Finite Group Databasebeta

← #20285632

Subgroups — #20285632

Order 6784.

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#16811170C211yesmaximal, characteristic, complemented
2#127×D53C211yesmaximal, characteristic, complemented
3#16810998C212yesmaximal, complemented
4#16810968C212yesmaximal, complemented
5#16810819C211yesmaximal, characteristic, complemented
6#16810818C211yesmaximal, characteristic, complemented
7#16810815C211yesmaximal, characteristic, complemented
8#16810758C212yesmaximal, complemented
9#16810755C212yesmaximal, complemented
10#16810675C211yesmaximal, characteristic, complemented
11C53×#459C211yesmaximal, characteristic, complemented
12C2×#11719945—21no—
13C2×D4×D53—21no—
14C2×D4×D53—21no—
15C2×D4×D53C2212yes—
16C2×D424C2212yescomplemented
17C2×#11719922C2212yescomplemented
18C2×#11719922C2212yescomplemented
19#13133216C2211yescharacteristic, complemented
20#13133214C2211yescharacteristic
21#13133213—21no—
22#13133213—21no—
23#13133213C2212yes—
24#13133213C2211yescharacteristic, complemented
25#13133203—21no—
26#13133192C2211yescharacteristic
27#13133190—21no—
28#13133188C2211yescharacteristic
29C4×D212C2211yescharacteristic, complemented
30C4×C22⋊D53—21no—
31C42×D53C2211yescharacteristic
32C4⋊#88322C2211yescharacteristic, complemented
33#13133139C2212yescomplemented
34#13133138C2212yescomplemented
35#13133137C2212yescomplemented
36#13133129C2212yes—
37C8×Dic53C2212yescomplemented
38C2×D8×C53C2212yescomplemented
39#127×C53C2211yescharacteristic
40C4×D4×C53C2211yescharacteristic, complemented
41C4⋊C8×C53C2211yescharacteristic, complemented
42#102×C53C2212yescomplemented
43C23×D53—22no—
44C2×D4×C53—21no—
45C2×D4×C53—21no—
46C2×D4×C53C2312yes—
47C2×D4×C53C2311yescharacteristic
48C22×C4×C53—21no—
49C2×C22⋊D53—21no—
50C2×C22⋊D53—21no—
51C2×C22⋊D53—21no—
52C2×C22⋊D53—22no—
53C2×C22⋊D53—22no—
54C2×C22⋊D53—22no—
55D4×D53—42no—
56D4×D53—42no—
57D4×D53—44no—
58D4×D53—44no—
59#11719945—44no—
60C2×D212—21no—
61C2×D212—21no—
62C2×D212—21no—
63C2×D212C2312yes—
64C2×D212C2311yescharacteristic
65C2×C4×D53—21no—
66C2×C4×D53D411yescharacteristic
67C2×C4×D53D411yescharacteristic
68C2×C4×D53C2311yescharacteristic
69C2×Dic106—21no—
70D8×C53—42no—
71C2×C8×C53C2312yes—
72C4⋊C4×C53C2311yescharacteristic
73C22⋊C4×C53—21no—
74C42×C53C2311yescharacteristic
75C22⋊Dic53—21no—
76#11719922—42no—
77#11719922—42no—
78#11719921—21no—
79#11719921—21no—
80C4⋊Dic53C2311yescharacteristic
81#11719919—21no—
82C4×Dic53D411yescharacteristic
83C4×Dic53D411yescharacteristic
84C4×Dic53C2311yescharacteristic
85C2×#88322C2312yes—
86D424—42no—
87C23×C53—21no—
88C23×C53—22no—
89C22×D53—21no—
90C22×D53—22no—
91C22×D53—44no—
92C22×D53—44no—
93C22×D53—44no—
94C22×D53C2×D411yescharacteristic
95D4×C53—22no—
96D4×C53—22no—
97D4×C53—21no—
98D4×C53—21no—
99D4×C53—42no—
100D4×C53—42no—
101C2×C4×C53—21no—
102C2×C4×C53—42no—
103C2×C4×C53C2×D411yescharacteristic
104C2×C4×C53C2×D411yescharacteristic
105C2×C4×C53C2411yescharacteristic
106C22⋊D53—44no—
107C22⋊D53—42no—
108C22⋊D53—42no—
109C22⋊D53—44no—
110C22⋊D53—44no—
111C22⋊D53—42no—
112C2×Dic53—21no—
113C2×Dic53C2×D411yescharacteristic
114C2×Dic53C2×D411yescharacteristic
115C2×Dic53C2×D411yescharacteristic
116D212—22no—
117D212—22no—
118D212—21no—
119D212—21no—
120D212—42no—
121D212—42no—
122D212—42no—
123C4×D53—21no—
124C4×D53—21no—
125C4×D53—22no—
126C4×D53—42no—
127C4×D53D814yescomplemented
128Dic106—42no—
129C8×C53—22no—
130#88322—22no—
131C22×C53—42no—
132C22×C53—41no—
133C22×C53—41no—
134C22×C53—42no—
135C22×C53—42no—
136C22×C53—41no—
137C22×C53C22×D411yescharacteristic
138C2×D53—21no—
139C2×D53—21no—
140C2×D53—42no—
141C2×D53—41no—
142C2×D53—41no—
143C2×D53—42no—
144C2×D53—42no—
145C2×D53—41no—
146C2×D53—88no—
147C2×D53C2×D812yes—
148C4×C53—21no—
149C4×C53—41no—
150C4×C53C2×D812yes—
151C4×C53H(5,𝔽2)11yescharacteristic
152C4×C53H(5,𝔽2)11yescharacteristic
153Dic53—21no—
154Dic53—41no—
155Dic53C2×D812yes—
156Dic53AGL(1,ℤ/8)12yes—
157#3205—531nomaximal, Sylow
158C2×C53—82no—
159C2×C53—81no—
160C2×C53#53511yescharacteristic
161C2×C53C22×D811yescharacteristic
162C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
163D53—22no—
164D53—81no—
165D53—82no—
166#532—531no—
167C2×#127—531no—
168#493—532no—
169#486—532no—
170#459—531no—
171#459—531no—
172#459—531no—
173#459D5311yescharacteristic, complemented
174#447—532no—
175#447—532no—
176#431—531no—
177C53#320511yescharacteristic, complemented, Sylow
178C2×#41—1061no—
179C22×D4—532no—
180C22×D4—1061no—
181C22×D4—1061no—
182C2×D8—532no—
183C2×D8—532no—
184C2×D8—532no—
185C2×D8C2×D5312yescomplemented
186C4⋊Q8—531no—
187#127—532no—
188#127—531no—
189#127—531no—
190#127—1061no—
191#127—1061no—
192#127C2×D5311yescharacteristic
193#124—1061no—
194#121—531no—
195#121—531no—
196#121—1061no—
197C4×D4—531no—
198C4×D4—1061no—
199C4×D4C2×D5311yescharacteristic, complemented
200C2×C42—531no—
201C4⋊C8—531no—
202C4⋊C8C2×D5311yescharacteristic, complemented
203#102—532no—
204#102—532no—
205#102—532no—
206#102C2×D5312yescomplemented
207C22⋊C8—532no—
208C4×C8—532no—
209C24—1062no—
210#41—2124no—
211C2×Q8—1061no—
212C2×D4—21no—
213C2×D4—21no—
214C2×D4—532no—
215C2×D4—531no—
216C2×D4—1061no—
217C2×D4—1061no—
218C2×D4—1061no—
219C2×D4—1062no—
220C2×D4—1062no—
221C2×D4—1062no—
222C2×D4—1061no—
223C2×D4—1061no—
224C2×D4—1061no—
225C2×D4—2124no—
226C2×D4—2122no—
227C2×D4—2122no—
228C2×D4—2124no—
229C2×D4C22×D5312yes—
230C2×D4C22×D5311yescharacteristic
231C22×C4—21no—
232C22×C4—531no—
233C22×C4—531no—
234C22×C4—531no—
235C22×C4—1061no—
236D8—42no—
237D8—2122no—
238D8—2122no—
239D8—2122no—
240C2×C8—532no—
241C2×C8C22×D5312yes—
242C4⋊C4—531no—
243C4⋊C4—1061no—
244C4⋊C4C22×D5311yescharacteristic
245C22⋊C4—21no—
246C22⋊C4—1061no—
247C22⋊C4—1061no—
248C22⋊C4—1061no—
249C42—531no—
250C42—531no—
251C42—531no—
252C42C22×D5311yescharacteristic
253C23—21no—
254C23—22no—
255C23—531no—
256C23—1061no—
257C23—1062no—
258C23—2124no—
259C23—2124no—
260C23—2124no—
261Q8—2122no—
262D4—21no—
263D4—22no—
264D4—21no—
265D4—22no—
266D4—42no—
267D4—42no—
268D4—1061no—
269D4—1062no—
270D4—1061no—
271D4—1062no—
272D4—2122no—
273D4—2122no—
274D4—2122no—
275D4—2124no—
276D4—2124no—
277D4—2122no—
278D4—2124no—
279D4—2122no—
280D4—2122no—
281C2×C4—21no—
282C2×C4—42no—
283C2×C4—531no—
284C2×C4—531no—
285C2×C4—534no—
286C2×C4—531no—
287C2×C4—1062no—
288C2×C4—1061no—
289C2×C4—1061no—
290C2×C4—1061no—
291C2×C4—2122no—
292C2×C4D4×D5311yescharacteristic
293C2×C4D4×D5311yescharacteristic
294C2×C4C23×D5311yescharacteristic
295C8—22no—
296C8—1062no—
297C22—41no—
298C22—42no—
299C22—42no—
300C22—41no—
301C22—42no—
302C22—41no—
303C22—532no—
304C22—1061no—
305C22—1061no—
306C22—4248no—
307C22—2121no—
308C22—2122no—
309C22—2122no—
310C22—2121no—
311C22—2122no—
312C22—2121no—
313C22C2×D4×D5311yescharacteristic
314C4—21no—
315C4—41no—
316C4—532no—
317C4—532no—
318C4—1061no—
319C4—2121no—
320C4D8×D5312yes—
321C4#1313326211yescharacteristic
322C4#1313326311yescharacteristic
323C2—82no—
324C2—81no—
325C2—1062no—
326C2—4242no—
327C2—4241no—
328C2#1681118911yescharacteristic
329C2C2×D8×D5311yescharacteristic
330C2AGL(1,ℤ/8)×D5311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#459C2
socleC22×C53C22×D4
hypercenter#459D53
hypocenterC53#3205
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #16811170C2semidirect
2. #127×D53C2semidirect
3. #16810998C2semidirect
4. #16810968C2semidirect
5. #16810819C2semidirect
6. #16810818C2semidirect
7. #16810815C2semidirect
8. #16810758C2semidirect
9. #16810755C2semidirect
10. #16810675C2semidirect
11. C53×#459C2semidirect
16. C2×D424C22semidirect
17. C2×#11719922C22semidirect
18. C2×#11719922C22semidirect
19. #13133216C22semidirect
24. #13133213C22semidirect
29. C4×D212C22semidirect
32. C4⋊#88322C22semidirect
33. #13133139C22semidirect
34. #13133138C22semidirect
35. #13133137C22semidirect
37. C8×Dic53C22semidirect
38. C2×D8×C53C22semidirect
40. C4×D4×C53C22semidirect
41. C4⋊C8×C53C22semidirect
42. #102×C53C22semidirect
127. C4×D53D8semidirect
173. #459D53semidirect
177. C53#3205semidirect
185. C2×D8C2×D53semidirect
199. C4×D4C2×D53semidirect
202. C4⋊C8C2×D53semidirect
206. #102C2×D53semidirect