Finite Group Databasebeta

← #24913156

Subgroups — #24913156

Order 10624.

476 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 139 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×#14581783C211yesmaximal, characteristic, complemented
2C2×#14581750C211yesmaximal, characteristic
3#19127640C211yesmaximal, characteristic, complemented
4#19127638C211yesmaximal, characteristic
5#19127627C211yesmaximal, characteristic
6#19127600C212yesmaximal, complemented
7#19127593C212yesmaximal
8#19127584C212yesmaximal, complemented
9#19127572C212yesmaximal
10#19127546C211yesmaximal, characteristic, complemented
11#398×C83C211yesmaximal, characteristic, complemented
12C22×C4×D83C2211yescharacteristic
13#14581783—24no—
14#14581750—24no—
15C2×C4⋊Dic83C2212yes—
16C2×C4⋊Dic83C2212yes—
17C2×C4⋊Dic83C2211yescharacteristic
18C2×C4⋊Dic83C2211yescharacteristic
19C2×Dic83⋊C4C2212yes—
20C2×(C2×D83)⋊C4C2212yes—
21C2×(C2×D83)⋊C4C2212yes—
22C2×(C2×D83)⋊C4C2211yescharacteristic
23C2×(C2×D83)⋊C4C2211yescharacteristic
24C2×C4×Dic83C2211yescharacteristic
25C2×C4×Dic83C2212yes—
26C2×C4×Dic83C2211yescharacteristic
27C83⋊#95C2212yes—
28C83⋊#95C2212yes—
29C83⋊#95C2212yes—
30C83⋊#95C2212yes—
31C83⋊#95C2211yescharacteristic
32C2×C4⋊C4×C83C2212yes—
33C2×C4⋊C4×C83C2211yescharacteristic
34C2×C4⋊C4×C83C2211yescharacteristic
35#95×C83C2212yes—
36#95×C83C2211yescharacteristic
37C23×D83C2311yescharacteristic
38C22×Dic83C2312yes—
39C22×Dic83C2311yescharacteristic
40C22×Dic83C2312yes—
41C22×Dic83C2311yescharacteristic
42C22×Dic83C2311yescharacteristic
43C2×C4×D83—22no—
44C2×C4×D83—22no—
45C2×C4×D83—22no—
46C4⋊Dic83—22no—
47C4⋊Dic83—24no—
48C4⋊Dic83—24no—
49C4⋊Dic83—22no—
50Dic83⋊C4—24no—
51(C2×D83)⋊C4—24no—
52(C2×D83)⋊C4—22no—
53(C2×D83)⋊C4—24no—
54(C2×D83)⋊C4—22no—
55C4×Dic83—24no—
56C4×Dic83—22no—
57C4×Dic83D414yes—
58C22×C4×C83C2311yescharacteristic
59C22×C4×C83C2312yes—
60C22×C4×C83C2311yescharacteristic
61C22×C4×C83C2311yescharacteristic
62C22×C4×C83C2312yes—
63C4⋊C4×C83—22no—
64C4⋊C4×C83—22no—
65C4⋊C4×C83—24no—
66C23×C83C2411yescharacteristic
67C22×D83—21no—
68C22×D83—22no—
69C22×D83—21no—
70C22×D83—22no—
71C22×D83#4112yes—
72C2×C4×C83—21no—
73C2×C4×C83—22no—
74C2×C4×C83—21no—
75C2×C4×C83—22no—
76C2×C4×C83—22no—
77C2×C4×C83—21no—
78C2×C4×C83—22no—
79C2×C4×C83—22no—
80C2×C4×C83—22no—
81C2×C4×C83—22no—
82C2×C4×C83C2×D412yes—
83C2×C4×C83#4112yes—
84C2×C4×C83#4114yes—
85C2×Dic83—22no—
86C2×Dic83—21no—
87C2×Dic83—21no—
88C2×Dic83—21no—
89C2×Dic83—22no—
90C2×Dic83—22no—
91C2×Dic83—22no—
92C2×Dic83—22no—
93C2×Dic83—22no—
94C2×Dic83—22no—
95C2×Dic83—22no—
96C2×Dic83C2×D412yes—
97C2×Dic83C2×D412yes—
98C4×D83—28no—
99C22×C83C22×D411yescharacteristic
100C22×C83C2×#4111yescharacteristic
101C22×C83C2×#4111yescharacteristic
102C22×C83C2×#4112yes—
103C22×C832−1+412yes—
104C2×D83—22no—
105C2×D83—24no—
106C2×D83—41no—
107C2×D83—41no—
108C2×D83—42no—
109C4×C83—22no—
110C4×C83—22no—
111C4×C83—24no—
112C4×C83—42no—
113C4×C83—41no—
114Dic83—22no—
115Dic83—22no—
116Dic83—41no—
117Dic83—42no—
118Dic83—42no—
119C2×C83#52511yescharacteristic
120C2×C83#53211yescharacteristic
121C2×C83#53611yescharacteristic
122C2×C83#55212yes—
123C2×C83#55212yes—
124D83—42no—
125#2611—831nomaximal, Sylow
126C83#261111yescharacteristic, complemented, Sylow
127C2×#125—831no—
128C2×#117—831no—
129#400—831no—
130#399—831no—
131#398D8311yescharacteristic, complemented
132#397—831no—
133#389—832no—
134#388—832no—
135#387—832no—
136#385—832no—
137#376—831no—
138C23×C4—831no—
139#125—1664no—
140#117—1664no—
141C2×C4⋊C4—831no—
142C2×C4⋊C4—832no—
143C2×C4⋊C4—832no—
144C2×C4⋊C4—831no—
145C2×C4⋊C4—832no—
146C2×C4⋊C4C2×D8311yescharacteristic
147C2×C4⋊C4C2×D8311yescharacteristic
148C2×C4⋊C4C2×D8312yes—
149C2×C22⋊C4—831no—
150C2×C22⋊C4—831no—
151C2×C22⋊C4—832no—
152C2×C22⋊C4—832no—
153C2×C42—831no—
154C2×C42—831no—
155C2×C42—832no—
156#95—832no—
157#95—831no—
158#95—832no—
159#95—832no—
160#95—832no—
161#95C2×D8311yescharacteristic
162#95C2×D8312yes—
163C24—831no—
164C22×C4—831no—
165C22×C4—831no—
166C22×C4—832no—
167C22×C4—831no—
168C22×C4—832no—
169C22×C4—1662no—
170C22×C4—1662no—
171C22×C4—1662no—
172C22×C4C22×D8311yescharacteristic
173C22×C4C22×D8312yes—
174C22×C4C22×D8311yescharacteristic
175C22×C4C22×D8311yescharacteristic
176C22×C4C22×D8312yes—
177C4⋊C4—22no—
178C4⋊C4—22no—
179C4⋊C4—24no—
180C4⋊C4—1662no—
181C4⋊C4—1664no—
182C4⋊C4—1664no—
183C4⋊C4—1662no—
184C4⋊C4—1664no—
185C22⋊C4—1662no—
186C22⋊C4—1664no—
187C22⋊C4—1662no—
188C22⋊C4—1664no—
189C42—834no—
190C42—1662no—
191C42—1664no—
192C23—832no—
193C23—1661no—
194C23—1661no—
195C23—1662no—
196C23—1662no—
197C23C23×D8311yescharacteristic
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—832no—
209C2×C4—832no—
210C2×C4—1668no—
211C2×C4—1661no—
212C2×C4—1662no—
213C2×C4—1662no—
214C2×C4—1662no—
215C2×C4—1662no—
216C2×C4—1662no—
217C2×C4—1662no—
218C2×C4—1662no—
219C2×C4—1661no—
220C2×C4—1661no—
221C2×C4—1662no—
222C2×C4D4×D8312yes—
223C2×C4#1305058912yes—
224C2×C4#1305059114yes—
225C22—3322no—
226C22—3321no—
227C22—3321no—
228C22—1664no—
229C22—1662no—
230C22C2×D4×D8311yescharacteristic
231C22#41×D8311yescharacteristic
232C22C2×#1305058911yescharacteristic
233C22C2×#1305059112yes—
234C22#1458183912yes—
235C4—24no—
236C4—22no—
237C4—22no—
238C4—42no—
239C4—41no—
240C4—3322no—
241C4—3321no—
242C4—3322no—
243C4—1662no—
244C4—1662no—
245C2—3322no—
246C2#1912821111yescharacteristic
247C2#1912825211yescharacteristic
248C2#1912827311yescharacteristic
249C2#1912838912yes—
250C2#1912839312yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#398×C83C2
prime radicalC23×D83C23
hypercenter#398D83
hypocenterC83#2611
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#14581783C2semidirect
3. #19127640C2semidirect
6. #19127600C2semidirect
8. #19127584C2semidirect
10. #19127546C2semidirect
11. #398×C83C2semidirect
126. C83#2611semidirect
131. #398D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817832
C2×#145817502
#191276402
#191276382
#191276272
#191276002
#191275932
#191275842
#191275722
#191275462
#398×C832
#261183