Finite Group Databasebeta

← #20284190

Subgroups — #20284190

Order 6784.

584 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 133 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×#13133241C212yesmaximal, complemented
2#127×D53C211yesmaximal, characteristic, complemented
3C4×#11719947C211yesmaximal, characteristic, complemented
4#16810824C211yesmaximal, characteristic, complemented
5#16810822C212yesmaximal, complemented
6#16810820C211yesmaximal, characteristic, complemented
7#16810818C211yesmaximal, characteristic, complemented
8#16810816C212yesmaximal, complemented
9#16810645C211yesmaximal, characteristic, complemented
10#16810626C212yesmaximal, complemented
11C53×#459C211yesmaximal, characteristic, complemented
12C2×#11719947C2211yescharacteristic
13C2×D4×D53—21no—
14C2×D4×D53—21no—
15C2×D4×D53C2212yes—
16#13133241—28no—
17C2×#11719913C2212yescomplemented
18C2×#11719923C2212yescomplemented
19C2×#11719922C2212yescomplemented
20C2×#11719912C2212yes—
21#13133214C2211yescharacteristic
22#13133213—21no—
23#13133213—21no—
24#13133213C2212yes—
25C4×Dic106C2211yescharacteristic, complemented
26#13133182C2211yescharacteristic, complemented
27D4×Dic53C2211yescharacteristic, complemented
28C4×C22⋊D53—21no—
29#13133177—21no—
30#13133175C2211yescharacteristic
31#13133173—21no—
32C2×C4×Dic53—21no—
33C42×D53C2211yescharacteristic
34C4⋊#88322C2211yescharacteristic, complemented
35#13133146C2212yescomplemented
36#13133139C2212yescomplemented
37#13133138C2212yescomplemented
38#13133137C2212yescomplemented
39C2×D8×C53C2212yescomplemented
40#127×C53C2211yescharacteristic
41C4×D4×C53C2211yescharacteristic, complemented
42C4⋊C8×C53C2211yescharacteristic, complemented
43#102×C53C2212yescomplemented
44C23×D53—22no—
45C2×D4×C53—21no—
46C2×D4×C53—21no—
47C2×D4×C53C2312yes—
48C2×D4×C53C2311yescharacteristic
49C22×C4×C53—21no—
50C2×C22⋊D53—21no—
51C2×C22⋊D53—22no—
52C2×C22⋊D53—22no—
53C2×C22⋊D53—22no—
54C22×Dic53—21no—
55C22×Dic53—21no—
56#11719947—22no—
57#11719947D414yescomplemented
58D4×D53—24no—
59D4×D53—24no—
60D4×D53—44no—
61D4×D53—44no—
62C2×D212—21no—
63C2×D212—21no—
64C2×D212C2312yes—
65C2×C4×D53D411yescharacteristic
66C2×C4×D53D411yescharacteristic
67C2×C4×D53C2311yescharacteristic
68C2×Dic106C2311yescharacteristic
69D8×C53—24no—
70C2×C8×C53C2312yes—
71C4⋊C4×C53C2311yescharacteristic
72C22⋊C4×C53—21no—
73C42×C53C2311yescharacteristic
74C22⋊Dic53—21no—
75#11719923—24no—
76#11719922—24no—
77#11719921—21no—
78C4⋊Dic53C2311yescharacteristic
79#11719919—21no—
80#11719919—21no—
81C4×Dic53—21no—
82C4×Dic53—21no—
83C4×Dic53—21no—
84C4×Dic53D411yescharacteristic
85C4×Dic53D411yescharacteristic
86C4×Dic53C2311yescharacteristic
87C2×#88322C2312yes—
88#11719913—24no—
89#11719912—24no—
90C23×C53—21no—
91C23×C53—22no—
92C22×D53—22no—
93C22×D53—44no—
94C22×D53—44no—
95C22×D53—44no—
96C22×D53C2×D411yescharacteristic
97D4×C53—22no—
98D4×C53—21no—
99D4×C53—22no—
100D4×C53—42no—
101D4×C53—42no—
102D4×C53C2×D412yescomplemented
103C2×C4×C53—21no—
104C2×C4×C53—42no—
105C2×C4×C53C2×D411yescharacteristic
106C2×C4×C53C2×D411yescharacteristic
107C2×C4×C53C2411yescharacteristic
108C22⋊D53—24no—
109C22⋊D53—44no—
110C22⋊D53—44no—
111C22⋊D53—44no—
112C2×Dic53—21no—
113C2×Dic53—24no—
114C2×Dic53—21no—
115C2×Dic53—42no—
116C2×Dic53C2×D411yescharacteristic
117C2×Dic53C2×D411yescharacteristic
118C2×Dic53C2×D411yescharacteristic
119D212—22no—
120D212—22no—
121D212—42no—
122D212—42no—
123C4×D53—22no—
124C4×D53—22no—
125C4×D53C2×D412yes—
126C4×D53#4112yes—
127Dic106—21no—
128Dic106C2×D412yescomplemented
129C8×C53—22no—
130#88322—22no—
131C22×C53—22no—
132C22×C53—41no—
133C22×C53—41no—
134C22×C53—42no—
135C22×C53—42no—
136C22×C53—42no—
137C22×C53C22×D411yescharacteristic
138C2×D53—21no—
139C2×D53—21no—
140C2×D53—42no—
141C2×D53—42no—
142C2×D53—42no—
143C2×D53—88no—
144C2×D53#12112yes—
145C4×C53—21no—
146C4×C53—41no—
147C4×C53AGL(1,ℤ/8)12yescomplemented
148C4×C53C22×D411yescharacteristic
149C4×C53C2×#4111yescharacteristic
150Dic53—22no—
151Dic53—21no—
152Dic53—41no—
153Dic53#12112yes—
154Dic53AGL(1,ℤ/8)12yescomplemented
155#3038—531nomaximal, Sylow
156C2×C53—42no—
157C2×C53—82no—
158C2×C53C2×#12111yescharacteristic
159C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
160C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
161D53—22no—
162D53—82no—
163C2×AGL(1,ℤ/8)—532no—
164C2×#127—531no—
165C4×#41—531no—
166#460—531no—
167#460—531no—
168#460—532no—
169#459—532no—
170#459—531no—
171#459D5311yescharacteristic, complemented
172#423—531no—
173#418—532no—
174C53#303811yescharacteristic, complemented, Sylow
175C2×#41—531no—
176C22×D4—532no—
177C22×D4—1061no—
178C22×D4—1061no—
179AGL(1,ℤ/8)—1068no—
180C2×SD16—532no—
181C2×SD16—532no—
182C2×D8—532no—
183C2×D8C2×D5312yescomplemented
184C2×M16—532no—
185#127—532no—
186#127—531no—
187#127—1061no—
188#127—1061no—
189#127C2×D5311yescharacteristic
190C4×Q8—531no—
191C4×Q8—531no—
192C4×D4—531no—
193C4×D4—1061no—
194C4×D4—1061no—
195C4×D4C2×D5311yescharacteristic, complemented
196#117—531no—
197#117—1061no—
198C2×C42—531no—
199C2×C42—1061no—
200C4⋊C8—532no—
201C4⋊C8—531no—
202C4⋊C8C2×D5311yescharacteristic, complemented
203#102—532no—
204#102—532no—
205#102—532no—
206#102C2×D5312yescomplemented
207C24—1062no—
208#41—534no—
209#41—1062no—
210C2×Q8—531no—
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—532no—
214C2×D4—1064no—
215C2×D4—1064no—
216C2×D4—1062no—
217C2×D4—1062no—
218C2×D4—1062no—
219C2×D4—1061no—
220C2×D4—1061no—
221C2×D4—1061no—
222C2×D4—2124no—
223C2×D4—2124no—
224C2×D4C22×D5312yes—
225C2×D4C22×D5311yescharacteristic
226C22×C4—21no—
227C22×C4—531no—
228C22×C4—531no—
229C22×C4—531no—
230C22×C4—1061no—
231C22×C4—1061no—
232SD16—1064no—
233SD16—1064no—
234D8—24no—
235D8—1064no—
236M16—1064no—
237C2×C8—532no—
238C2×C8C22×D5312yes—
239C4⋊C4—531no—
240C4⋊C4—1061no—
241C4⋊C4—1061no—
242C4⋊C4C22×D5311yescharacteristic
243C22⋊C4—21no—
244C22⋊C4—1061no—
245C22⋊C4—1061no—
246C42—531no—
247C42—531no—
248C42—531no—
249C42—1061no—
250C42—1061no—
251C42—1061no—
252C42C22×D5311yescharacteristic
253C23—21no—
254C23—22no—
255C23—531no—
256C23—1062no—
257C23—2124no—
258C23—2124no—
259C23—2124no—
260Q8—532no—
261Q8—1061no—
262D4—22no—
263D4—21no—
264D4—22no—
265D4—42no—
266D4—42no—
267D4—1064no—
268D4—1062no—
269D4—1062no—
270D4—2122no—
271D4—2124no—
272D4—2124no—
273D4—2124no—
274D4—2122no—
275D4D4×D5312yescomplemented
276C2×C4—21no—
277C2×C4—42no—
278C2×C4—531no—
279C2×C4—531no—
280C2×C4—532no—
281C2×C4—532no—
282C2×C4—531no—
283C2×C4—1062no—
284C2×C4—1064no—
285C2×C4—1061no—
286C2×C4—1062no—
287C2×C4—1061no—
288C2×C4—2122no—
289C2×C4D4×D5311yescharacteristic
290C2×C4D4×D5311yescharacteristic
291C2×C4C23×D5311yescharacteristic
292C8—22no—
293C8—1062no—
294C22—22no—
295C22—41no—
296C22—41no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—532no—
301C22—1061no—
302C22—1061no—
303C22—4248no—
304C22—2122no—
305C22—2122no—
306C22—2122no—
307C22C2×D4×D5311yescharacteristic
308C4—21no—
309C4—41no—
310C4—532no—
311C4—532no—
312C4—1062no—
313C4—1061no—
314C4—2121no—
315C4#1313324112yescomplemented
316C4C2×D4×D5311yescharacteristic
317C4#41×D5311yescharacteristic
318C2—42no—
319C2—82no—
320C2—1062no—
321C2—4242no—
322C2#121×D5311yescharacteristic
323C2AGL(1,ℤ/8)×D5311yescharacteristic
324C2C2×#1313324111yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#459C2
socleC22×C53C22×D4
hypercenter#459D53
hypocenterC53#3038
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. C2×#13133241C2semidirect
2. #127×D53C2semidirect
3. C4×#11719947C2semidirect
4. #16810824C2semidirect
5. #16810822C2semidirect
6. #16810820C2semidirect
7. #16810818C2semidirect
8. #16810816C2semidirect
9. #16810645C2semidirect
10. #16810626C2semidirect
11. C53×#459C2semidirect
17. C2×#11719913C22semidirect
18. C2×#11719923C22semidirect
19. C2×#11719922C22semidirect
25. C4×Dic106C22semidirect
26. #13133182C22semidirect
27. D4×Dic53C22semidirect
34. C4⋊#88322C22semidirect
35. #13133146C22semidirect
36. #13133139C22semidirect
37. #13133138C22semidirect
38. #13133137C22semidirect
39. C2×D8×C53C22semidirect
41. C4×D4×C53C22semidirect
42. C4⋊C8×C53C22semidirect
43. #102×C53C22semidirect
57. #11719947D4semidirect
102. D4×C53C2×D4semidirect
128. Dic106C2×D4semidirect
147. C4×C53AGL(1,ℤ/8)semidirect
154. Dic53AGL(1,ℤ/8)semidirect
171. #459D53semidirect
174. C53#3038semidirect
183. C2×D8C2×D53semidirect
195. C4×D4C2×D53semidirect
202. C4⋊C8C2×D53semidirect
206. #102C2×D53semidirect
275. D4D4×D53semidirect
315. C4#13133241semidirect