Finite Group Databasebeta

← #20285782

Subgroups — #20285782

Order 6784.

396 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 109 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#16811234C211yesmaximal, characteristic, complemented
2#16811191C211yesmaximal, characteristic
3#16810997C211yesmaximal, characteristic, complemented
4#16810988C211yesmaximal, characteristic, complemented
5#16810977C211yesmaximal, characteristic, complemented
6#16810847C212yesmaximal, complemented
7#16810821C212yesmaximal, complemented
8#16810784C212yesmaximal, complemented
9#16810772C212yesmaximal, complemented
10#16810685C211yesmaximal, characteristic, complemented
11C53×#497C211yesmaximal, characteristic, complemented
12C2×#11719949C2211yescharacteristic
13C2×D4×D53C2211yescharacteristic
14C2×#11719924C2212yes—
15C2×#11719924C2212yescomplemented
16#13133214C2211yescharacteristic
17#13133213C2211yescharacteristic
18#13133212—22no—
19#13133209C2211yescharacteristic, complemented
20#13133207—22no—
21#13133206C2211yescharacteristic
22#13133193C2212yes—
23#13133190—22no—
24#13133187—22no—
25Q8×Dic53C2212yescomplemented
26#13133174C2211yescharacteristic
27#13133146C2212yescomplemented
28#13133141C2212yescomplemented
29#13133139C2212yes—
30#13133139C2212yescomplemented
31#13133138C2212yes—
32#13133129C2212yes—
33C4×#88322C2211yescharacteristic, complemented
34C2×Q16×C53C2212yescomplemented
35C2×SD16×C53C2212yes—
36C4⋊Q8×C53C2211yescharacteristic, complemented
37#124×C53C2211yescharacteristic
38#97×C53C2211yescharacteristic, complemented
39C23×D53—21no—
40C2×Q8×C53C2311yescharacteristic
41C2×Q8×C53C2312yes—
42C2×D4×C53C2311yescharacteristic
43C2×C22⋊D53—21no—
44C2×C22⋊D53—22no—
45#11719949—42no—
46D4×D53—42no—
47C2×D212—21no—
48C2×D212—22no—
49C2×D212C2311yescharacteristic
50C2×D212C2311yescharacteristic
51C2×C4×D53—21no—
52C2×C4×D53C2311yescharacteristic
53Q16×C53—42no—
54SD16×C53—42no—
55C2×C8×C53C2312yes—
56C4⋊C4×C53—22no—
57C22⋊C4×C53—22no—
58C42×C53C2311yescharacteristic
59#11719924—42no—
60#11719924—42no—
61#11719921—22no—
62#11719921—22no—
63#11719921D412yes—
64C4⋊Dic53—22no—
65C4⋊Dic53C2312yes—
66#11719919—22no—
67#11719919—22no—
68#11719919D412yes—
69C4×Dic53—22no—
70C4×Dic53C2311yescharacteristic
71C2×#88322C2312yes—
72C23×C53—21no—
73C22×D53—21no—
74C22×D53—21no—
75C22×D53—42no—
76C22×D53—82no—
77C22×D53C2×D411yescharacteristic
78Q8×C53—22no—
79Q8×C53—22no—
80Q8×C53—22no—
81D4×C53—22no—
82C2×C4×C53—21no—
83C2×C4×C53—22no—
84C2×C4×C53C2×D412yes—
85C2×C4×C53C2411yescharacteristic
86C22⋊D53—42no—
87C22⋊D53—82no—
88C2×Dic53—22no—
89C2×Dic53C2×D411yescharacteristic
90C2×Dic53C2×D412yes—
91D212—22no—
92D212—22no—
93D212—42no—
94D212—82no—
95C4×D53—22no—
96C4×D53—42no—
97C8×C53—22no—
98#88322—22no—
99C22×C53—41no—
100C22×C53—42no—
101C22×C53C22×D411yescharacteristic
102C2×D53—21no—
103C2×D53—22no—
104C2×D53—41no—
105C2×D53—41no—
106C2×D53—42no—
107C2×D53—42no—
108C2×D53—84no—
109C4×C53—22no—
110C4×C53—41no—
111C4×C53—42no—
112C4×C53H(5,𝔽2)12yes—
113Dic53—21no—
114Dic53—22no—
115Dic53—42no—
116#3222—531nomaximal, Sylow
117C2×C53—81no—
118C2×C53#53511yescharacteristic
119C2×C53#57712yes—
120D53—41no—
121D53—81no—
122D53—81no—
123#541—531no—
124#535—531no—
125#497D5311yescharacteristic, complemented
126#492—531no—
127#490—531no—
128#488—531no—
129#464—532no—
130#460—532no—
131#452—532no—
132#450—532no—
133#433—531no—
134C53#322211yescharacteristic, complemented, Sylow
135C2×#41—531no—
136C22×D4—531no—
137C2×Q16C2×D5312yescomplemented
138C2×SD16—532no—
139C2×SD16—532no—
140C2×SD16C2×D5312yes—
141C4⋊Q8C2×D5311yescharacteristic, complemented
142#127—531no—
143#127—531no—
144#126—1062no—
145#125—531no—
146#125—1062no—
147#124—531no—
148#124C2×D5311yescharacteristic
149C22⋊Q8—532no—
150#121—1062no—
151#120—1062no—
152C4×Q8—532no—
153#117—531no—
154C4⋊C8—532no—
155#103—532no—
156#102—532no—
157#102—532no—
158#102—532no—
159C22⋊C8—532no—
160#97C2×D5311yescharacteristic, complemented
161C4×C8—531no—
162C24—1061no—
163#41—2122no—
164C2×Q8C22×D5311yescharacteristic
165C2×Q8C22×D5312yes—
166C2×D4—531no—
167C2×D4—531no—
168C2×D4—1061no—
169C2×D4—1062no—
170C2×D4—1061no—
171C2×D4—1062no—
172C2×D4—2122no—
173C2×D4C22×D5311yescharacteristic
174C22×C4—531no—
175C22×C4—1061no—
176Q16—42no—
177SD16—42no—
178SD16—2122no—
179SD16—2122no—
180C2×C8—532no—
181C2×C8C22×D5312yes—
182C4⋊C4—22no—
183C4⋊C4—532no—
184C4⋊C4—532no—
185C4⋊C4—1062no—
186C4⋊C4—1062no—
187C4⋊C4—1062no—
188C22⋊C4—22no—
189C22⋊C4—532no—
190C22⋊C4—1062no—
191C22⋊C4—1062no—
192C42—531no—
193C42—1062no—
194C42C22×D5311yescharacteristic
195C23—21no—
196C23—531no—
197C23—1061no—
198C23—1061no—
199C23—4242no—
200C23—2122no—
201Q8—22no—
202Q8—22no—
203Q8—22no—
204D4—22no—
205D4—1062no—
206D4—1062no—
207D4—4242no—
208D4—4242no—
209D4—2122no—
210D4—2122no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—532no—
214C2×C4—531no—
215C2×C4—1062no—
216C2×C4—1062no—
217C2×C4—2122no—
218C2×C4D4×D5312yes—
219C2×C4C23×D5311yescharacteristic
220C8—22no—
221C8—1062no—
222C22—42no—
223C22—41no—
224C22—1062no—
225C22—1061no—
226C22—4244no—
227C22—2122no—
228C22—2122no—
229C22—2121no—
230C22—2121no—
231C22C2×D4×D5311yescharacteristic
232C4—22no—
233C4—41no—
234C4—42no—
235C4—1062no—
236C4—1061no—
237C4—2122no—
238C4#1313326212yes—
239C2—81no—
240C2—4241no—
241C2—4241no—
242C2—2121no—
243C2#1681118611yescharacteristic
244C2#1681143212yes—
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#497C2
socleC22×C53C22×D4
hypercenter#497D53
hypocenterC53#3222
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. #16811234C2semidirect
3. #16810997C2semidirect
4. #16810988C2semidirect
5. #16810977C2semidirect
6. #16810847C2semidirect
7. #16810821C2semidirect
8. #16810784C2semidirect
9. #16810772C2semidirect
10. #16810685C2semidirect
11. C53×#497C2semidirect
15. C2×#11719924C22semidirect
19. #13133209C22semidirect
25. Q8×Dic53C22semidirect
27. #13133146C22semidirect
28. #13133141C22semidirect
30. #13133139C22semidirect
33. C4×#88322C22semidirect
34. C2×Q16×C53C22semidirect
36. C4⋊Q8×C53C22semidirect
38. #97×C53C22semidirect
125. #497D53semidirect
134. C53#3222semidirect
137. C2×Q16C2×D53semidirect
141. C4⋊Q8C2×D53semidirect
160. #97C2×D53semidirect