Finite Group Databasebeta

← #24912454

Subgroups — #24912454

Order 10624.

604 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 151 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×#14581791C211yesmaximal, characteristic, complemented
2C2×#14581778C212yesmaximal
3C2×(C22⋊D83)⋊C4C211yesmaximal, characteristic
4#19127628C211yesmaximal, characteristic, complemented
5#19127627C211yesmaximal, characteristic, complemented
6#19127626C211yesmaximal, characteristic, complemented
7#19127605C212yesmaximal, complemented
8#19127560C212yesmaximal, complemented
9#19127559C212yesmaximal, complemented
10#19127552C211yesmaximal, characteristic, complemented
11#397×C83C211yesmaximal, characteristic, complemented
12C22×Dic166C2212yes—
13C22×C22⋊D83C2211yescharacteristic
14C23×Dic83C2211yescharacteristic
15C22×C4×D83C2211yescharacteristic
16#14581791—24no—
17#14581778—28no—
18(C22⋊D83)⋊C4—24no—
19C2×C4⋊Dic83C2211yescharacteristic
20C2×Dic83⋊C4C2212yes—
21C2×Dic83⋊C4C2212yes—
22C2×Dic83⋊C4C2211yescharacteristic
23C2×C22⋊Dic83C2212yes—
24C2×(C2×D83)⋊C4C2212yes—
25C2×(C2×D83)⋊C4C2212yes—
26C2×(C2×D83)⋊C4C2211yescharacteristic
27C2×C4×Dic83C2212yes—
28C2×C4×Dic83C2211yescharacteristic
29C83⋊#95C2211yescharacteristic
30C83⋊#95C2212yes—
31C83⋊#95C2212yes—
32C83⋊#95C2212yes—
33C2×C4⋊C4×C83C2211yescharacteristic
34C2×C22⋊C4×C83C2212yes—
35C2×C22⋊C4×C83C2211yescharacteristic
36#95×C83C2211yescharacteristic
37#95×C83C2212yes—
38C23×D83C2311yescharacteristic
39C2×Dic166—24no—
40C2×Dic166—24no—
41C2×Dic166—24no—
42C2×C22⋊D83—22no—
43C2×C22⋊D83—22no—
44C2×C22⋊D83D414yes—
45C22×Dic83—22no—
46C22×Dic83—24no—
47C22×Dic83C2311yescharacteristic
48C22×Dic83C2312yes—
49C22×Dic83C2312yes—
50C22×Dic83C2311yescharacteristic
51C22×Dic83C2311yescharacteristic
52C2×C4×D83—22no—
53C2×C4×D83—24no—
54C4⋊Dic83—22no—
55Dic83⋊C4—24no—
56Dic83⋊C4—22no—
57Dic83⋊C4—24no—
58C22⋊Dic83—24no—
59(C2×D83)⋊C4—24no—
60(C2×D83)⋊C4—24no—
61(C2×D83)⋊C4—22no—
62C4×Dic83—24no—
63C4×Dic83D414yes—
64C24×C83C2311yescharacteristic
65C22×C4×C83C2311yescharacteristic
66C22×C4×C83C2312yes—
67C22×C4×C83C2312yes—
68C22×C4×C83C2311yescharacteristic
69C4⋊C4×C83—22no—
70C22⋊C4×C83—24no—
71C22⋊C4×C83—22no—
72C23×C83—21no—
73C23×C83—22no—
74C23×C83—21no—
75C23×C83—22no—
76C23×C83C2×D412yes—
77C23×C83C2411yescharacteristic
78C22×D83—21no—
79C22×D83—22no—
80C22×D83—21no—
81C22×D83—22no—
82C22×D83C2×D412yes—
83C2×C4×C83—22no—
84C2×C4×C83—22no—
85C2×C4×C83—22no—
86C2×C4×C83—21no—
87C2×C4×C83—22no—
88C2×C4×C83—22no—
89C2×C4×C83—22no—
90C2×C4×C83—22no—
91C2×C4×C83—22no—
92C2×C4×C83C2×D412yes—
93C22⋊D83—28no—
94C2×Dic83—22no—
95C2×Dic83—22no—
96C2×Dic83—21no—
97C2×Dic83—22no—
98C2×Dic83—22no—
99C2×Dic83—22no—
100C2×Dic83—22no—
101C2×Dic83—22no—
102C2×Dic83—44no—
103C2×Dic83C2×D412yes—
104C2×Dic83C2×D412yes—
105C2×Dic83C2×D412yes—
106C2×Dic83#4112yes—
107C2×Dic83#4114yes—
108C4×D83—44no—
109Dic166—48no—
110C22×C83—22no—
111C22×C83—24no—
112C22×C83—41no—
113C22×C83—41no—
114C22×C83—42no—
115C22×C83C22×D411yescharacteristic
116C22×C83C22×D411yescharacteristic
117C22×C83C2×#4111yescharacteristic
118C22×C83C2×#4112yes—
119C22×C832−1+412yes—
120C2×D83—22no—
121C2×D83—24no—
122C2×D83—41no—
123C2×D83—41no—
124C2×D83—42no—
125C4×C83—22no—
126C4×C83—42no—
127C4×C83—41no—
128C4×C83—42no—
129Dic83—42no—
130Dic83—41no—
131Dic83—42no—
132Dic83#12114yes—
133Dic83#12414yescomplemented
134C2×C83—42no—
135C2×C83C2×#12111yescharacteristic
136C2×C83C2×#12411yescharacteristic
137C2×C83#53611yescharacteristic
138C2×C83#54712yes—
139C2×C83#55612yes—
140D83—42no—
141#2547—831nomaximal, Sylow
142C83#254711yescharacteristic, complemented, Sylow
143C2×C4⋊Q8—831no—
144C2×#124—832no—
145C2×C4×D4—831no—
146#397—831no—
147#397—831no—
148#397—831no—
149#397D8311yescharacteristic, complemented
150#391—832no—
151#380—832no—
152#380—832no—
153C4×C4⋊C4—831no—
154C22×Q8—832no—
155C22×D4—831no—
156C23×C4—831no—
157C23×C4—831no—
158C4⋊Q8—1664no—
159#124—1668no—
160C4×D4—1664no—
161C2×C4⋊C4—831no—
162C2×C4⋊C4—832no—
163C2×C4⋊C4—831no—
164C2×C4⋊C4—832no—
165C2×C4⋊C4C2×D8311yescharacteristic
166C2×C22⋊C4—832no—
167C2×C22⋊C4—831no—
168C2×C22⋊C4—832no—
169C2×C22⋊C4—832no—
170C2×C22⋊C4C2×D8311yescharacteristic
171C2×C22⋊C4C2×D8312yes—
172C2×C42—831no—
173C2×C42—832no—
174#95—831no—
175#95—832no—
176#95—832no—
177#95—832no—
178#95C2×D8311yescharacteristic
179#95C2×D8312yes—
180C24—831no—
181C24C22×D8311yescharacteristic
182C2×Q8—1664no—
183C2×Q8—1664no—
184C2×Q8—1664no—
185C2×D4—834no—
186C2×D4—1662no—
187C2×D4—1662no—
188C22×C4—831no—
189C22×C4—831no—
190C22×C4—831no—
191C22×C4—832no—
192C22×C4—832no—
193C22×C4—1662no—
194C22×C4—1664no—
195C22×C4—1662no—
196C22×C4—1664no—
197C22×C4C22×D8311yescharacteristic
198C22×C4C22×D8312yes—
199C22×C4C22×D8312yes—
200C22×C4C22×D8311yescharacteristic
201C4⋊C4—22no—
202C4⋊C4—1662no—
203C4⋊C4—1664no—
204C4⋊C4—1664no—
205C4⋊C4—1662no—
206C22⋊C4—24no—
207C22⋊C4—22no—
208C22⋊C4—1664no—
209C22⋊C4—1664no—
210C22⋊C4—1664no—
211C22⋊C4—1662no—
212C42—834no—
213C42—1664no—
214C23—21no—
215C23—22no—
216C23—21no—
217C23—22no—
218C23—832no—
219C23—1661no—
220C23—1662no—
221C23—1661no—
222C23—1662no—
223C23D4×D8312yes—
224C23C23×D8311yescharacteristic
225Q8—3328no—
226D4—1668no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—3324no—
237C2×C4—3324no—
238C2×C4—832no—
239C2×C4—832no—
240C2×C4—832no—
241C2×C4—834no—
242C2×C4—832no—
243C2×C4—1662no—
244C2×C4—1662no—
245C2×C4—1661no—
246C2×C4—1662no—
247C2×C4—1662no—
248C2×C4—1662no—
249C2×C4—1662no—
250C2×C4—1662no—
251C2×C4D4×D8312yes—
252C22—24no—
253C22—22no—
254C22—42no—
255C22—41no—
256C22—41no—
257C22—3322no—
258C22—3321no—
259C22—3321no—
260C22—1664no—
261C22—1662no—
262C22C2×D4×D8311yescharacteristic
263C22C2×D4×D8311yescharacteristic
264C22#41×D8312yes—
265C22#41×D8311yescharacteristic
266C22#1458183912yes—
267C4—22no—
268C4—42no—
269C4—42no—
270C4—41no—
271C4—3322no—
272C4—3322no—
273C4—3321no—
274C4—834no—
275C4—834no—
276C2—42no—
277C2—3322no—
278C2#121×D8311yescharacteristic
279C2#124×D8311yescharacteristic
280C2#1912827311yescharacteristic
281C2#1912834912yes—
282C2#1912841812yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#397×C83C2
prime radicalC22×C22⋊D83C22
hypercenter#397D83
hypocenterC83#2547
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#14581791C2semidirect
4. #19127628C2semidirect
5. #19127627C2semidirect
6. #19127626C2semidirect
7. #19127605C2semidirect
8. #19127560C2semidirect
9. #19127559C2semidirect
10. #19127552C2semidirect
11. #397×C83C2semidirect
133. Dic83#124semidirect
142. C83#2547semidirect
149. #397D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817912
C2×#145817782
C2×(C22⋊D83)⋊C42
#191276282
#191276272
#191276262
#191276052
#191275602
#191275592
#191275522
#397×C832
#254783