Finite Group Databasebeta

← #24913336

Subgroups — #24913336

Order 10624.

512 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 131 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×(C22⋊D83)⋊C4C211yesmaximal, characteristic
3#19127638C211yesmaximal, characteristic, complemented
4#19127637C211yesmaximal, characteristic, complemented
5#19127635C212yesmaximal, complemented
6#19127627C211yesmaximal, characteristic, complemented
7#19127616C212yesmaximal
8#19127576C212yesmaximal, complemented
9#19127575C212yesmaximal, complemented
10#19127568C211yesmaximal, characteristic, complemented
11#397×C83C211yesmaximal, characteristic, complemented
12C22×C22⋊D83C2211yescharacteristic
13C23×Dic83C2211yescharacteristic
14C22×C4×D83C2211yescharacteristic
15#14581783—24no—
16(C22⋊D83)⋊C4—24no—
17C2×C4⋊Dic83C2212yes—
18C2×C4⋊Dic83C2211yescharacteristic
19C2×C4⋊Dic83C2212yes—
20C2×Dic83⋊C4C2212yes—
21C2×Dic83⋊C4C2211yescharacteristic
22C2×Dic83⋊C4C2212yes—
23C2×C22⋊Dic83C2212yes—
24C2×(C2×D83)⋊C4C2212yes—
25C2×(C2×D83)⋊C4C2211yescharacteristic
26C2×(C2×D83)⋊C4C2212yes—
27C2×C4×Dic83C2211yescharacteristic
28C83⋊#95C2212yes—
29C83⋊#95C2212yes—
30C83⋊#95C2211yescharacteristic
31C83⋊#95C2212yes—
32C2×C4⋊C4×C83C2211yescharacteristic
33C2×C22⋊C4×C83C2212yes—
34C2×C22⋊C4×C83C2211yescharacteristic
35#95×C83C2211yescharacteristic
36#95×C83C2212yes—
37C23×D83C2311yescharacteristic
38C2×C22⋊D83—22no—
39C2×C22⋊D83—22no—
40C2×C22⋊D83—22no—
41C22×Dic83—22no—
42C22×Dic83—42no—
43C22×Dic83C2311yescharacteristic
44C22×Dic83C2312yes—
45C22×Dic83C2312yes—
46C22×Dic83C2311yescharacteristic
47C22×Dic83C2311yescharacteristic
48C2×C4×D83—22no—
49C2×C4×D83—42no—
50C4⋊Dic83—22no—
51C4⋊Dic83—24no—
52C4⋊Dic83—42no—
53Dic83⋊C4—24no—
54Dic83⋊C4—22no—
55Dic83⋊C4—42no—
56C22⋊Dic83—42no—
57(C2×D83)⋊C4—24no—
58(C2×D83)⋊C4—22no—
59(C2×D83)⋊C4—42no—
60C4×Dic83D414yes—
61C24×C83C2311yescharacteristic
62C22×C4×C83C2311yescharacteristic
63C22×C4×C83C2312yes—
64C22×C4×C83C2312yes—
65C22×C4×C83C2311yescharacteristic
66C4⋊C4×C83—22no—
67C22⋊C4×C83—24no—
68C22⋊C4×C83—22no—
69C23×C83—21no—
70C23×C83—21no—
71C23×C83—22no—
72C23×C83—22no—
73C23×C83#4112yes—
74C23×C83C2411yescharacteristic
75C22×D83—21no—
76C22×D83—21no—
77C22×D83—22no—
78C22×D83—22no—
79C22×D83#4112yes—
80C2×C4×C83—22no—
81C2×C4×C83—21no—
82C2×C4×C83—22no—
83C2×C4×C83—22no—
84C2×C4×C83—22no—
85C2×C4×C83—22no—
86C2×C4×C83—22no—
87C2×C4×C83—22no—
88C2×C4×C83—22no—
89C2×C4×C83C2×D412yes—
90C22⋊D83—28no—
91C2×Dic83—21no—
92C2×Dic83—21no—
93C2×Dic83—22no—
94C2×Dic83—21no—
95C2×Dic83—22no—
96C2×Dic83—22no—
97C2×Dic83—22no—
98C2×Dic83—22no—
99C2×Dic83—22no—
100C2×Dic83—22no—
101C2×Dic83—22no—
102C2×Dic83—44no—
103C2×Dic83C2×D412yes—
104C2×Dic83C2×D412yes—
105C4×D83—44no—
106C22×C83—22no—
107C22×C83—24no—
108C22×C83—41no—
109C22×C83—41no—
110C22×C83—42no—
111C22×C83C22×D411yescharacteristic
112C22×C83C2×#4111yescharacteristic
113C22×C83C2×#4111yescharacteristic
114C22×C83H(5,𝔽2)12yes—
115C22×C832−1+412yes—
116C2×D83—22no—
117C2×D83—24no—
118C2×D83—41no—
119C2×D83—41no—
120C2×D83—42no—
121C4×C83—22no—
122C4×C83—42no—
123C4×C83—41no—
124C4×C83—42no—
125Dic83—22no—
126Dic83—22no—
127Dic83—42no—
128Dic83—41no—
129Dic83—42no—
130C2×C83—42no—
131C2×C83#52511yescharacteristic
132C2×C83#53511yescharacteristic
133C2×C83#53611yescharacteristic
134C2×C83#53912yes—
135C2×C83#53912yes—
136D83—42no—
137#2633—831nomaximal, Sylow
138C83#263311yescharacteristic, complemented, Sylow
139C2×#125—831no—
140C2×C4×D4—831no—
141#399—831no—
142#399—831no—
143#398—832no—
144#397—831no—
145#397D8311yescharacteristic, complemented
146#394—832no—
147#385—832no—
148#385—832no—
149#383—831no—
150C22×D4—831no—
151C23×C4—831no—
152C23×C4—831no—
153#125—1664no—
154C4×D4—1664no—
155C2×C4⋊C4—832no—
156C2×C4⋊C4—832no—
157C2×C4⋊C4—831no—
158C2×C4⋊C4—832no—
159C2×C4⋊C4—832no—
160C2×C4⋊C4—831no—
161C2×C4⋊C4C2×D8311yescharacteristic
162C2×C22⋊C4—832no—
163C2×C22⋊C4—832no—
164C2×C22⋊C4—831no—
165C2×C22⋊C4—832no—
166C2×C22⋊C4C2×D8311yescharacteristic
167C2×C22⋊C4C2×D8312yes—
168C2×C42—831no—
169#95—832no—
170#95—832no—
171#95—831no—
172#95—832no—
173#95C2×D8311yescharacteristic
174#95C2×D8312yes—
175C24—831no—
176C24C22×D8311yescharacteristic
177C2×D4—1662no—
178C2×D4—1662no—
179C2×D4—1662no—
180C22×C4—3322no—
181C22×C4—3322no—
182C22×C4—831no—
183C22×C4—831no—
184C22×C4—832no—
185C22×C4—831no—
186C22×C4—832no—
187C22×C4—1662no—
188C22×C4—1662no—
189C22×C4C22×D8312yes—
190C22×C4C22×D8312yes—
191C22×C4C22×D8311yescharacteristic
192C22×C4C22×D8311yescharacteristic
193C4⋊C4—22no—
194C4⋊C4—3322no—
195C4⋊C4—3322no—
196C4⋊C4—1662no—
197C4⋊C4—1664no—
198C4⋊C4—1664no—
199C4⋊C4—1662no—
200C22⋊C4—24no—
201C22⋊C4—22no—
202C22⋊C4—3322no—
203C22⋊C4—3322no—
204C22⋊C4—1664no—
205C22⋊C4—1662no—
206C42—834no—
207C23—21no—
208C23—21no—
209C23—22no—
210C23—22no—
211C23—832no—
212C23—1661no—
213C23—1661no—
214C23—1662no—
215C23—1662no—
216C23#1305058912yes—
217C23C23×D8311yescharacteristic
218D4—1668no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—3324no—
229C2×C4—3324no—
230C2×C4—832no—
231C2×C4—832no—
232C2×C4—1661no—
233C2×C4—1662no—
234C2×C4—1662no—
235C2×C4—1662no—
236C2×C4—1662no—
237C2×C4—1662no—
238C2×C4—1662no—
239C2×C4—1662no—
240C2×C4—1661no—
241C2×C4—1661no—
242C2×C4—1662no—
243C2×C4D4×D8312yes—
244C22—22no—
245C22—24no—
246C22—41no—
247C22—41no—
248C22—42no—
249C22—3321no—
250C22—3321no—
251C22—3322no—
252C22—1662no—
253C22—1664no—
254C22C2×D4×D8311yescharacteristic
255C22#41×D8311yescharacteristic
256C22C2×#1305058911yescharacteristic
257C22#1458183812yes—
258C22#1458183912yes—
259C4—22no—
260C4—42no—
261C4—41no—
262C4—42no—
263C4—3321no—
264C4—3322no—
265C4—3322no—
266C4—1662no—
267C4—1662no—
268C2—42no—
269C2—3322no—
270C2#1912821111yescharacteristic
271C2#1912826911yescharacteristic
272C2#1912827311yescharacteristic
273C2#1912829212yes—
274C2#1912829512yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#397×C83C2
prime radicalC22×C22⋊D83C22
hypercenter#397D83
hypocenterC83#2633
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#14581783C2semidirect
3. #19127638C2semidirect
4. #19127637C2semidirect
5. #19127635C2semidirect
6. #19127627C2semidirect
8. #19127576C2semidirect
9. #19127575C2semidirect
10. #19127568C2semidirect
11. #397×C83C2semidirect
138. C83#2633semidirect
145. #397D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817832
C2×(C22⋊D83)⋊C42
#191276382
#191276372
#191276352
#191276272
#191276162
#191275762
#191275752
#191275682
#397×C832
#263383