Finite Group Databasebeta

← #20254305

Subgroups — #20254305

Order 6016.

532 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 123 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
1#15689670C211yesmaximal, characteristic, complemented
2#127×D47C211yesmaximal, characteristic, complemented
3#15689498C212yesmaximal, complemented
4#15689468C212yesmaximal, complemented
5#15689319C211yesmaximal, characteristic, complemented
6#15689318C211yesmaximal, characteristic, complemented
7#15689315C211yesmaximal, characteristic, complemented
8#15689258C212yesmaximal, complemented
9#15689255C212yesmaximal, complemented
10#15689175C211yesmaximal, characteristic, complemented
11C47×#459C211yesmaximal, characteristic, complemented
12C2×#10625350—21no—
13C2×D4×D47—21no—
14C2×D4×D47—21no—
15C2×D4×D47C2212yes—
16C2×D376C2212yescomplemented
17C2×#10625334C2212yescomplemented
18C2×#10625334C2212yescomplemented
19#13094023C2211yescharacteristic, complemented
20#13094021C2211yescharacteristic
21#13094020—21no—
22#13094020—21no—
23#13094020C2212yes—
24#13094020C2211yescharacteristic, complemented
25#13094010—21no—
26#13093999C2211yescharacteristic
27#13093997—21no—
28#13093995C2211yescharacteristic
29C4×D188C2211yescharacteristic, complemented
30C4×C22⋊D47—21no—
31C42×D47C2211yescharacteristic
32C4⋊(C47⋊C8)C2211yescharacteristic, complemented
33#13093946C2212yescomplemented
34#13093945C2212yescomplemented
35#13093944C2212yescomplemented
36(C2×D47)⋊C8C2212yes—
37C8×Dic47C2212yescomplemented
38C2×D8×C47C2212yescomplemented
39#127×C47C2211yescharacteristic
40C4×D4×C47C2211yescharacteristic, complemented
41C4⋊C8×C47C2211yescharacteristic, complemented
42#102×C47C2212yescomplemented
43C23×D47—22no—
44C2×D4×C47—21no—
45C2×D4×C47—21no—
46C2×D4×C47C2311yescharacteristic
47C2×D4×C47C2312yes—
48C22×C4×C47—21no—
49C2×C22⋊D47—21no—
50C2×C22⋊D47—21no—
51C2×C22⋊D47—21no—
52C2×C22⋊D47—22no—
53C2×C22⋊D47—22no—
54C2×C22⋊D47—22no—
55D4×D47—42no—
56D4×D47—42no—
57D4×D47—44no—
58D4×D47—44no—
59#10625350—44no—
60C2×D188—21no—
61C2×D188—21no—
62C2×D188—21no—
63C2×D188C2312yes—
64C2×D188C2311yescharacteristic
65C2×C4×D47—21no—
66C2×C4×D47D411yescharacteristic
67C2×C4×D47D411yescharacteristic
68C2×C4×D47C2311yescharacteristic
69C2×Dic94—21no—
70D8×C47—42no—
71C2×C8×C47C2312yes—
72C4⋊C4×C47C2311yescharacteristic
73C22⋊C4×C47—21no—
74C42×C47C2311yescharacteristic
75C22⋊Dic47—21no—
76#10625334—42no—
77#10625334—42no—
78(C2×D47)⋊C4—21no—
79(C2×D47)⋊C4—21no—
80C4⋊Dic47C2311yescharacteristic
81Dic47⋊C4—21no—
82C4×Dic47D411yescharacteristic
83C4×Dic47D411yescharacteristic
84C4×Dic47C2311yescharacteristic
85C2×C47⋊C8C2312yes—
86D376—42no—
87C23×C47—21no—
88C23×C47—22no—
89C22×D47—21no—
90C22×D47—22no—
91C22×D47—44no—
92C22×D47—44no—
93C22×D47—44no—
94C22×D47C2×D411yescharacteristic
95D4×C47—21no—
96D4×C47—22no—
97D4×C47—21no—
98D4×C47—22no—
99D4×C47—42no—
100D4×C47—42no—
101C2×C4×C47—21no—
102C2×C4×C47—42no—
103C2×C4×C47C2×D411yescharacteristic
104C2×C4×C47C2×D411yescharacteristic
105C2×C4×C47C2411yescharacteristic
106C22⋊D47—44no—
107C22⋊D47—42no—
108C22⋊D47—42no—
109C22⋊D47—44no—
110C22⋊D47—44no—
111C22⋊D47—42no—
112C2×Dic47—21no—
113C2×Dic47C2×D411yescharacteristic
114C2×Dic47C2×D411yescharacteristic
115C2×Dic47C2×D411yescharacteristic
116D188—21no—
117D188—22no—
118D188—21no—
119D188—22no—
120D188—42no—
121D188—42no—
122D188—42no—
123C4×D47—21no—
124C4×D47—21no—
125C4×D47—22no—
126C4×D47—42no—
127C4×D47D814yescomplemented
128Dic94—42no—
129C8×C47—22no—
130C47⋊C8—22no—
131C22×C47—42no—
132C22×C47—41no—
133C22×C47—41no—
134C22×C47—42no—
135C22×C47—42no—
136C22×C47—41no—
137C22×C47C22×D411yescharacteristic
138C2×D47—21no—
139C2×D47—21no—
140C2×D47—42no—
141C2×D47—41no—
142C2×D47—41no—
143C2×D47—42no—
144C2×D47—42no—
145C2×D47—41no—
146C2×D47—88no—
147C2×D47C2×D812yes—
148C4×C47—21no—
149C4×C47—41no—
150C4×C47C2×D812yes—
151C4×C47H(5,𝔽2)11yescharacteristic
152C4×C47H(5,𝔽2)11yescharacteristic
153Dic47—21no—
154Dic47—41no—
155Dic47C2×D812yes—
156Dic47AGL(1,ℤ/8)12yes—
157#3205—471nomaximal, Sylow
158C2×C47—82no—
159C2×C47—81no—
160C2×C47#53511yescharacteristic
161C2×C47C22×D811yescharacteristic
162C2×C47C2×AGL(1,ℤ/8)11yescharacteristic
163D47—22no—
164D47—81no—
165D47—82no—
166#532—471no—
167C2×#127—471no—
168#493—472no—
169#486—472no—
170#459—471no—
171#459—471no—
172#459—471no—
173#459D4711yescharacteristic, complemented
174#447—472no—
175#447—472no—
176#431—471no—
177C47#320511yescharacteristic, complemented, Sylow
178C2×#41—941no—
179C22×D4—472no—
180C22×D4—941no—
181C22×D4—941no—
182C2×D8—472no—
183C2×D8—472no—
184C2×D8—472no—
185C2×D8C2×D4712yescomplemented
186C4⋊Q8—471no—
187#127—472no—
188#127—471no—
189#127—471no—
190#127—941no—
191#127—941no—
192#127C2×D4711yescharacteristic
193#124—941no—
194#121—471no—
195#121—471no—
196#121—941no—
197C4×D4—471no—
198C4×D4—941no—
199C4×D4C2×D4711yescharacteristic, complemented
200C2×C42—471no—
201C4⋊C8—471no—
202C4⋊C8C2×D4711yescharacteristic, complemented
203#102—472no—
204#102—472no—
205#102—472no—
206#102C2×D4712yescomplemented
207C22⋊C8—472no—
208C4×C8—472no—
209C24—942no—
210#41—1884no—
211C2×Q8—941no—
212C2×D4—21no—
213C2×D4—21no—
214C2×D4—472no—
215C2×D4—471no—
216C2×D4—941no—
217C2×D4—941no—
218C2×D4—941no—
219C2×D4—942no—
220C2×D4—942no—
221C2×D4—942no—
222C2×D4—941no—
223C2×D4—941no—
224C2×D4—941no—
225C2×D4—1884no—
226C2×D4—1882no—
227C2×D4—1882no—
228C2×D4—1884no—
229C2×D4C22×D4712yes—
230C2×D4C22×D4711yescharacteristic
231C22×C4—21no—
232C22×C4—471no—
233C22×C4—471no—
234C22×C4—471no—
235C22×C4—941no—
236D8—42no—
237D8—1882no—
238D8—1882no—
239D8—1882no—
240C2×C8—472no—
241C2×C8C22×D4712yes—
242C4⋊C4—471no—
243C4⋊C4—941no—
244C4⋊C4C22×D4711yescharacteristic
245C22⋊C4—21no—
246C22⋊C4—941no—
247C22⋊C4—941no—
248C22⋊C4—941no—
249C42—471no—
250C42—471no—
251C42—471no—
252C42C22×D4711yescharacteristic
253C23—21no—
254C23—22no—
255C23—471no—
256C23—941no—
257C23—942no—
258C23—1884no—
259C23—1884no—
260C23—1884no—
261Q8—1882no—
262D4—21no—
263D4—22no—
264D4—21no—
265D4—22no—
266D4—42no—
267D4—42no—
268D4—941no—
269D4—942no—
270D4—941no—
271D4—942no—
272D4—1882no—
273D4—1882no—
274D4—1882no—
275D4—1884no—
276D4—1884no—
277D4—1882no—
278D4—1884no—
279D4—1882no—
280D4—1882no—
281C2×C4—21no—
282C2×C4—42no—
283C2×C4—471no—
284C2×C4—471no—
285C2×C4—474no—
286C2×C4—471no—
287C2×C4—942no—
288C2×C4—941no—
289C2×C4—941no—
290C2×C4—941no—
291C2×C4—1882no—
292C2×C4D4×D4711yescharacteristic
293C2×C4D4×D4711yescharacteristic
294C2×C4C23×D4711yescharacteristic
295C8—22no—
296C8—942no—
297C22—41no—
298C22—42no—
299C22—42no—
300C22—41no—
301C22—42no—
302C22—41no—
303C22—472no—
304C22—941no—
305C22—941no—
306C22—3768no—
307C22—1881no—
308C22—1882no—
309C22—1882no—
310C22—1881no—
311C22—1882no—
312C22—1881no—
313C22C2×D4×D4711yescharacteristic
314C4—21no—
315C4—41no—
316C4—472no—
317C4—472no—
318C4—941no—
319C4—1881no—
320C4D8×D4712yes—
321C4#1309406911yescharacteristic
322C4#1309407011yescharacteristic
323C2—82no—
324C2—81no—
325C2—942no—
326C2—3762no—
327C2—3761no—
328C2#1568968911yescharacteristic
329C2C2×D8×D4711yescharacteristic
330C2AGL(1,ℤ/8)×D4711yescharacteristic
subgroupG/H
centerC22C2×D4×D47
commutatorC2×C4×C47C24
FrattiniC2×C4C23×D47
FittingC47×#459C2
socleC22×C47C22×D4
hypercenter#459D47
hypocenterC47#3205
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #15689670C2semidirect
2. #127×D47C2semidirect
3. #15689498C2semidirect
4. #15689468C2semidirect
5. #15689319C2semidirect
6. #15689318C2semidirect
7. #15689315C2semidirect
8. #15689258C2semidirect
9. #15689255C2semidirect
10. #15689175C2semidirect
11. C47×#459C2semidirect
16. C2×D376C22semidirect
17. C2×#10625334C22semidirect
18. C2×#10625334C22semidirect
19. #13094023C22semidirect
24. #13094020C22semidirect
29. C4×D188C22semidirect
32. C4⋊(C47⋊C8)C22semidirect
33. #13093946C22semidirect
34. #13093945C22semidirect
35. #13093944C22semidirect
37. C8×Dic47C22semidirect
38. C2×D8×C47C22semidirect
40. C4×D4×C47C22semidirect
41. C4⋊C8×C47C22semidirect
42. #102×C47C22semidirect
127. C4×D47D8semidirect
173. #459D47semidirect
177. C47#3205semidirect
185. C2×D8C2×D47semidirect
199. C4×D4C2×D47semidirect
202. C4⋊C8C2×D47semidirect
206. #102C2×D47semidirect