Finite Group Databasebeta

← #24910850

Subgroups — #24910850

Order 10624.

786 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 157 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×#14581776C211yesmaximal, characteristic, complemented
2C2×#14581761C211yesmaximal, characteristic
3C2×#14581748C211yesmaximal, characteristic, complemented
4C22×C22⋊Dic83C211yesmaximal, characteristic, complemented
5#19127624C212yesmaximal, complemented
6#19127616C212yesmaximal, complemented
7#19127601C212yesmaximal, complemented
8#19127594C212yesmaximal, complemented
9#19127578C211yesmaximal, characteristic, complemented
10#19127573C211yesmaximal, characteristic, complemented
11#379×C83C211yesmaximal, characteristic, complemented
12C22×C22⋊D83C2211yescharacteristic
13C22×C22⋊D83C2212yes—
14C23×Dic83C2211yescharacteristic
15C23×Dic83C2211yescharacteristic
16#14581776—24no—
17#14581761—24no—
18#14581748—24no—
19C2×C4⋊Dic83C2211yescharacteristic, complemented
20C2×C4⋊Dic83C2211yescharacteristic, complemented
21C2×Dic83⋊C4C2212yescomplemented
22C2×Dic83⋊C4C2212yescomplemented
23C2×C22⋊Dic83—24no—
24C2×C22⋊Dic83C2212yes—
25C2×C22⋊Dic83C2212yes—
26C2×C22⋊Dic83C2212yes—
27C2×C22⋊Dic83C2211yescharacteristic
28C2×(C2×D83)⋊C4C2211yescharacteristic, complemented
29C2×(C2×D83)⋊C4C2211yescharacteristic, complemented
30C2×(C2×D83)⋊C4C2212yescomplemented
31C2×C4×Dic83C2212yescomplemented
32C83⋊#95C2212yescomplemented
33C83⋊#95C2212yescomplemented
34C25×C83C2211yescharacteristic
35C2×C22⋊C4×C83C2211yescharacteristic
36C2×C22⋊C4×C83C2211yescharacteristic
37C2×C22⋊C4×C83C2212yes—
38C2×C22⋊C4×C83C2212yes—
39C23×D83C2311yescharacteristic
40C2×C22⋊D83—22no—
41C2×C22⋊D83—22no—
42C2×C22⋊D83—24no—
43C2×C22⋊D83—22no—
44C2×C22⋊D83—24no—
45C2×C22⋊D83—42no—
46C22×Dic83—22no—
47C22×Dic83—22no—
48C22×Dic83—22no—
49C22×Dic83—42no—
50C22×Dic83D414yescomplemented
51C22×Dic83C2312yes—
52C22×Dic83C2311yescharacteristic
53C22×Dic83C2311yescharacteristic
54C22×Dic83C2311yescharacteristic
55C22×Dic83C2312yes—
56C4⋊Dic83—22no—
57C4⋊Dic83—22no—
58Dic83⋊C4—24no—
59Dic83⋊C4—42no—
60C22⋊Dic83—24no—
61C22⋊Dic83—24no—
62C22⋊Dic83—24no—
63C22⋊Dic83D414yes—
64(C2×D83)⋊C4—22no—
65(C2×D83)⋊C4—22no—
66(C2×D83)⋊C4—42no—
67C4×Dic83—24no—
68C24×C83—22no—
69C24×C83—24no—
70C24×C83—22no—
71C24×C83—22no—
72C24×C83—42no—
73C24×C83C2312yes—
74C24×C83C2311yescharacteristic
75C22×C4×C83C2311yescharacteristic
76C22×C4×C83C2312yes—
77C22×C4×C83C2311yescharacteristic
78C22⋊C4×C83—22no—
79C22⋊C4×C83—22no—
80C22⋊C4×C83—24no—
81C22⋊C4×C83—42no—
82C23×C83—22no—
83C23×C83—22no—
84C23×C83—28no—
85C23×C83—22no—
86C23×C83—22no—
87C23×C83—22no—
88C23×C83—22no—
89C23×C83—22no—
90C23×C83—22no—
91C23×C83—48no—
92C23×C83—48no—
93C23×C83—44no—
94C23×C83—44no—
95C23×C83C2×D412yes—
96C23×C83C2×D412yes—
97C23×C83#4114yes—
98C23×C83#4112yes—
99C23×C83C2411yescharacteristic
100C22×D83—21no—
101C22×D83—21no—
102C22×D83—22no—
103C22×D83—21no—
104C22×D83—22no—
105C2×C4×C83—22no—
106C2×C4×C83—21no—
107C2×C4×C83—21no—
108C2×C4×C83—22no—
109C2×C4×C83—22no—
110C2×C4×C83—22no—
111C2×C4×C83—22no—
112C22⋊D83—48no—
113C22⋊D83—44no—
114C2×Dic83—28no—
115C2×Dic83—21no—
116C2×Dic83—21no—
117C2×Dic83—21no—
118C2×Dic83—21no—
119C2×Dic83—22no—
120C2×Dic83—22no—
121C2×Dic83—22no—
122C2×Dic83—21no—
123C2×Dic83—21no—
124C2×Dic83—21no—
125C2×Dic83—22no—
126C2×Dic83—22no—
127C2×Dic83—44no—
128C2×Dic83C2×D414yes—
129C2×Dic83C2×D412yes—
130C2×Dic83C2×D412yes—
131C22×C83—22no—
132C22×C83—22no—
133C22×C83—24no—
134C22×C83—24no—
135C22×C83—24no—
136C22×C83—24no—
137C22×C83—24no—
138C22×C83—416no—
139C22×C83—42no—
140C22×C83—42no—
141C22×C83—42no—
142C22×C83—42no—
143C22×C83#12114yes—
144C22×C83C22×D411yescharacteristic
145C22×C83C22×D411yescharacteristic
146C22×C83C2×#4112yes—
147C22×C83C2×#4111yescharacteristic
148C22×C83H(5,𝔽2)12yes—
149C2×D83—41no—
150C2×D83—41no—
151C2×D83—41no—
152C2×D83—42no—
153C2×D83—42no—
154C4×C83—41no—
155C4×C83—41no—
156C4×C83—42no—
157Dic83—22no—
158Dic83—22no—
159Dic83—24no—
160Dic83—41no—
161Dic83—42no—
162C2×C83—24no—
163C2×C83—44no—
164C2×C83C2×#12111yescharacteristic
165C2×C83#52511yescharacteristic
166C2×C83#53511yescharacteristic
167C2×C83#54612yes—
168C2×C83#55112yes—
169D83—81no—
170#2411—831nomaximal, Sylow
171C83#241111yescharacteristic, complemented, Sylow
172C2×#123—831no—
173C2×#120—831no—
174C2×#117—831no—
175C22×C22⋊C4—831no—
176#396—832no—
177#394—832no—
178#390—832no—
179#388—832no—
180#386—831no—
181#385—831no—
182#379D8311yescharacteristic, complemented
183C25C2×D8311yescharacteristic
184C22×D4—832no—
185C22×D4—831no—
186C23×C4—831no—
187C23×C4—831no—
188#123—1664no—
189#120—1664no—
190#117—1664no—
191C2×C4⋊C4—832no—
192C2×C4⋊C4—832no—
193C2×C4⋊C4—831no—
194C2×C4⋊C4—831no—
195C2×C22⋊C4—832no—
196C2×C22⋊C4—831no—
197C2×C22⋊C4—832no—
198C2×C22⋊C4—832no—
199C2×C22⋊C4—832no—
200C2×C22⋊C4—831no—
201C2×C22⋊C4—831no—
202C2×C22⋊C4—1664no—
203C2×C22⋊C4C2×D8311yescharacteristic
204C2×C22⋊C4C2×D8311yescharacteristic
205C2×C22⋊C4C2×D8312yes—
206C2×C22⋊C4C2×D8312yes—
207C2×C42—832no—
208#95—832no—
209#95—832no—
210C24—22no—
211C24—24no—
212C24—22no—
213C24—22no—
214C24—42no—
215C24—831no—
216C24C22×D8312yes—
217C24C22×D8311yescharacteristic
218C2×D4—3322no—
219C2×D4—1664no—
220C2×D4—1662no—
221C2×D4—1662no—
222C2×D4—1664no—
223C2×D4—1662no—
224C22×C4—3322no—
225C22×C4—834no—
226C22×C4—832no—
227C22×C4—831no—
228C22×C4—831no—
229C22×C4—832no—
230C22×C4—831no—
231C22×C4—1662no—
232C22×C4—1662no—
233C22×C4—1662no—
234C22×C4C22×D8311yescharacteristic
235C22×C4C22×D8312yes—
236C22×C4C22×D8311yescharacteristic
237C4⋊C4—3322no—
238C4⋊C4—1664no—
239C4⋊C4—1662no—
240C4⋊C4—1662no—
241C22⋊C4—22no—
242C22⋊C4—22no—
243C22⋊C4—24no—
244C22⋊C4—42no—
245C22⋊C4—3322no—
246C22⋊C4—834no—
247C22⋊C4—1662no—
248C22⋊C4—1662no—
249C22⋊C4—1664no—
250C22⋊C4—1664no—
251C22⋊C4—1664no—
252C42—1664no—
253C23—22no—
254C23—22no—
255C23—28no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—48no—
263C23—48no—
264C23—44no—
265C23—44no—
266C23—1661no—
267C23—1661no—
268C23—1662no—
269C23—1661no—
270C23—1662no—
271C23D4×D8312yes—
272C23D4×D8312yes—
273C23#1305058914yes—
274C23#1305059012yes—
275C23C23×D8311yescharacteristic
276D4—3324no—
277D4—3328no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—3324no—
286C2×C4—834no—
287C2×C4—832no—
288C2×C4—832no—
289C2×C4—1662no—
290C2×C4—1661no—
291C2×C4—1661no—
292C2×C4—1661no—
293C2×C4—1662no—
294C2×C4—1662no—
295C2×C4—1668no—
296C2×C4—1661no—
297C2×C4—1661no—
298C2×C4—1662no—
299C2×C4—1661no—
300C2×C4—1661no—
301C2×C4—1662no—
302C22—24no—
303C22—24no—
304C22—22no—
305C22—22no—
306C22—24no—
307C22—24no—
308C22—24no—
309C22—416no—
310C22—42no—
311C22—42no—
312C22—42no—
313C22—42no—
314C22—3322no—
315C22—3322no—
316C22—3321no—
317C22—3321no—
318C22—3321no—
319C22#1458176514yes—
320C22C2×D4×D8311yescharacteristic
321C22C2×D4×D8311yescharacteristic
322C22C2×#1305058912yes—
323C22C2×#1305059011yescharacteristic
324C22#1458183712yes—
325C4—42no—
326C4—41no—
327C4—41no—
328C4—3321no—
329C4—3322no—
330C4—1664no—
331C4—1662no—
332C4—1662no—
333C2—24no—
334C2—44no—
335C2—6641no—
336C2C2×#1458176511yescharacteristic
337C2#1912820711yescharacteristic
338C2#1912826611yescharacteristic
339C2#1912833812yes—
340C2#1912837712yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#379×C83C2
prime radicalC2×#14581761C2
hypercenter#379D83
hypocenterC83#2411
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
1. C2×#14581776C2semidirect
3. C2×#14581748C2semidirect
4. C22×C22⋊Dic83C2semidirect
5. #19127624C2semidirect
6. #19127616C2semidirect
7. #19127601C2semidirect
8. #19127594C2semidirect
9. #19127578C2semidirect
10. #19127573C2semidirect
11. #379×C83C2semidirect
19. C2×C4⋊Dic83C22semidirect
20. C2×C4⋊Dic83C22semidirect
21. C2×Dic83⋊C4C22semidirect
22. C2×Dic83⋊C4C22semidirect
28. C2×(C2×D83)⋊C4C22semidirect
29. C2×(C2×D83)⋊C4C22semidirect
30. C2×(C2×D83)⋊C4C22semidirect
31. C2×C4×Dic83C22semidirect
32. C83⋊#95C22semidirect
33. C83⋊#95C22semidirect
50. C22×Dic83D4semidirect
171. C83#2411semidirect
182. #379D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817762
C2×#145817612
C2×#145817482
C22×C22⋊Dic832
#191276242
#191276162
#191276012
#191275942
#191275782
#191275732
#379×C832
#241183