Finite Group Databasebeta

← #20246773

Subgroups — #20246773

Order 6016.

786 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 165 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×#13094007C211yesmaximal, characteristic, complemented
2C2×#13093993C211yesmaximal, characteristic
3C2×#13093983C211yesmaximal, characteristic, complemented
4#15689046C212yesmaximal, complemented
5#15689039C212yesmaximal, complemented
6#15689023C212yesmaximal, complemented
7#15689017C212yesmaximal, complemented
8#15689002C211yesmaximal, characteristic, complemented
9#15688996C211yesmaximal, characteristic, complemented
10#15688979C211yesmaximal, characteristic, complemented
11C22×C22⋊C4×C47C211yesmaximal, characteristic, complemented
12C22×C22⋊D47C2211yescharacteristic
13C22×C22⋊D47C2212yes—
14#13094007—24no—
15#13093993—24no—
16#13093983—24no—
17C2×C4⋊Dic47C2212yescomplemented
18C2×Dic47⋊C4C2211yescharacteristic, complemented
19C2×Dic47⋊C4C2211yescharacteristic, complemented
20C2×Dic47⋊C4C2212yescomplemented
21C2×C22⋊Dic47C2211yescharacteristic
22C2×C22⋊Dic47C2211yescharacteristic
23C2×C22⋊Dic47C2212yes—
24C2×C22⋊Dic47C2212yes—
25C2×C22⋊Dic47C2212yes—
26C2×C22⋊Dic47C2211yescharacteristic
27C2×(C2×D47)⋊C4C2211yescharacteristic, complemented
28C2×(C2×D47)⋊C4C2211yescharacteristic, complemented
29C2×(C2×D47)⋊C4C2212yescomplemented
30C2×C4×Dic47C2212yescomplemented
31C47⋊#95C2212yescomplemented
32C47⋊#95C2212yescomplemented
33C25×C47C2211yescharacteristic
34C23×C4×C47C2211yescharacteristic
35C23×C4×C47C2211yescharacteristic
36C2×C22⋊C4×C47—24no—
37C2×C22⋊C4×C47C2212yes—
38C2×C22⋊C4×C47C2212yes—
39C24×C47—22no—
40C24×C47—24no—
41C24×C47—22no—
42C24×C47—22no—
43C24×C47—42no—
44C24×C47C2312yes—
45C24×C47C2311yescharacteristic
46C23×D47C2311yescharacteristic
47C22×C4×C47—22no—
48C22×C4×C47—22no—
49C22×C4×C47—22no—
50C22×C4×C47—42no—
51C22×C4×C47D414yescomplemented
52C22×C4×C47C2312yes—
53C22×C4×C47C2311yescharacteristic
54C22×C4×C47C2311yescharacteristic
55C2×C22⋊D47—22no—
56C2×C22⋊D47—22no—
57C2×C22⋊D47—24no—
58C2×C22⋊D47—22no—
59C2×C22⋊D47—24no—
60C2×C22⋊D47—42no—
61C22×Dic47C2311yescharacteristic
62C22×Dic47C2311yescharacteristic
63C22×Dic47C2312yes—
64C22×Dic47C2311yescharacteristic
65C22×Dic47C2312yes—
66C22⋊C4×C47—24no—
67C22⋊C4×C47—24no—
68C22⋊Dic47—22no—
69C22⋊Dic47—22no—
70C22⋊Dic47—24no—
71C22⋊Dic47—24no—
72C22⋊Dic47—42no—
73C22⋊Dic47D414yes—
74(C2×D47)⋊C4—22no—
75(C2×D47)⋊C4—22no—
76(C2×D47)⋊C4—42no—
77C4⋊Dic47—24no—
78Dic47⋊C4—22no—
79Dic47⋊C4—22no—
80Dic47⋊C4—42no—
81C4×Dic47—24no—
82C23×C47—22no—
83C23×C47—22no—
84C23×C47—28no—
85C23×C47—22no—
86C23×C47—22no—
87C23×C47—22no—
88C23×C47—22no—
89C23×C47—22no—
90C23×C47—22no—
91C23×C47—48no—
92C23×C47—48no—
93C23×C47—44no—
94C23×C47—44no—
95C23×C47C2×D412yes—
96C23×C47C2×D412yes—
97C23×C47#4114yes—
98C23×C47#4112yes—
99C23×C47C2411yescharacteristic
100C22×D47—21no—
101C22×D47—21no—
102C22×D47—21no—
103C22×D47—22no—
104C22×D47—22no—
105C2×C4×C47—28no—
106C2×C4×C47—21no—
107C2×C4×C47—21no—
108C2×C4×C47—21no—
109C2×C4×C47—21no—
110C2×C4×C47—22no—
111C2×C4×C47—22no—
112C2×C4×C47—22no—
113C2×C4×C47—44no—
114C2×C4×C47C2×D412yes—
115C2×C4×C47C2×D412yes—
116C22⋊D47—48no—
117C22⋊D47—44no—
118C2×Dic47—22no—
119C2×Dic47—22no—
120C2×Dic47—22no—
121C2×Dic47—21no—
122C2×Dic47—21no—
123C2×Dic47—21no—
124C2×Dic47—21no—
125C2×Dic47—22no—
126C2×Dic47—22no—
127C2×Dic47—21no—
128C2×Dic47—22no—
129C2×Dic47—22no—
130C2×Dic47C2×D414yes—
131C22×C47—22no—
132C22×C47—22no—
133C22×C47—24no—
134C22×C47—24no—
135C22×C47—24no—
136C22×C47—24no—
137C22×C47—24no—
138C22×C47—416no—
139C22×C47—42no—
140C22×C47—42no—
141C22×C47—42no—
142C22×C47—42no—
143C22×C47#12114yes—
144C22×C47C22×D411yescharacteristic
145C22×C47C22×D411yescharacteristic
146C22×C47C2×#4111yescharacteristic
147C22×C47C2×#4112yes—
148C22×C47H(5,𝔽2)12yes—
149C2×D47—41no—
150C2×D47—41no—
151C2×D47—41no—
152C2×D47—42no—
153C2×D47—42no—
154C4×C47—22no—
155C4×C47—22no—
156C4×C47—42no—
157Dic47—24no—
158Dic47—41no—
159Dic47—41no—
160Dic47—41no—
161Dic47—42no—
162#2411—471nomaximal, Sylow
163C2×C47—24no—
164C2×C47—44no—
165C2×C47C2×#12111yescharacteristic
166C2×C47#52511yescharacteristic
167C2×C47#53511yescharacteristic
168C2×C47#54612yes—
169C2×C47#55112yes—
170D47—81no—
171C2×#123—471no—
172C2×#120—471no—
173C2×#117—471no—
174C22×C22⋊C4D4711yescharacteristic, complemented
175#396—472no—
176#394—472no—
177#390—472no—
178#388—472no—
179#386—471no—
180#385—471no—
181#379—471no—
182C47#241111yescharacteristic, complemented, Sylow
183C25C2×D4711yescharacteristic
184C22×D4—471no—
185C22×D4—472no—
186C23×C4C2×D4711yescharacteristic
187C23×C4C2×D4711yescharacteristic
188#123—944no—
189#120—944no—
190#117—944no—
191C2×C4⋊C4—472no—
192C2×C4⋊C4—471no—
193C2×C4⋊C4—471no—
194C2×C4⋊C4—472no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—471no—
197C2×C22⋊C4—471no—
198C2×C22⋊C4—472no—
199C2×C22⋊C4—471no—
200C2×C22⋊C4—472no—
201C2×C22⋊C4—471no—
202C2×C22⋊C4—471no—
203C2×C22⋊C4—472no—
204C2×C22⋊C4—472no—
205C2×C22⋊C4C2×D4712yes—
206C2×C22⋊C4C2×D4712yes—
207C2×C42—472no—
208#95—472no—
209#95—472no—
210C24—24no—
211C24—22no—
212C24—22no—
213C24—22no—
214C24—42no—
215C24—471no—
216C24C22×D4712yes—
217C24C22×D4711yescharacteristic
218C2×D4—944no—
219C2×D4—944no—
220C2×D4—942no—
221C2×D4—942no—
222C2×D4—942no—
223C2×D4—1882no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—42no—
228C22×C4—472no—
229C22×C4—471no—
230C22×C4—471no—
231C22×C4—472no—
232C22×C4—471no—
233C22×C4C22⋊D4714yescomplemented
234C22×C4C22×D4712yes—
235C22×C4C22×D4711yescharacteristic
236C22×C4C22×D4711yescharacteristic
237C4⋊C4—944no—
238C4⋊C4—942no—
239C4⋊C4—942no—
240C4⋊C4—1882no—
241C22⋊C4—24no—
242C22⋊C4—24no—
243C22⋊C4—474no—
244C22⋊C4—944no—
245C22⋊C4—944no—
246C22⋊C4—942no—
247C22⋊C4—942no—
248C22⋊C4—942no—
249C22⋊C4—942no—
250C22⋊C4—1882no—
251C22⋊C4—1882no—
252C42—944no—
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—941no—
267C23—941no—
268C23—941no—
269C23—942no—
270C23—942no—
271C23#1062535014yes—
272C23D4×D4712yes—
273C23#1062535212yes—
274C23C2×C22⋊D4712yes—
275C23C23×D4711yescharacteristic
276D4—1888no—
277D4—1884no—
278C2×C4—28no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—44no—
287C2×C4—474no—
288C2×C4—942no—
289C2×C4—942no—
290C2×C4—942no—
291C2×C4—941no—
292C2×C4—941no—
293C2×C4—941no—
294C2×C4—941no—
295C2×C4—941no—
296C2×C4—942no—
297C2×C4—942no—
298C2×C4—942no—
299C2×C4—942no—
300C2×C4C2×C22⋊D4712yes—
301C2×C4C2×C22⋊D4712yes—
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—1882no—
315C22—1882no—
316C22—1881no—
317C22—1881no—
318C22—1881no—
319C22#1309399614yes—
320C22C2×D4×D4711yescharacteristic
321C22C22×C22⋊D4711yescharacteristic
322C22C2×#1062535211yescharacteristic
323C22C2×#1062535012yes—
324C22#1309406912yes—
325C4—22no—
326C4—22no—
327C4—42no—
328C4—944no—
329C4—1882no—
330C4—1881no—
331C4—1881no—
332C4—1881no—
333C2—24no—
334C2—44no—
335C2—3761no—
336C2C2×#1309399611yescharacteristic
337C2#1568963411yescharacteristic
338C2#1568969011yescharacteristic
339C2#1568976312yes—
340C2#1568979912yes—
subgroupG/H
center, FrattiniC23C23×D47
commutator, socleC23×C47C24
FittingC22×C22⋊C4×C47C2
hypercenterC22×C22⋊C4D47
hypocenterC47#2411
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#13094007C2semidirect
3. C2×#13093983C2semidirect
4. #15689046C2semidirect
5. #15689039C2semidirect
6. #15689023C2semidirect
7. #15689017C2semidirect
8. #15689002C2semidirect
9. #15688996C2semidirect
10. #15688979C2semidirect
11. C22×C22⋊C4×C47C2semidirect
17. C2×C4⋊Dic47C22semidirect
18. C2×Dic47⋊C4C22semidirect
19. C2×Dic47⋊C4C22semidirect
20. C2×Dic47⋊C4C22semidirect
27. C2×(C2×D47)⋊C4C22semidirect
28. C2×(C2×D47)⋊C4C22semidirect
29. C2×(C2×D47)⋊C4C22semidirect
30. C2×C4×Dic47C22semidirect
31. C47⋊#95C22semidirect
32. C47⋊#95C22semidirect
51. C22×C4×C47D4semidirect
174. C22×C22⋊C4D47semidirect
182. C47#2411semidirect
233. C22×C4C22⋊D47semidirect