Finite Group Databasebeta

← #20254192

Subgroups — #20254192

Order 6016.

462 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 115 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×#13094000C211yesmaximal, characteristic, complemented
2#121×D47C211yesmaximal, characteristic
3#15689457C212yesmaximal, complemented
4#15689437C211yesmaximal, characteristic, complemented
5#15689434C211yesmaximal, characteristic, complemented
6#15689372C212yesmaximal, complemented
7#15689366C212yesmaximal, complemented
8#15689272C212yesmaximal, complemented
9#15689262C211yesmaximal, characteristic, complemented
10#15689085C211yesmaximal, characteristic, complemented
11C47×#448C211yesmaximal, characteristic, complemented
12C2×D4×D47—21no—
13C2×D4×D47C2211yescharacteristic
14C22×C4×D47C2211yescharacteristic
15C2×#10625336C2212yescomplemented
16C2×#10625324C2212yescomplemented
17#13094000—21no—
18#13094000—21no—
19#13094000C2212yes—
20#13094000C2212yescomplemented
21#13093999C2211yescharacteristic
22#13093998C2211yescharacteristic
23#13093997—21no—
24#13093996—21no—
25#13093995C2211yescharacteristic
26C2×C4⋊Dic47C2211yescharacteristic, complemented
27C2×Dic47⋊C4—21no—
28C4⋊C4×D47C2211yescharacteristic
29C2×(C2×D47)⋊C4—21no—
30C22⋊C4×D47—21no—
31#13093958C2212yescomplemented
32#13093957C2212yescomplemented
33#13093948C2212yescomplemented
34#13093946C2212yescomplemented
35#13093945C2212yescomplemented
36C22⋊(C47⋊C8)C2211yescharacteristic, complemented
37(C2×D47)⋊C8C2212yes—
38C22×Q8×C47C2211yescharacteristic, complemented
39C2×SD16×C47C2212yescomplemented
40#121×C47C2211yescharacteristic
41#103×C47C2212yescomplemented
42C22⋊C8×C47C2211yescharacteristic, complemented
43C23×D47—21no—
44C23×D47D411yescharacteristic
45C2×Q8×C47—21no—
46C2×Q8×C47—42no—
47C2×Q8×C47C2312yes—
48C2×D4×C47—21no—
49C2×D4×C47C2311yescharacteristic
50C22×C4×C47—21no—
51C22×C4×C47C2311yescharacteristic
52C2×C22⋊D47—21no—
53C2×C22⋊D47—21no—
54C2×C22⋊D47—21no—
55C22×Dic47—21no—
56C22×Dic47D411yescharacteristic
57D4×D47—42no—
58D4×D47—44no—
59C2×D188—21no—
60C2×D188C2311yescharacteristic
61C2×C4×D47—21no—
62C2×C4×D47—42no—
63C2×C4×D47D411yescharacteristic
64C2×C4×D47D411yescharacteristic
65C2×C4×D47C2311yescharacteristic
66C2×C4×D47C2311yescharacteristic
67SD16×C47—42no—
68C2×C8×C47C2312yes—
69C4⋊C4×C47C2311yescharacteristic
70C22⋊C4×C47—21no—
71C22⋊Dic47—21no—
72#10625336—42no—
73(C2×D47)⋊C4—22no—
74(C2×D47)⋊C4—22no—
75(C2×D47)⋊C4—21no—
76(C2×D47)⋊C4—21no—
77C4⋊Dic47—21no—
78C4⋊Dic47C2312yes—
79C4⋊Dic47C2311yescharacteristic
80Dic47⋊C4—21no—
81Dic47⋊C4—22no—
82Dic47⋊C4—22no—
83C2×C47⋊C8C2312yes—
84#10625324—42no—
85C23×C47—21no—
86C23×C47C2×D411yescharacteristic
87C22×D47—22no—
88C22×D47—22no—
89C22×D47—21no—
90C22×D47—41no—
91C22×D47—42no—
92C22×D47—42no—
93C22×D47—42no—
94C22×D47C2×D411yescharacteristic
95C22×D47C2×D411yescharacteristic
96Q8×C47—22no—
97Q8×C47—22no—
98Q8×C47—42no—
99D4×C47—22no—
100D4×C47—42no—
101C2×C4×C47—22no—
102C2×C4×C47—22no—
103C2×C4×C47—21no—
104C2×C4×C47—42no—
105C2×C4×C47C2×D411yescharacteristic
106C2×C4×C47C2411yescharacteristic
107C22⋊D47—42no—
108C22⋊D47—42no—
109C22⋊D47—42no—
110C2×Dic47—22no—
111C2×Dic47—21no—
112C2×Dic47—41no—
113C2×Dic47—42no—
114C2×Dic47C2×D411yescharacteristic
115C2×Dic47C2×D411yescharacteristic
116D188—22no—
117D188—42no—
118C4×D47—22no—
119C4×D47—22no—
120C4×D47—41no—
121C4×D47—41no—
122C4×D47—42no—
123C8×C47—22no—
124C47⋊C8—22no—
125C22×C47—21no—
126C22×C47—21no—
127C22×C47—41no—
128C22×C47—41no—
129C22×C47—41no—
130C22×C47AGL(1,ℤ/8)12yes—
131C22×C47C22×D411yescharacteristic
132C2×D47—21no—
133C2×D47—21no—
134C2×D47—21no—
135C2×D47—21no—
136C2×D47—21no—
137C2×D47—44no—
138C2×D47—41no—
139C2×D47—41no—
140C2×D47—41no—
141C2×D47—84no—
142C2×D47#13712yes—
143C4×C47—21no—
144C4×C47—42no—
145C4×C47—41no—
146C4×C47H(5,𝔽2)12yes—
147Dic47—21no—
148Dic47—21no—
149Dic47—41no—
150Dic47—42no—
151#3193—471nomaximal, Sylow
152C2×C47—22no—
153C2×C47—81no—
154C2×C47#53411yescharacteristic
155C2×C47C2×AGL(1,ℤ/8)11yescharacteristic
156C2×C47C2×#13711yescharacteristic
157D47—22no—
158D47—41no—
159D47—81no—
160C2×C22⋊Q8—471no—
161C2×#121—471no—
162#484—472no—
163#480—471no—
164#480—471no—
165#468—472no—
166#468—472no—
167#450—472no—
168#448—471no—
169#448D4711yescharacteristic, complemented
170#407—471no—
171C47#319311yescharacteristic, complemented, Sylow
172C22×Q8C2×D4711yescharacteristic, complemented
173C22×D4—471no—
174C22×D4—941no—
175C23×C4—471no—
176C2×SD16—472no—
177C2×SD16C2×D4712yescomplemented
178C2×M16—472no—
179C22⋊Q8—472no—
180C22⋊Q8—472no—
181C22⋊Q8—941no—
182C22⋊Q8—941no—
183#121—471no—
184#121—471no—
185#121—471no—
186#121—941no—
187#121—941no—
188#121C2×D4711yescharacteristic
189C2×C4⋊C4—471no—
190C2×C4⋊C4—471no—
191C2×C4⋊C4—941no—
192C2×C22⋊C4—941no—
193C2×C22⋊C4—941no—
194#107—472no—
195#107—472no—
196#103—472no—
197#103C2×D4712yescomplemented
198#102—472no—
199#102—472no—
200C22⋊C8—471no—
201C22⋊C8—472no—
202C22⋊C8C2×D4711yescharacteristic, complemented
203C24—471no—
204C24—941no—
205C2×Q8—21no—
206C2×Q8—42no—
207C2×Q8C22×D4712yes—
208C2×D4—21no—
209C2×D4—471no—
210C2×D4—941no—
211C2×D4—941no—
212C2×D4—941no—
213C2×D4—941no—
214C2×D4—1882no—
215C2×D4—1884no—
216C2×D4C22×D4711yescharacteristic
217C22×C4—21no—
218C22×C4—471no—
219C22×C4—471no—
220C22×C4—471no—
221C22×C4—471no—
222C22×C4—471no—
223C22×C4—941no—
224C22×C4—941no—
225C22×C4—1882no—
226C22×C4C22×D4711yescharacteristic
227SD16—42no—
228SD16—1882no—
229M16—1882no—
230C2×C8—472no—
231C2×C8C22×D4712yes—
232C4⋊C4—472no—
233C4⋊C4—471no—
234C4⋊C4—941no—
235C4⋊C4—942no—
236C4⋊C4—942no—
237C4⋊C4—941no—
238C4⋊C4C22×D4711yescharacteristic
239C22⋊C4—21no—
240C22⋊C4—941no—
241C22⋊C4—942no—
242C22⋊C4—942no—
243C22⋊C4—941no—
244C22⋊C4—941no—
245C23—21no—
246C23—471no—
247C23—471no—
248C23—942no—
249C23—942no—
250C23—941no—
251C23—1881no—
252C23—1882no—
253C23—1882no—
254C23—1882no—
255C23D4×D4711yescharacteristic
256Q8—22no—
257Q8—22no—
258Q8—42no—
259D4—22no—
260D4—42no—
261D4—942no—
262D4—1882no—
263D4—1882no—
264D4—1882no—
265D4—1882no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—42no—
270C2×C4—471no—
271C2×C4—471no—
272C2×C4—942no—
273C2×C4—942no—
274C2×C4—942no—
275C2×C4—941no—
276C2×C4—1881no—
277C2×C4—1881no—
278C2×C4—1882no—
279C2×C4—1881no—
280C2×C4—1882no—
281C2×C4D4×D4711yescharacteristic
282C2×C4C23×D4711yescharacteristic
283C8—22no—
284C8—942no—
285C22—21no—
286C22—21no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—472no—
291C22—941no—
292C22—941no—
293C22—941no—
294C22—941no—
295C22—941no—
296C22—3764no—
297C22—1884no—
298C22—1881no—
299C22—1881no—
300C22—1881no—
301C22#1309405112yes—
302C22C2×D4×D4711yescharacteristic
303C4—21no—
304C4—41no—
305C4—42no—
306C4—941no—
307C4—941no—
308C4—1881no—
309C4—1882no—
310C4#1309406912yes—
311C2—22no—
312C2—81no—
313C2—942no—
314C2—3761no—
315C2—1881no—
316C2#1568968311yescharacteristic
317C2C2×#1309405111yescharacteristic
318C2#137×D4711yescharacteristic
subgroupG/H
centerC22C2×D4×D47
commutatorC2×C4×C47C24
FrattiniC2×C4C23×D47
FittingC47×#448C2
socleC22×C47C22×D4
hypercenter#448D47
hypocenterC47#3193
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C2×#13094000C2semidirect
3. #15689457C2semidirect
4. #15689437C2semidirect
5. #15689434C2semidirect
6. #15689372C2semidirect
7. #15689366C2semidirect
8. #15689272C2semidirect
9. #15689262C2semidirect
10. #15689085C2semidirect
11. C47×#448C2semidirect
15. C2×#10625336C22semidirect
16. C2×#10625324C22semidirect
20. #13094000C22semidirect
26. C2×C4⋊Dic47C22semidirect
31. #13093958C22semidirect
32. #13093957C22semidirect
33. #13093948C22semidirect
34. #13093946C22semidirect
35. #13093945C22semidirect
36. C22⋊(C47⋊C8)C22semidirect
38. C22×Q8×C47C22semidirect
39. C2×SD16×C47C22semidirect
41. #103×C47C22semidirect
42. C22⋊C8×C47C22semidirect
169. #448D47semidirect
171. C47#3193semidirect
172. C22×Q8C2×D47semidirect
177. C2×SD16C2×D47semidirect
197. #103C2×D47semidirect
202. C22⋊C8C2×D47semidirect