Finite Group Databasebeta

← #20286387

Subgroups — #20286387

Order 6784.

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

normal subgroups only

This group has 620 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×D8×D53C212yesmaximal, complemented
2D4×D212C211yesmaximal, characteristic, complemented
3#16811258C211yesmaximal, characteristic, complemented
4#16810999C211yesmaximal, characteristic, complemented
5#16810858C212yesmaximal, complemented
6#16810819C211yesmaximal, characteristic, complemented
7#16810758C212yesmaximal, complemented
8#16810755C212yesmaximal, complemented
9C4×#11719922C211yesmaximal, characteristic, complemented
10#16810688C211yesmaximal, characteristic, complemented
11C53×#459C211yesmaximal, characteristic, complemented
12C22×D212—21no—
13C2×D4×D53—21no—
14C2×D4×D53C2212yescomplemented
15C2×D4×D53C2212yescomplemented
16C2×D424C2212yescomplemented
17C2×#11719922C2211yescharacteristic
18C2×#11719922C2212yescomplemented
19C2×#11719922C2212yescomplemented
20D8×D53—28no—
21C2×C8×D53C2212yescomplemented
22#13133214C2211yescharacteristic, complemented
23#13133192C2212yescomplemented
24#13133191—21no—
25#13133188—21no—
26#13133188C2212yescomplemented
27#13133187—22no—
28#13133185—22no—
29C4×D212C2211yescharacteristic
30C4×D212C2211yescharacteristic, complemented
31#13133150C2211yescharacteristic, complemented
32#13133139C2212yescomplemented
33#13133138C2211yescharacteristic, complemented
34#13133137C2211yescharacteristic, complemented
35#13133129C2212yescomplemented
36C4×#88322C2211yescharacteristic, complemented
37C2×D8×C53C2212yescomplemented
38#127×C53C2211yescharacteristic, complemented
39C4×D4×C53C2211yescharacteristic, complemented
40C4⋊C8×C53C2211yescharacteristic, complemented
41#102×C53C2212yescomplemented
42C23×D53—22no—
43C23×D53—22no—
44C2×D4×C53—21no—
45C2×D4×C53—21no—
46C2×D4×C53C2312yes—
47C2×D4×C53C2311yescharacteristic
48C22×C4×C53—21no—
49C2×C22⋊D53—22no—
50C2×C22⋊D53—22no—
51C2×C22⋊D53—22no—
52D4×D53—24no—
53D4×D53—24no—
54D4×D53—42no—
55D4×D53—42no—
56D4×D53—44no—
57C2×D212—22no—
58C2×D212—21no—
59C2×D212—21no—
60C2×D212—44no—
61C2×D212D411yescharacteristic
62C2×D212C2312yes—
63C2×D212C2311yescharacteristic
64C2×C4×D53—21no—
65C2×C4×D53C2312yes—
66D8×C53—24no—
67C2×C8×C53C2312yes—
68C4⋊C4×C53C2311yescharacteristic
69C22⋊C4×C53—21no—
70C42×C53C2311yescharacteristic
71C22⋊Dic53—22no—
72#11719922—24no—
73#11719922—42no—
74#11719922D414yescomplemented
75#11719921—21no—
76#11719921—22no—
77#11719921D412yes—
78C4⋊Dic53D411yescharacteristic
79C4⋊Dic53C2311yescharacteristic
80C2×#88322C2311yescharacteristic
81C2×#88322C2311yescharacteristic
82D424—24no—
83C8×D53—24no—
84C23×C53—21no—
85C23×C53—22no—
86C22×D53—21no—
87C22×D53—22no—
88C22×D53—44no—
89C22×D53—44no—
90C22×D53—44no—
91C22×D53—44no—
92C22×D53—82no—
93C22×D53—82no—
94C22×D53C2×D412yes—
95D4×C53—22no—
96D4×C53—21no—
97D4×C53—22no—
98D4×C53—42no—
99D4×C53—81no—
100D4×C53C2×D412yescomplemented
101C2×C4×C53—21no—
102C2×C4×C53—42no—
103C2×C4×C53C2×D411yescharacteristic
104C2×C4×C53C2×D411yescharacteristic
105C2×C4×C53C2411yescharacteristic
106C22⋊D53—44no—
107C22⋊D53—44no—
108C22⋊D53—44no—
109C2×Dic53—21no—
110C2×Dic53C2×D412yes—
111D212—22no—
112D212—21no—
113D212—22no—
114D212—44no—
115D212—42no—
116D212—81no—
117D212D814yescomplemented
118D212C2×D412yescomplemented
119C4×D53—22no—
120C4×D53—22no—
121C4×D53—42no—
122C8×C53—22no—
123#88322—21no—
124#88322C2×D412yescomplemented
125C22×C53—22no—
126C22×C53—41no—
127C22×C53—41no—
128C22×C53—42no—
129C22×C53—42no—
130C22×C53—42no—
131C22×C53C22×D411yescharacteristic
132C2×D53—22no—
133C2×D53—22no—
134C2×D53—22no—
135C2×D53—41no—
136C2×D53—41no—
137C2×D53—42no—
138C2×D53—42no—
139C2×D53—42no—
140C2×D53—88no—
141C2×D53—88no—
142C2×D53C2×D814yescomplemented
143C4×C53—21no—
144C4×C53—41no—
145C4×C53C2×D812yescomplemented
146C4×C53C22×D411yescharacteristic
147C4×C53H(5,𝔽2)11yescharacteristic
148Dic53—22no—
149Dic53—41no—
150D4×D8—531nomaximal, Sylow
151C2×C53—42no—
152C2×C53—82no—
153C2×C53D4211yescharacteristic
154C2×C53C22×D811yescharacteristic
155C2×C53#57611yescharacteristic
156D53—24no—
157D53—42no—
158D53—82no—
159C22×D8—532no—
160D42—531no—
161D42—531no—
162#493—531no—
163#466—532no—
164#459—531no—
165#459D5311yescharacteristic, complemented
166#447—532no—
167#447—532no—
168C4×D8—531no—
169C8×D4—531no—
170C53D4×D811yescharacteristic, complemented, Sylow
171C22×D4—532no—
172C22×D4—532no—
173C22×D4—1061no—
174C22×D4—1061no—
175C2×D8—531no—
176C2×D8—532no—
177C2×D8—532no—
178C2×D8—532no—
179C2×D8—1068no—
180C2×D8C2×D5312yescomplemented
181C22×C8—532no—
182#127—531no—
183#127C2×D5311yescharacteristic, complemented
184#121—532no—
185#121—532no—
186#121—1061no—
187#121—1061no—
188#120—1062no—
189#120—1062no—
190C4×D4—531no—
191C4×D4—531no—
192C4×D4C2×D5311yescharacteristic, complemented
193#107—531no—
194C4⋊C8C2×D5311yescharacteristic, complemented
195#102—531no—
196#102—532no—
197#102—531no—
198#102C2×D5312yescomplemented
199C22⋊C8—532no—
200C4×C8—531no—
201C24—1062no—
202C24—1062no—
203C2×D4—21no—
204C2×D4—21no—
205C2×D4—531no—
206C2×D4—531no—
207C2×D4—532no—
208C2×D4—1062no—
209C2×D4—1062no—
210C2×D4—1062no—
211C2×D4—1064no—
212C2×D4—1064no—
213C2×D4—1062no—
214C2×D4—1061no—
215C2×D4—1061no—
216C2×D4—2124no—
217C2×D4—2122no—
218C2×D4—2122no—
219C2×D4—2124no—
220C2×D4C22×D5312yes—
221C2×D4C22×D5311yescharacteristic
222C22×C4—21no—
223C22×C4—532no—
224C22×C4—1061no—
225D8—24no—
226D8—534no—
227D8—1064no—
228D8—1064no—
229D8—2122no—
230C2×C8—531no—
231C2×C8—531no—
232C2×C8—1064no—
233C2×C8C22×D5312yes—
234C4⋊C4—531no—
235C4⋊C4—531no—
236C4⋊C4C22×D5311yescharacteristic
237C22⋊C4—21no—
238C22⋊C4—532no—
239C22⋊C4—1062no—
240C22⋊C4—1062no—
241C22⋊C4—1061no—
242C42C22×D5311yescharacteristic
243C23—21no—
244C23—22no—
245C23—532no—
246C23—1061no—
247C23—1062no—
248C23—4242no—
249C23—4242no—
250C23—2124no—
251C23—2124no—
252C23—2124no—
253C23—2124no—
254D4—22no—
255D4—21no—
256D4—22no—
257D4—42no—
258D4—81no—
259D4—534no—
260D4—532no—
261D4—1062no—
262D4—1061no—
263D4—1062no—
264D4—4241no—
265D4—2124no—
266D4—2124no—
267D4—2124no—
268D4—2124no—
269D4—2122no—
270D4D4×D5312yescomplemented
271C2×C4—21no—
272C2×C4—42no—
273C2×C4—532no—
274C2×C4—1062no—
275C2×C4—1062no—
276C2×C4—1061no—
277C2×C4—2122no—
278C2×C4D4×D5311yescharacteristic
279C2×C4D4×D5311yescharacteristic
280C2×C4C23×D5311yescharacteristic
281C8—22no—
282C8—532no—
283C8—1061no—
284C22—22no—
285C22—41no—
286C22—41no—
287C22—42no—
288C22—42no—
289C22—42no—
290C22—534no—
291C22—1062no—
292C22—1062no—
293C22—1062no—
294C22—4248no—
295C22—4248no—
296C22—2122no—
297C22—2122no—
298C22—2122no—
299C22—2121no—
300C22—2121no—
301C22C2×D4×D5311yescharacteristic
302C4—21no—
303C4—41no—
304C4—1062no—
305C4—2121no—
306C4D8×D5312yescomplemented
307C4C2×D4×D5311yescharacteristic
308C4#1313326311yescharacteristic
309C2—42no—
310C2—82no—
311C2—1064no—
312C2—4242no—
313C2—2122no—
314C2#1681125711yescharacteristic
315C2C2×D8×D5311yescharacteristic
316C2#1681142711yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#459C2
socleC22×C53C22×D4
hypercenter#459D53
hypocenterC53D4×D8
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×D8×D53C2semidirect
2. D4×D212C2semidirect
3. #16811258C2semidirect
4. #16810999C2semidirect
5. #16810858C2semidirect
6. #16810819C2semidirect
7. #16810758C2semidirect
8. #16810755C2semidirect
9. C4×#11719922C2semidirect
10. #16810688C2semidirect
11. C53×#459C2semidirect
14. C2×D4×D53C22semidirect
15. C2×D4×D53C22semidirect
16. C2×D424C22semidirect
18. C2×#11719922C22semidirect
19. C2×#11719922C22semidirect
21. C2×C8×D53C22semidirect
22. #13133214C22semidirect
23. #13133192C22semidirect
26. #13133188C22semidirect
30. C4×D212C22semidirect
31. #13133150C22semidirect
32. #13133139C22semidirect
33. #13133138C22semidirect
34. #13133137C22semidirect
35. #13133129C22semidirect
36. C4×#88322C22semidirect
37. C2×D8×C53C22semidirect
38. #127×C53C22semidirect
39. C4×D4×C53C22semidirect
40. C4⋊C8×C53C22semidirect
41. #102×C53C22semidirect
74. #11719922D4semidirect
100. D4×C53C2×D4semidirect
117. D212D8semidirect
118. D212C2×D4semidirect
124. #88322C2×D4semidirect
142. C2×D53C2×D8semidirect
145. C4×C53C2×D8semidirect
165. #459D53semidirect
170. C53D4×D8semidirect
180. C2×D8C2×D53semidirect
183. #127C2×D53semidirect
192. C4×D4C2×D53semidirect
194. C4⋊C8C2×D53semidirect
198. #102C2×D53semidirect
270. D4D4×D53semidirect
306. C4D8×D53semidirect