Finite Group Databasebeta

← #24910700

Subgroups — #24910700

Order 10624.

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

normal subgroups only

This group has 696 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×#14581754C211yesmaximal, characteristic, complemented
2C2×(C22⋊D83)⋊C4C212yesmaximal, complemented
3C2×#14581748C211yesmaximal, characteristic, complemented
4#19127632C211yesmaximal, characteristic, complemented
5#19127625C212yesmaximal, complemented
6#19127619C211yesmaximal, characteristic, complemented
7#19127616C212yesmaximal, complemented
8#19127610C211yesmaximal, characteristic
9#19127552C211yesmaximal, characteristic, complemented
10#19127548C212yesmaximal, complemented
11#397×C83C211yesmaximal, characteristic, complemented
12C22×D332C2211yescharacteristic
13C22×C22⋊D83C2212yes—
14C23×Dic83C2211yescharacteristic
15C22×C4×D83C2212yes—
16#14581754—24no—
17(C22⋊D83)⋊C4—28no—
18#14581748—24no—
19C2×Dic83⋊C4C2212yes—
20C2×Dic83⋊C4C2211yescharacteristic
21C2×Dic83⋊C4C2211yescharacteristic
22C2×Dic83⋊C4C2212yes—
23C2×C22⋊Dic83C2211yescharacteristic
24C2×(C2×D83)⋊C4C2212yes—
25C2×(C2×D83)⋊C4C2212yes—
26C2×(C2×D83)⋊C4C2212yes—
27C2×(C2×D83)⋊C4C2212yes—
28C2×C4×Dic83C2212yes—
29C2×C4×Dic83C2211yescharacteristic
30C2×C4×Dic83C2211yescharacteristic
31C83⋊#95C2211yescharacteristic
32C83⋊#95C2212yes—
33C2×C4⋊C4×C83C2211yescharacteristic
34C2×C22⋊C4×C83C2211yescharacteristic
35C2×C22⋊C4×C83C2212yes—
36#95×C83C2211yescharacteristic
37#95×C83C2212yes—
38C23×D83C2312yes—
39C2×D332—24no—
40C2×D332—22no—
41C2×C22⋊D83—24no—
42C2×C22⋊D83—24no—
43C2×C22⋊D83—24no—
44C22×Dic83—22no—
45C22×Dic83—22no—
46C22×Dic83D414yescomplemented
47C22×Dic83C2311yescharacteristic
48C22×Dic83C2311yescharacteristic
49C22×Dic83C2311yescharacteristic
50C22×Dic83C2312yes—
51C22×Dic83C2311yescharacteristic
52C2×C4×D83—24no—
53C2×C4×D83—24no—
54C2×C4×D83—24no—
55Dic83⋊C4—22no—
56Dic83⋊C4—22no—
57Dic83⋊C4—24no—
58Dic83⋊C4—24no—
59C22⋊Dic83—22no—
60(C2×D83)⋊C4—24no—
61(C2×D83)⋊C4—24no—
62(C2×D83)⋊C4—24no—
63(C2×D83)⋊C4—24no—
64C4×Dic83—22no—
65C4×Dic83—24no—
66C4×Dic83D414yescomplemented
67C24×C83C2311yescharacteristic
68C22×C4×C83C2312yes—
69C22×C4×C83C2312yes—
70C22×C4×C83C2311yescharacteristic
71C22×C4×C83C2311yescharacteristic
72C4⋊C4×C83—22no—
73C22⋊C4×C83—24no—
74C22⋊C4×C83—22no—
75C23×C83—21no—
76C23×C83—21no—
77C23×C83—22no—
78C23×C83—22no—
79C23×C83C2×D412yes—
80C23×C83C2411yescharacteristic
81C22×D83—22no—
82C22×D83—22no—
83C22×D83—22no—
84C22×D83—22no—
85C22×D83—22no—
86C22×D83—22no—
87C22×D83#4114yes—
88C2×C4×C83—22no—
89C2×C4×C83—22no—
90C2×C4×C83—21no—
91C2×C4×C83—22no—
92C2×C4×C83—22no—
93C2×C4×C83—22no—
94C2×C4×C83—22no—
95C2×C4×C83—22no—
96C2×C4×C83—22no—
97C2×C4×C83C2×D412yes—
98C22⋊D83—216no—
99C2×Dic83—22no—
100C2×Dic83—21no—
101C2×Dic83—22no—
102C2×Dic83—21no—
103C2×Dic83—22no—
104C2×Dic83—28no—
105C2×Dic83—21no—
106C2×Dic83—22no—
107C2×Dic83C2×D412yes—
108C2×Dic83C2×D412yes—
109C2×Dic83C2×D412yes—
110C2×Dic83C2×D412yes—
111C2×Dic83#4114yes—
112C2×Dic83#4112yes—
113D332—28no—
114C4×D83—48no—
115C22×C83—22no—
116C22×C83—24no—
117C22×C83—41no—
118C22×C83—42no—
119C22×C83—41no—
120C22×C83C22×D411yescharacteristic
121C22×C83C22×D411yescharacteristic
122C22×C83C2×#4111yescharacteristic
123C22×C83C2×#4112yes—
124C22×C83C2×#4112yes—
125C2×D83—24no—
126C2×D83—24no—
127C2×D83—24no—
128C2×D83—42no—
129C2×D83—42no—
130C2×D83—42no—
131C2×D83—42no—
132C4×C83—22no—
133C4×C83—42no—
134C4×C83—42no—
135C4×C83—41no—
136Dic83—22no—
137Dic83—24no—
138Dic83—22no—
139Dic83—41no—
140Dic83#12114yes—
141C2×C83—42no—
142C2×C83C2×#12111yescharacteristic
143C2×C83#52512yes—
144C2×C83#52511yescharacteristic
145C2×C83#52912yes—
146C2×C83#53211yescharacteristic
147D83—44no—
148#2392—831nomaximal, Sylow
149C83#239211yescharacteristic, complemented, Sylow
150C2×C4×D4—831no—
151C2×C4×D4—832no—
152C2×#117—831no—
153#398—831no—
154#397D8311yescharacteristic, complemented
155#396—832no—
156#394—832no—
157#394—831no—
158#392—831no—
159C4×C4⋊C4—831no—
160C4×C22⋊C4—832no—
161C22×D4—832no—
162C22×D4—831no—
163C23×C4—831no—
164C23×C4—832no—
165C4×D4—1664no—
166C4×D4—1668no—
167#117—1664no—
168C2×C4⋊C4—831no—
169C2×C4⋊C4—831no—
170C2×C4⋊C4—832no—
171C2×C4⋊C4—832no—
172C2×C4⋊C4C2×D8311yescharacteristic
173C2×C22⋊C4—831no—
174C2×C22⋊C4—832no—
175C2×C22⋊C4—832no—
176C2×C22⋊C4—832no—
177C2×C22⋊C4—832no—
178C2×C22⋊C4C2×D8311yescharacteristic
179C2×C22⋊C4C2×D8312yes—
180C2×C42—831no—
181C2×C42—831no—
182C2×C42—832no—
183#95—832no—
184#95—831no—
185#95C2×D8312yes—
186#95C2×D8311yescharacteristic
187C24—832no—
188C24C22×D8311yescharacteristic
189C2×D4—1664no—
190C2×D4—1664no—
191C2×D4—1664no—
192C2×D4—1664no—
193C2×D4—1662no—
194C22×C4—831no—
195C22×C4—834no—
196C22×C4—832no—
197C22×C4—831no—
198C22×C4—831no—
199C22×C4—831no—
200C22×C4—1664no—
201C22×C4—1664no—
202C22×C4—1664no—
203C22×C4—1662no—
204C22×C4—1662no—
205C22×C4C22×D8312yes—
206C22×C4C22×D8312yes—
207C22×C4C22×D8311yescharacteristic
208C22×C4C22×D8311yescharacteristic
209C4⋊C4—22no—
210C4⋊C4—1662no—
211C4⋊C4—1664no—
212C4⋊C4—1664no—
213C4⋊C4—1662no—
214C22⋊C4—24no—
215C22⋊C4—22no—
216C22⋊C4—1664no—
217C22⋊C4—1664no—
218C22⋊C4—1664no—
219C22⋊C4—1664no—
220C22⋊C4—1662no—
221C42—834no—
222C42—1662no—
223C42—1664no—
224C23—21no—
225C23—21no—
226C23—22no—
227C23—22no—
228C23—834no—
229C23—1662no—
230C23—1662no—
231C23—1662no—
232C23—1662no—
233C23—1662no—
234C23—1662no—
235C23D4×D8312yes—
236C23C23×D8311yescharacteristic
237D4—1668no—
238D4—16616no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—3328no—
249C2×C4—832no—
250C2×C4—832no—
251C2×C4—832no—
252C2×C4—832no—
253C2×C4—832no—
254C2×C4—834no—
255C2×C4—1661no—
256C2×C4—1662no—
257C2×C4—1668no—
258C2×C4—1662no—
259C2×C4—1661no—
260C2×C4—1662no—
261C2×C4—1661no—
262C2×C4—1662no—
263C2×C4D4×D8312yes—
264C22—22no—
265C22—24no—
266C22—41no—
267C22—42no—
268C22—41no—
269C22—3322no—
270C22—3322no—
271C22—3322no—
272C22—3322no—
273C22—1664no—
274C22—1664no—
275C22—1664no—
276C22C2×D4×D8311yescharacteristic
277C22C2×D4×D8311yescharacteristic
278C22#41×D8311yescharacteristic
279C22#41×D8312yes—
280C22#41×D8312yes—
281C4—22no—
282C4—42no—
283C4—42no—
284C4—41no—
285C4—3321no—
286C4—834no—
287C4—1662no—
288C4—1664no—
289C4—1662no—
290C2—42no—
291C2—3324no—
292C2#121×D8311yescharacteristic
293C2#1912820911yescharacteristic
294C2#1912820912yes—
295C2#1912823412yes—
296C2#1912825011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#397×C83C2
prime radical#19127610C2
hypercenter#397D83
hypocenterC83#2392
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#14581754C2semidirect
2. C2×(C22⋊D83)⋊C4C2semidirect
3. C2×#14581748C2semidirect
4. #19127632C2semidirect
5. #19127625C2semidirect
6. #19127619C2semidirect
7. #19127616C2semidirect
9. #19127552C2semidirect
10. #19127548C2semidirect
11. #397×C83C2semidirect
46. C22×Dic83D4semidirect
66. C4×Dic83D4semidirect
149. C83#2392semidirect
154. #397D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817542
C2×(C22⋊D83)⋊C42
C2×#145817482
#191276322
#191276252
#191276192
#191276162
#191276102
#191275522
#191275482
#397×C832
#239283