Finite Group Databasebeta

← #20286525

Subgroups — #20286525

Order 6784.

494 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 125 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#16811276C211yesmaximal, characteristic, complemented
2D4×D212C211yesmaximal, characteristic, complemented
3#16810971C211yesmaximal, characteristic, complemented
4#16810938C212yesmaximal, complemented
5#16810860C212yesmaximal, complemented
6#16810819C211yesmaximal, characteristic, complemented
7#16810758C212yesmaximal, complemented
8C4×#11719922C211yesmaximal, characteristic, complemented
9#16810688C211yesmaximal, characteristic, complemented
10#102×D53C212yesmaximal, complemented
11C53×#474C211yesmaximal, characteristic, complemented
12C2×#11719949—21no—
13C22×D212—21no—
14C2×D4×D53C2212yes—
15C2×D424C2212yescomplemented
16C2×#11719922C2211yescharacteristic
17C2×C8×D53C2212yescomplemented
18#13133214C2211yescharacteristic
19#13133202—22no—
20#13133193—21no—
21#13133192C2212yes—
22#13133192C2212yescomplemented
23#13133191—21no—
24#13133187—22no—
25C4×D212C2211yescharacteristic
26C4×D212C2211yescharacteristic, complemented
27C4⋊C4×D53C2212yescomplemented
28#13133150C2211yescharacteristic, complemented
29#13133139C2212yescomplemented
30#13133138C2211yescharacteristic
31#13133137C2212yescomplemented
32#13133137C2211yescharacteristic, complemented
33#13133137C2212yescomplemented
34#13133129C2212yescomplemented
35C4×#88322C2211yescharacteristic, complemented
36C4⋊Q8×C53C2211yescharacteristic, complemented
37C4×D4×C53C2211yescharacteristic
38#107×C53C2212yescomplemented
39C4⋊C8×C53C2211yescharacteristic, complemented
40#102×C53C2212yes—
41C23×D53—22no—
42C2×Q8×C53—21no—
43C2×D4×C53C2311yescharacteristic
44C22×C4×C53—21no—
45C2×C22⋊D53—22no—
46#11719949—44no—
47D4×D53—24no—
48D4×D53—42no—
49C2×D212—22no—
50C2×D212—22no—
51C2×D212—21no—
52C2×D212—21no—
53C2×D212—44no—
54C2×D212D411yescharacteristic
55C2×D212C2312yes—
56C2×D212C2311yescharacteristic
57C2×C4×D53—22no—
58C2×C4×D53—21no—
59C2×C4×D53C2312yes—
60C2×C8×C53C2312yes—
61C4⋊C4×C53—21no—
62C4⋊C4×C53C2312yes—
63C4⋊C4×C53C2311yescharacteristic
64C22⋊C4×C53—21no—
65C42×C53C2311yescharacteristic
66#11719922—22no—
67#11719921—21no—
68#11719921—22no—
69#11719921—22no—
70#11719921D412yes—
71C4⋊Dic53D411yescharacteristic
72C4⋊Dic53C2311yescharacteristic
73#11719919—22no—
74C2×#88322C2311yescharacteristic
75C2×#88322C2311yescharacteristic
76D424—42no—
77C8×D53—42no—
78C23×C53—21no—
79C22×D53—21no—
80C22×D53—22no—
81C22×D53—44no—
82C22×D53—44no—
83C22×D53—82no—
84C22×D53C2×D412yes—
85Q8×C53—42no—
86D4×C53—21no—
87D4×C53#4112yescomplemented
88C2×C4×C53—22no—
89C2×C4×C53—21no—
90C2×C4×C53—42no—
91C2×C4×C53C2×D411yescharacteristic
92C2×C4×C53C2×D411yescharacteristic
93C2×C4×C53C2411yescharacteristic
94C22⋊D53—44no—
95C2×Dic53—21no—
96C2×Dic53C2×D412yes—
97D212—22no—
98D212—21no—
99D212—22no—
100D212—21no—
101D212—44no—
102D212—44no—
103D212—42no—
104D212—81no—
105D212D814yescomplemented
106C4×D53—22no—
107C4×D53—22no—
108C4×D53—44no—
109C4×D53—42no—
110C8×C53—22no—
111#88322—21no—
112#88322—21no—
113C22×C53—22no—
114C22×C53—41no—
115C22×C53—41no—
116C22×C53C22×D411yescharacteristic
117C2×D53—22no—
118C2×D53—22no—
119C2×D53—42no—
120C2×D53—41no—
121C2×D53—41no—
122C2×D53—42no—
123C2×D53—42no—
124C2×D53—41no—
125C2×D53—88no—
126C2×D53C2×D814yes—
127C4×C53—21no—
128C4×C53—42no—
129C4×C53—41no—
130C4×C53C2×D812yes—
131C4×C53C2×#4111yescharacteristic
132C4×C53H(5,𝔽2)11yescharacteristic
133Dic53—22no—
134Dic53—41no—
135#3294—531nomaximal, Sylow
136C2×C53—42no—
137C2×C53#54611yescharacteristic
138C2×C53C22×D811yescharacteristic
139C2×C53#57711yescharacteristic
140D53—24no—
141D53—81no—
142D53—82no—
143#547—531no—
144D42—531no—
145#486—531no—
146#480—532no—
147#474D5311yescharacteristic, complemented
148#466—532no—
149#459—531no—
150#447—532no—
151C4×D8—531no—
152C8×D4—531no—
153C2×#102—532no—
154C53#329411yescharacteristic, complemented, Sylow
155C2×#41—1061no—
156C22×D4—532no—
157C22×D4—1061no—
158C2×D8—531no—
159C2×D8—532no—
160C22×C8—532no—
161C4⋊Q8C2×D5311yescharacteristic, complemented
162#127—531no—
163#123—1062no—
164C22⋊Q8—1061no—
165#121—532no—
166#121—532no—
167#121—1061no—
168#120—1062no—
169C4×D4—531no—
170C4×D4—531no—
171C4×D4C2×D5311yescharacteristic
172C2×C4⋊C4—532no—
173#107—531no—
174#107C2×D5312yescomplemented
175C4⋊C8C2×D5311yescharacteristic, complemented
176#102—531no—
177#102—532no—
178#102—532no—
179#102—531no—
180#102—532no—
181#102C2×D5312yes—
182C22⋊C8—532no—
183C4×C8—531no—
184C24—1062no—
185#41—2124no—
186C2×Q8—21no—
187C2×D4—531no—
188C2×D4—532no—
189C2×D4—531no—
190C2×D4—1062no—
191C2×D4—1062no—
192C2×D4—1062no—
193C2×D4—1064no—
194C2×D4—1061no—
195C2×D4—1061no—
196C2×D4—2124no—
197C2×D4—2122no—
198C2×D4C22×D5311yescharacteristic
199C22×C4—21no—
200C22×C4—532no—
201C22×C4—1062no—
202C22×C4—1061no—
203D8—1062no—
204D8—2122no—
205C2×C8—531no—
206C2×C8—531no—
207C2×C8—2122no—
208C2×C8C22×D5312yes—
209C4⋊C4—21no—
210C4⋊C4—531no—
211C4⋊C4—531no—
212C4⋊C4—1062no—
213C4⋊C4C22×D5312yes—
214C4⋊C4C22×D5311yescharacteristic
215C22⋊C4—21no—
216C22⋊C4—532no—
217C22⋊C4—1062no—
218C22⋊C4—1062no—
219C22⋊C4—1061no—
220C42C22×D5311yescharacteristic
221C23—21no—
222C23—532no—
223C23—1061no—
224C23—1062no—
225C23—4242no—
226C23—2124no—
227C23—2124no—
228Q8—42no—
229D4—21no—
230D4—534no—
231D4—1061no—
232D4—1062no—
233D4—1061no—
234D4—1062no—
235D4—4241no—
236D4—2124no—
237D4—2124no—
238D4—2124no—
239D4—2122no—
240D4#1171994912yescomplemented
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—532no—
245C2×C4—1062no—
246C2×C4—1062no—
247C2×C4—1061no—
248C2×C4—2124no—
249C2×C4—2122no—
250C2×C4D4×D5311yescharacteristic
251C2×C4D4×D5311yescharacteristic
252C2×C4C23×D5311yescharacteristic
253C8—22no—
254C8—1061no—
255C8—1061no—
256C22—22no—
257C22—41no—
258C22—41no—
259C22—534no—
260C22—1062no—
261C22—1062no—
262C22—4248no—
263C22—2122no—
264C22—2121no—
265C22—2121no—
266C22—2122no—
267C22—2122no—
268C22—2121no—
269C22C2×D4×D5311yescharacteristic
270C4—21no—
271C4—41no—
272C4—42no—
273C4—1062no—
274C4—2121no—
275C4D8×D5312yes—
276C4C2×#1171994911yescharacteristic
277C4#1313326311yescharacteristic
278C2—42no—
279C2—1064no—
280C2—4242no—
281C2—4241no—
282C2#1681126411yescharacteristic
283C2C2×D8×D5311yescharacteristic
284C2#1681143211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#474C2
socleC22×C53C22×D4
hypercenter#474D53
hypocenterC53#3294
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #16811276C2semidirect
2. D4×D212C2semidirect
3. #16810971C2semidirect
4. #16810938C2semidirect
5. #16810860C2semidirect
6. #16810819C2semidirect
7. #16810758C2semidirect
8. C4×#11719922C2semidirect
9. #16810688C2semidirect
10. #102×D53C2semidirect
11. C53×#474C2semidirect
15. C2×D424C22semidirect
17. C2×C8×D53C22semidirect
22. #13133192C22semidirect
26. C4×D212C22semidirect
27. C4⋊C4×D53C22semidirect
28. #13133150C22semidirect
29. #13133139C22semidirect
31. #13133137C22semidirect
32. #13133137C22semidirect
33. #13133137C22semidirect
34. #13133129C22semidirect
35. C4×#88322C22semidirect
36. C4⋊Q8×C53C22semidirect
38. #107×C53C22semidirect
39. C4⋊C8×C53C22semidirect
87. D4×C53#41semidirect
105. D212D8semidirect
147. #474D53semidirect
154. C53#3294semidirect
161. C4⋊Q8C2×D53semidirect
174. #107C2×D53semidirect
175. C4⋊C8C2×D53semidirect
240. D4#11719949semidirect