Finite Group Databasebeta

← #20284192

Subgroups — #20284192

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×#13133244C212yesmaximal, complemented
2#127×D53C211yesmaximal, characteristic, complemented
3C4×#11719949C211yesmaximal, characteristic, complemented
4#16810823C211yesmaximal, characteristic, complemented
5#16810821C212yesmaximal, complemented
6#16810819C211yesmaximal, characteristic, complemented
7#16810817C212yesmaximal, complemented
8#16810815C211yesmaximal, characteristic, complemented
9#16810645C211yesmaximal, characteristic, complemented
10#16810626C212yesmaximal, complemented
11C53×#460C211yesmaximal, characteristic, complemented
12C2×#11719949C2211yescharacteristic
13C2×D4×D53—21no—
14C2×D4×D53—21no—
15C2×D4×D53C2212yes—
16#13133244—28no—
17C2×#11719924C2212yescomplemented
18C2×D424C2212yescomplemented
19C2×#11719922C2212yescomplemented
20C2×#11719912C2212yes—
21#13133214C2211yescharacteristic
22#13133213—21no—
23#13133213—21no—
24#13133213C2212yes—
25Q8×Dic53C2211yescharacteristic, complemented
26C4×D212—21no—
27C4×D212C2211yescharacteristic, complemented
28#13133179—21no—
29#13133179C2211yescharacteristic, complemented
30#13133175—21no—
31#13133175C2211yescharacteristic
32C42×D53—21no—
33C42×D53C2211yescharacteristic
34C4⋊#88322C2211yescharacteristic, complemented
35#13133146C2212yescomplemented
36#13133139C2212yescomplemented
37#13133138C2212yescomplemented
38#13133137C2212yescomplemented
39C2×SD16×C53C2212yescomplemented
40#127×C53C2211yescharacteristic
41C4×Q8×C53C2211yescharacteristic, complemented
42C4⋊C8×C53C2211yescharacteristic, complemented
43#102×C53C2212yescomplemented
44C23×D53—22no—
45C2×Q8×C53C2311yescharacteristic
46C2×D4×C53—21no—
47C2×D4×C53—21no—
48C2×D4×C53C2312yes—
49C2×C22⋊D53—22no—
50C2×C22⋊D53—22no—
51C2×C22⋊D53—22no—
52#11719949—22no—
53#11719949D414yescomplemented
54D4×D53—24no—
55D4×D53—24no—
56D4×D53—44no—
57D4×D53—44no—
58C2×D212—21no—
59C2×D212—21no—
60C2×D212—21no—
61C2×D212C2312yes—
62C2×D212C2311yescharacteristic
63C2×C4×D53—21no—
64C2×C4×D53—21no—
65C2×C4×D53—21no—
66C2×C4×D53D411yescharacteristic
67C2×C4×D53D411yescharacteristic
68C2×C4×D53C2311yescharacteristic
69SD16×C53—24no—
70C2×C8×C53C2312yes—
71C4⋊C4×C53—21no—
72C4⋊C4×C53C2311yescharacteristic
73C42×C53—21no—
74C42×C53C2311yescharacteristic
75#11719924—24no—
76#11719922—24no—
77#11719921—21no—
78#11719921—21no—
79#11719921—21no—
80C4⋊Dic53—21no—
81C4⋊Dic53C2311yescharacteristic
82C4×Dic53—21no—
83C4×Dic53—21no—
84C4×Dic53D411yescharacteristic
85C4×Dic53D411yescharacteristic
86C4×Dic53C2311yescharacteristic
87C2×#88322C2312yes—
88D424—24no—
89#11719912—24no—
90C23×C53—22no—
91C22×D53—21no—
92C22×D53—22no—
93C22×D53—44no—
94C22×D53—44no—
95C22×D53—44no—
96C22×D53C2×D411yescharacteristic
97Q8×C53—21no—
98Q8×C53C2×D412yescomplemented
99D4×C53—22no—
100D4×C53—22no—
101D4×C53—42no—
102D4×C53—42no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53C2×D411yescharacteristic
106C2×C4×C53C2×D411yescharacteristic
107C2×C4×C53C2411yescharacteristic
108C22⋊D53—44no—
109C22⋊D53—44no—
110C22⋊D53—44no—
111C2×Dic53—21no—
112C2×Dic53C2×D411yescharacteristic
113C2×Dic53C2×D411yescharacteristic
114C2×Dic53C2×D411yescharacteristic
115D212—24no—
116D212—22no—
117D212—21no—
118D212—22no—
119D212—42no—
120D212—42no—
121D212C2×D412yescomplemented
122C4×D53—22no—
123C4×D53—24no—
124C4×D53—22no—
125C4×D53—42no—
126C4×D53—42no—
127C4×D53C2×D412yes—
128C4×D53#4112yes—
129C8×C53—22no—
130#88322—22no—
131C22×C53—42no—
132C22×C53—42no—
133C22×C53—42no—
134C22×C53C22×D411yescharacteristic
135C2×D53—21no—
136C2×D53—21no—
137C2×D53—22no—
138C2×D53—41no—
139C2×D53—41no—
140C2×D53—42no—
141C2×D53—42no—
142C2×D53—42no—
143C2×D53—88no—
144C2×D53#12112yes—
145C4×C53—22no—
146C4×C53—21no—
147C4×C53—41no—
148C4×C53AGL(1,ℤ/8)12yescomplemented
149C4×C53C22×D411yescharacteristic
150C4×C53C2×#4111yescharacteristic
151Dic53—21no—
152Dic53—41no—
153Dic53#12112yes—
154Dic53AGL(1,ℤ/8)12yescomplemented
155#3038—531nomaximal, Sylow
156C2×C53—82no—
157C2×C53C2×#12111yescharacteristic
158C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
159C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
160D53—22no—
161D53—42no—
162D53—82no—
163C2×AGL(1,ℤ/8)—532no—
164C2×#127—531no—
165C4×#41—531no—
166#460—532no—
167#460—531no—
168#460D5311yescharacteristic, complemented
169#459—531no—
170#459—531no—
171#459—532no—
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×SD16C2×D5312yescomplemented
182C2×D8—532no—
183C2×D8—532no—
184C2×M16—532no—
185#127—532no—
186#127—531no—
187#127—1061no—
188#127—1061no—
189#127C2×D5311yescharacteristic
190C4×Q8—531no—
191C4×Q8C2×D5311yescharacteristic, complemented
192C4×D4—531no—
193C4×D4—531no—
194C4×D4—1061no—
195C4×D4—1061no—
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×Q8C22×D5311yescharacteristic
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—531no—
214C2×D4—532no—
215C2×D4—1061no—
216C2×D4—1064no—
217C2×D4—1064no—
218C2×D4—1062no—
219C2×D4—1062no—
220C2×D4—1062no—
221C2×D4—1061no—
222C2×D4—1061no—
223C2×D4—2124no—
224C2×D4—2124no—
225C2×D4C22×D5312yes—
226C22×C4—531no—
227C22×C4—531no—
228C22×C4—531no—
229C22×C4—1061no—
230C22×C4—1061no—
231C22×C4—1061no—
232SD16—24no—
233SD16—1064no—
234D8—1064no—
235D8—1064no—
236M16—1064no—
237C2×C8—532no—
238C2×C8C22×D5312yes—
239C4⋊C4—21no—
240C4⋊C4—531no—
241C4⋊C4—1061no—
242C4⋊C4C22×D5311yescharacteristic
243C22⋊C4—1061no—
244C22⋊C4—1061no—
245C22⋊C4—1061no—
246C42—21no—
247C42—531no—
248C42—531no—
249C42—531no—
250C42—1061no—
251C42—1061no—
252C42C22×D5311yescharacteristic
253C23—22no—
254C23—531no—
255C23—1061no—
256C23—1062no—
257C23—2124no—
258C23—2124no—
259C23—2124no—
260Q8—21no—
261Q8D4×D5312yescomplemented
262D4—22no—
263D4—22no—
264D4—42no—
265D4—42no—
266D4—532no—
267D4—1064no—
268D4—1062no—
269D4—1061no—
270D4—1062no—
271D4—2122no—
272D4—2124no—
273D4—2124no—
274D4—2124no—
275D4—2122no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—531no—
279C2×C4—531no—
280C2×C4—532no—
281C2×C4—532no—
282C2×C4—531no—
283C2×C4—1062no—
284C2×C4—1064no—
285C2×C4—1062no—
286C2×C4—1061no—
287C2×C4—2122no—
288C2×C4—2122no—
289C2×C4D4×D5311yescharacteristic
290C2×C4D4×D5311yescharacteristic
291C2×C4C23×D5311yescharacteristic
292C8—22no—
293C8—1062no—
294C22—42no—
295C22—42no—
296C22—42no—
297C22—532no—
298C22—1061no—
299C22—1061no—
300C22—1062no—
301C22—4248no—
302C22—2122no—
303C22—2122no—
304C22—2122no—
305C22—2121no—
306C22—2121no—
307C22C2×D4×D5311yescharacteristic
308C4—22no—
309C4—21no—
310C4—41no—
311C4—532no—
312C4—532no—
313C4—1061no—
314C4—2121no—
315C4#1313324412yescomplemented
316C4C2×D4×D5311yescharacteristic
317C4#41×D5311yescharacteristic
318C2—82no—
319C2—1062no—
320C2—4242no—
321C2—2122no—
322C2#121×D5311yescharacteristic
323C2AGL(1,ℤ/8)×D5311yescharacteristic
324C2C2×#1313324411yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#460C2
socleC22×C53C22×D4
hypercenter#460D53
hypocenterC53#3038
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×#13133244C2semidirect
2. #127×D53C2semidirect
3. C4×#11719949C2semidirect
4. #16810823C2semidirect
5. #16810821C2semidirect
6. #16810819C2semidirect
7. #16810817C2semidirect
8. #16810815C2semidirect
9. #16810645C2semidirect
10. #16810626C2semidirect
11. C53×#460C2semidirect
17. C2×#11719924C22semidirect
18. C2×D424C22semidirect
19. C2×#11719922C22semidirect
25. Q8×Dic53C22semidirect
27. C4×D212C22semidirect
29. #13133179C22semidirect
34. C4⋊#88322C22semidirect
35. #13133146C22semidirect
36. #13133139C22semidirect
37. #13133138C22semidirect
38. #13133137C22semidirect
39. C2×SD16×C53C22semidirect
41. C4×Q8×C53C22semidirect
42. C4⋊C8×C53C22semidirect
43. #102×C53C22semidirect
53. #11719949D4semidirect
98. Q8×C53C2×D4semidirect
121. D212C2×D4semidirect
148. C4×C53AGL(1,ℤ/8)semidirect
154. Dic53AGL(1,ℤ/8)semidirect
168. #460D53semidirect
174. C53#3038semidirect
181. C2×SD16C2×D53semidirect
191. C4×Q8C2×D53semidirect
202. C4⋊C8C2×D53semidirect
206. #102C2×D53semidirect
261. Q8D4×D53semidirect
315. C4#13133244semidirect