Finite Group Databasebeta

← #20285100

Subgroups — #20285100

Order 6784.

400 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 117 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#16811178C211yesmaximal, characteristic, complemented
2#124×D53C211yesmaximal, characteristic
3#16810985C211yesmaximal, characteristic, complemented
4#16810983C211yesmaximal, characteristic, complemented
5#16810982C211yesmaximal, characteristic, complemented
6#16810979C212yesmaximal, complemented
7#16810976C212yesmaximal, complemented
8#16810630C212yesmaximal, complemented
9#16810626C212yesmaximal
10#16810575C211yesmaximal, characteristic, complemented
11C53×#489C211yesmaximal, characteristic, complemented
12C2×Q8×D53C2211yescharacteristic
13C2×D4×D53C2211yescharacteristic
14C2×#11719912C2212yes—
15#13133217C2211yescharacteristic, complemented
16#13133208C2212yescomplemented
17#13133206C2211yescharacteristic
18#13133205C2211yescharacteristic
19#13133204C2211yescharacteristic
20#13133203—21no—
21#13133203—21no—
22#13133197—21no—
23#13133193—21no—
24Q8×Dic53—21no—
25#13133182—21no—
26#13133175C2212yes—
27C22⋊C4×D53—21no—
28C22⋊C4×D53—21no—
29C42×D53C2211yescharacteristic
30#13133142C2212yescomplemented
31#13133141C2212yescomplemented
32#13133140C2212yescomplemented
33#13133139C2212yes—
34#13133138C2212yes—
35#13133137C2212yes—
36#13133128C2211yescharacteristic, complemented
37C8×Dic53C2212yescomplemented
38C4⋊Q8×C53C2211yescharacteristic, complemented
39#124×C53C2211yescharacteristic
40#103×C53C2212yescomplemented
41#102×C53C2212yes—
42#97×C53C2211yescharacteristic, complemented
43C23×D53—21no—
44C2×Q8×C53—21no—
45C2×Q8×C53C2311yescharacteristic
46C2×D4×C53C2311yescharacteristic
47C2×C22⋊D53—21no—
48Q8×D53—24no—
49D4×D53—24no—
50C2×D212C2311yescharacteristic
51C2×C4×D53—21no—
52C2×C4×D53D411yescharacteristic
53C2×C4×D53D411yescharacteristic
54C2×C4×D53C2311yescharacteristic
55C2×Dic106—21no—
56C2×Dic106—21no—
57C2×Dic106C2311yescharacteristic
58C2×C8×C53C2312yes—
59C4⋊C4×C53—21no—
60C4⋊C4×C53C2312yes—
61C22⋊C4×C53—21no—
62C22⋊C4×C53—21no—
63C42×C53C2311yescharacteristic
64C22⋊Dic53—21no—
65C22⋊Dic53—21no—
66#11719921—22no—
67#11719921—21no—
68#11719921—21no—
69#11719921—21no—
70#11719921—21no—
71C4⋊Dic53—21no—
72C4⋊Dic53—22no—
73C4⋊Dic53C2312yes—
74#11719919—22no—
75C4×Dic53—22no—
76C4×Dic53D411yescharacteristic
77C4×Dic53D411yescharacteristic
78C4×Dic53C2311yescharacteristic
79C2×#88322C2312yes—
80#11719912—24no—
81C23×C53—21no—
82C22×D53—21no—
83C22×D53—42no—
84C22×D53—42no—
85C22×D53—42no—
86C22×D53C2×D411yescharacteristic
87Q8×C53—22no—
88Q8×C53—42no—
89D4×C53—22no—
90C2×C4×C53—21no—
91C2×C4×C53—22no—
92C2×C4×C53C2×D411yescharacteristic
93C2×C4×C53C2×D411yescharacteristic
94C2×C4×C53C2411yescharacteristic
95C22⋊D53—42no—
96C2×Dic53—21no—
97C2×Dic53—22no—
98C2×Dic53C2×D411yescharacteristic
99C2×Dic53C2×D411yescharacteristic
100C2×Dic53C2×D411yescharacteristic
101D212—22no—
102C4×D53—41no—
103C4×D53—41no—
104C4×D53—42no—
105C4×D53#4114yes—
106Dic106—22no—
107Dic106—42no—
108Dic106—42no—
109C8×C53—22no—
110#88322—22no—
111C22×C53—41no—
112C22×C53—41no—
113C22×C53—41no—
114C22×C53C22×D411yescharacteristic
115C2×D53—21no—
116C2×D53—21no—
117C2×D53—41no—
118C2×D53—41no—
119C2×D53—41no—
120C2×D53—84no—
121C2×D53#12412yes—
122C4×C53—21no—
123C4×C53—21no—
124C4×C53—41no—
125C4×C53—42no—
126C4×C53C2×#4112yes—
127Dic53—21no—
128Dic53—21no—
129Dic53—41no—
130Dic53—42no—
131Dic53#12412yes—
132#3140—531nomaximal, Sylow
133C2×C53—81no—
134C2×C53C2×#12411yescharacteristic
135C2×C53#57511yescharacteristic
136C2×C53#57511yescharacteristic
137D53—22no—
138D53—81no—
139#533—531no—
140C2×#124—531no—
141#489—531no—
142#489—531no—
143#489—531no—
144#489D5311yescharacteristic, complemented
145#488—532no—
146#488—532no—
147#419—532no—
148#418—532no—
149C4×M16—531no—
150C53#314011yescharacteristic, complemented, Sylow
151C22×Q8—531no—
152C22×D4—531no—
153C2×M16—532no—
154C4⋊Q8—531no—
155C4⋊Q8C2×D5311yescharacteristic, complemented
156#125—532no—
157#124—531no—
158#124—531no—
159#124—531no—
160#124—1061no—
161#124—1061no—
162#124C2×D5311yescharacteristic
163C22⋊Q8—1061no—
164C22⋊Q8—1061no—
165C4×Q8—1061no—
166C4×Q8—1061no—
167#117—532no—
168C2×C22⋊C4—1061no—
169C2×C22⋊C4—1061no—
170C2×C42—531no—
171#103—532no—
172#103—532no—
173#103—532no—
174#103C2×D5312yescomplemented
175#102—532no—
176#102—532no—
177#102—532no—
178#102C2×D5312yes—
179#97—531no—
180#97C2×D5311yescharacteristic, complemented
181C4×C8—532no—
182C24—1061no—
183C2×Q8—21no—
184C2×Q8—531no—
185C2×Q8—1061no—
186C2×Q8—1064no—
187C2×Q8—1061no—
188C2×Q8C22×D5311yescharacteristic
189C2×D4—531no—
190C2×D4—1064no—
191C2×D4—1061no—
192C2×D4C22×D5311yescharacteristic
193C22×C4—531no—
194C22×C4—531no—
195C22×C4—531no—
196C22×C4—1061no—
197M16—1064no—
198C2×C8—532no—
199C2×C8C22×D5312yes—
200C4⋊C4—21no—
201C4⋊C4—532no—
202C4⋊C4—1062no—
203C4⋊C4—1062no—
204C4⋊C4—1061no—
205C4⋊C4C22×D5312yes—
206C22⋊C4—21no—
207C22⋊C4—21no—
208C22⋊C4—1061no—
209C22⋊C4—1061no—
210C22⋊C4—1062no—
211C22⋊C4—1061no—
212C22⋊C4—1061no—
213C22⋊C4—1061no—
214C22⋊C4—1061no—
215C42—531no—
216C42—531no—
217C42—531no—
218C42—1062no—
219C42C22×D5311yescharacteristic
220C23—21no—
221C23—531no—
222C23—1061no—
223C23—2122no—
224C23—2122no—
225C23—2122no—
226Q8—22no—
227Q8—42no—
228Q8—1062no—
229Q8—2122no—
230Q8—2122no—
231D4—22no—
232D4—1062no—
233D4—2122no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—531no—
237C2×C4—534no—
238C2×C4—531no—
239C2×C4—531no—
240C2×C4—1062no—
241C2×C4—1061no—
242C2×C4—2121no—
243C2×C4—2121no—
244C2×C4—2122no—
245C2×C4D4×D5311yescharacteristic
246C2×C4D4×D5311yescharacteristic
247C2×C4C23×D5311yescharacteristic
248C8—22no—
249C8—1062no—
250C22—41no—
251C22—41no—
252C22—41no—
253C22—532no—
254C22—1061no—
255C22—1061no—
256C22—4244no—
257C22—2121no—
258C22—2121no—
259C22—2121no—
260C22C2×D4×D5311yescharacteristic
261C4—21no—
262C4—21no—
263C4—41no—
264C4—42no—
265C4—532no—
266C4—1061no—
267C4—1061no—
268C4—2121no—
269C4—2122no—
270C4#41×D5312yes—
271C2—81no—
272C2—1062no—
273C2—4241no—
274C2#124×D5311yescharacteristic
275C2#1681141911yescharacteristic
276C2#1681142211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#489C2
socleC22×C53C22×D4
hypercenter#489D53
hypocenterC53#3140
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #16811178C2semidirect
3. #16810985C2semidirect
4. #16810983C2semidirect
5. #16810982C2semidirect
6. #16810979C2semidirect
7. #16810976C2semidirect
8. #16810630C2semidirect
10. #16810575C2semidirect
11. C53×#489C2semidirect
15. #13133217C22semidirect
16. #13133208C22semidirect
30. #13133142C22semidirect
31. #13133141C22semidirect
32. #13133140C22semidirect
36. #13133128C22semidirect
37. C8×Dic53C22semidirect
38. C4⋊Q8×C53C22semidirect
40. #103×C53C22semidirect
42. #97×C53C22semidirect
144. #489D53semidirect
150. C53#3140semidirect
155. C4⋊Q8C2×D53semidirect
174. #103C2×D53semidirect
180. #97C2×D53semidirect