊C4×C83—22no—
68C4⋊C4×C83—24no—
69C23×C83C2411yescharacteristic
70C22×D83—21no—
71C22×D83—22no—
72C22×D83—22no—
73C22×D83—22no—
74C22×D83—22no—
75C22×D83—21no—
76C22×D83—21no—
77C22×D83—21no—
78C22×D83C2×D412yes—
79C22×D83#4112yes—
80C2×C4×C83—21no—
81C2×C4×C83—22no—
82C2×C4×C83—22no—
83C2×C4×C83—22no—
84C2×C4×C83—21no—
85C2×C4×C83—22no—
86C2×C4×C83—21no—
87C2×C4×C83—22no—
88C2×C4×C83—22no—
89C2×C4×C83—22no—
90C2×C4×C83C2×D412yes—
91C2×C4×C83#4112yes—
92C2×C4×C83#4114yes—
93C2×Dic83—21no—
94C2×Dic83—22no—
95C2×Dic83—22no—
96C2×Dic83—22no—
97C2×Dic83—22no—
98C2×Dic83—22no—
99C2×Dic83—22no—
100C2×Dic83—22no—
101C2×Dic83C2×D412yes—
102C2×Dic83#4114yes—
103D332—28no—
104C4×D83—28no—
105C4×D83—44no—
106C22×C83C22×D411yescharacteristic
107C22×C83C2×#4112yes—
108C22×C83C2×#4111yescharacteristic
109C22×C83C2×#4111yescharacteristic
110C22×C83C2×#4112yes—
111C2×D83—22no—
112C2×D83—22no—
113C2×D83—22no—
114C2×D83—24no—
115C2×D83—22no—
116C2×D83—41no—
117C2×D83—42no—
118C2×D83—42no—
119C2×D83—42no—
120C2×D83—41no—
121C4×C83—22no—
122C4×C83—22no—
123C4×C83—24no—
124C4×C83—41no—
125C4×C83—42no—
126Dic83—42no—
127Dic83—41no—
128Dic83—42no—
129Dic83#12314yes—
130C2×C83C2×#12311yescharacteristic
131C2×C83#52511yescharacteristic
132C2×C83#52912yes—
133C2×C83#52912yes—
134C2×C83#53211yescharacteristic
135D83—42no—
136D83—42no—
137#2395—831nomaximal, Sylow
138C83#239511yescharacteristic, complemented, Sylow
139C2×C4×D4—831no—
140C2×#117—831no—
141#400—831no—
142#399—831no—
143#398D8311yescharacteristic, complemented
144#397—831no—
145#397—832no—
146#394—832no—
147C4×C4⋊C4—832no—
148C4×C22⋊C4—832no—
149C4×C22⋊C4—831no—
150C22×D4—831no—
151C23×C4—831no—
152C23×C4—831no—
153C4×D4—1664no—
154#117—1664no—
155C2×C4⋊C4—831no—
156C2×C4⋊C4—832no—
157C2×C4⋊C4C2×D8311yescharacteristic
158C2×C4⋊C4C2×D8311yescharacteristic
159C2×C4⋊C4C2×D8312yes—
160C2×C22⋊C4—832no—
161C2×C22⋊C4—831no—
162C2×C22⋊C4—831no—
163C2×C22⋊C4—832no—
164C2×C22⋊C4—832no—
165C2×C22⋊C4—832no—
166C2×C42—832no—
167C2×C42—831no—
168C2×C42—832no—
169#95—831no—
170#95—832no—
171#95—832no—
172#95—832no—
173#95C2×D8312yes—
174#95C2×D8311yescharacteristic
175C24—831no—
176C24—831no—
177C2×D4—1662no—
178C2×D4—1662no—
179C2×D4—1662no—
180C22×C4—831no—
181C22×C4—834no—
182C22×C4—832no—
183C22×C4—831no—
184C22×C4—832no—
185C22×C4—1662no—
186C22×C4—1664no—
187C22×C4—1664no—
188C22×C4C22×D8311yescharacteristic
189C22×C4C22×D8312yes—
190C22×C4C22×D8311yescharacteristic
191C22×C4C22×D8311yescharacteristic
192C22×C4C22×D8312yes—
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—24no—
196C4⋊C4—1662no—
197C4⋊C4—1664no—
198C22⋊C4—1664no—
199C22⋊C4—1662no—
200C22⋊C4—1664no—
201C22⋊C4—1662no—
202C22⋊C4—1664no—
203C22⋊C4—1664no—
204C42—1664no—
205C42—1662no—
206C42—1664no—
207C23—832no—
208C23—832no—
209C23—1661no—
210C23—1662no—
211C23—1662no—
212C23—1661no—
213C23—1661no—
214C23—1661no—
215C23—1662no—
216C23—1662no—
217C23C23×D8311yescharacteristic
218D4—1668no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—3324no—
230C2×C4—832no—
231C2×C4—834no—
232C2×C4—1668no—
233C2×C4—1661no—
234C2×C4—1662no—
235C2×C4—1662no—
236C2×C4—1662no—
237C2×C4—1662no—
238C2×C4—1662no—
239C2×C4—1662no—
240C2×C4—1662no—
241C2×C4D4×D8312yes—
242C2×C4#1305058912yes—
243C2×C4#1305059114yes—
244C22—3321no—
245C22—3322no—
246C22—3322no—
247C22—3322no—
248C22—3321no—
249C22—1662no—
250C22—1662no—
251C22—1662no—
252C22—1664no—
253C22—1662no—
254C22C2×D4×D8311yescharacteristic
255C22#41×D8312yes—
256C22#41×D8311yescharacteristic
257C22C2×#1305058911yescharacteristic
258C22C2×#1305059112yes—
259C4—22no—
260C4—22no—
261C4—24no—
262C4—41no—
263C4—42no—
264C4—3322no—
265C4—3321no—
266C4—3322no—
267C4—834no—
268C2—3322no—
269C2—3322no—
270C2#123×D8311yescharacteristic
271C2#1912821111yescharacteristic
272C2#1912823612yes—
273C2#1912823712yes—
274C2#1912825211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#398×C83C2
prime radicalC22×D332C22
hypercenter#398D83
hypocenterC83#2395
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
2. C2×#14581750C2semidirect
3. #19127640C2semidirect
4. #19127638C2semidirect
5. #19127628C2semidirect
6. #19127627C2semidirect
8. C4⋊C4×Dic83C2semidirect
9. #19127548C2semidirect
10. #19127548C2semidirect
11. #398×C83C2semidirect
138. C83#2395semidirect
143. #398D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817542
C2×#145817502
#191276402
#191276382
#191276282
#191276272
#191276182
C4⋊C4×Dic832
#191275482
#398×C832
#239583