Finite Group Databasebeta

← #20285780

Subgroups — #20285780

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#16811186C211yesmaximal, characteristic
3#16811015C211yesmaximal, characteristic, complemented
4#16810987C211yesmaximal, characteristic, complemented
5#16810978C211yesmaximal, characteristic, complemented
6#16810843C212yesmaximal, complemented
7#16810821C212yesmaximal, complemented
8#16810782C212yesmaximal, complemented
9#16810772C212yesmaximal, complemented
10#16810682C211yesmaximal, characteristic, complemented
11C53×#492C211yesmaximal, characteristic, complemented
12C2×#11719947C2211yescharacteristic
13C2×D4×D53C2211yescharacteristic
14C2×#11719925C2212yescomplemented
15C2×#11719924C2212yes—
16#13133213C2211yescharacteristic
17#13133212—22no—
18#13133209C2211yescharacteristic, complemented
19#13133207—22no—
20#13133205C2211yescharacteristic
21#13133204C2211yescharacteristic
22#13133193C2212yes—
23#13133189—22no—
24#13133185—22no—
25Q8×Dic53C2212yescomplemented
26#13133174C2211yescharacteristic
27#13133146C2212yescomplemented
28#13133142C2212yescomplemented
29#13133139C2212yes—
30#13133138C2212yes—
31#13133138C2212yescomplemented
32#13133129C2212yes—
33#13133128C2211yescharacteristic, complemented
34C2×SD16×C53C2212yes—
35C2×SD16×C53C2212yescomplemented
36C4⋊Q8×C53C2211yescharacteristic, complemented
37#127×C53C2211yescharacteristic
38C4×C8×C53C2211yescharacteristic, complemented
39C23×D53—21no—
40C2×Q8×C53C2312yes—
41C2×D4×C53—22no—
42C2×D4×C53C2311yescharacteristic
43C2×D4×C53C2311yescharacteristic
44C2×C22⋊D53—21no—
45C2×C22⋊D53—21no—
46C2×C22⋊D53—22no—
47C22×Dic53—21no—
48#11719947—42no—
49D4×D53—42no—
50C2×D212C2311yescharacteristic
51C2×C4×D53C2311yescharacteristic
52C2×Dic106C2311yescharacteristic
53SD16×C53—42no—
54SD16×C53—42no—
55C2×C8×C53C2312yes—
56C4⋊C4×C53—22no—
57C42×C53C2311yescharacteristic
58C22⋊Dic53—22no—
59#11719925—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—
73C23×C53—21no—
74C22×D53—21no—
75C22×D53—42no—
76C22×D53—82no—
77C22×D53C2×D411yescharacteristic
78Q8×C53—22no—
79Q8×C53—22no—
80D4×C53—22no—
81D4×C53—22no—
82D4×C53—82no—
83C2×C4×C53—22no—
84C2×C4×C53C2×D412yes—
85C2×C4×C53C2411yescharacteristic
86C22⋊D53—42no—
87C22⋊D53—42no—
88C22⋊D53—82no—
89C2×Dic53—21no—
90C2×Dic53—22no—
91C2×Dic53—42no—
92C2×Dic53C2×D411yescharacteristic
93C2×Dic53C2×D412yes—
94D212—22no—
95C4×D53—22no—
96Dic106—22no—
97C8×C53—22no—
98#88322—22no—
99C22×C53—41no—
100C22×C53—41no—
101C22×C53—42no—
102C22×C53—42no—
103C22×C53C22×D411yescharacteristic
104C2×D53—21no—
105C2×D53—22no—
106C2×D53—41no—
107C2×D53—42no—
108C2×D53—84no—
109C4×C53—22no—
110C4×C53—42no—
111C4×C53H(5,𝔽2)12yes—
112Dic53—21no—
113Dic53—22no—
114Dic53—41no—
115Dic53—42no—
116#3222—531nomaximal, Sylow
117C2×C53—81no—
118C2×C53—81no—
119C2×C53#53511yescharacteristic
120C2×C53#57712yes—
121D53—41no—
122D53—81no—
123#541—531no—
124#535—531no—
125#497—531no—
126#492D5311yescharacteristic, complemented
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×Q16—532no—
138C2×SD16—532no—
139C2×SD16C2×D5312yes—
140C2×SD16C2×D5312yescomplemented
141C4⋊Q8C2×D5311yescharacteristic, complemented
142#127—531no—
143#127C2×D5311yescharacteristic
144#126—1062no—
145#125—531no—
146#125—1062no—
147#124—531no—
148#124—531no—
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#97—531no—
161C4×C8C2×D5311yescharacteristic, complemented
162C24—1061no—
163#41—2122no—
164C2×Q8—531no—
165C2×Q8C22×D5312yes—
166C2×D4—22no—
167C2×D4—531no—
168C2×D4—1062no—
169C2×D4—1061no—
170C2×D4—1061no—
171C2×D4—2122no—
172C2×D4C22×D5311yescharacteristic
173C2×D4C22×D5311yescharacteristic
174C22×C4—531no—
175C22×C4—1061no—
176Q16—2122no—
177SD16—42no—
178SD16—42no—
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—532no—
189C22⋊C4—1062no—
190C22⋊C4—1062no—
191C22⋊C4—1062no—
192C42—531no—
193C42—1062no—
194C42C22×D5311yescharacteristic
195C23—21no—
196C23—21no—
197C23—531no—
198C23—1061no—
199C23—4242no—
200C23—2122no—
201Q8—22no—
202Q8—22no—
203Q8—1062no—
204D4—22no—
205D4—22no—
206D4—82no—
207D4—1062no—
208D4—4242no—
209D4—2122no—
210D4—2122no—
211C2×C4—22no—
212C2×C4—532no—
213C2×C4—531no—
214C2×C4—1062no—
215C2×C4—1062no—
216C2×C4—1061no—
217C2×C4—2122no—
218C2×C4D4×D5312yes—
219C2×C4C23×D5311yescharacteristic
220C8—22no—
221C8—1062no—
222C22—42no—
223C22—42no—
224C22—41no—
225C22—41no—
226C22—1062no—
227C22—1061no—
228C22—4244no—
229C22—2122no—
230C22—2121no—
231C22C2×D4×D5311yescharacteristic
232C4—22no—
233C4—42no—
234C4—1062no—
235C4—1061no—
236C4—2122no—
237C4—2121no—
238C4#1313326212yes—
239C2—81no—
240C2—81no—
241C2—4241no—
242C2—2121no—
243C2#1681118611yescharacteristic
244C2#1681142912yes—
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#492C2
socleC22×C53C22×D4
hypercenter#492D53
hypocenterC53#3222
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. #16811234C2semidirect
3. #16811015C2semidirect
4. #16810987C2semidirect
5. #16810978C2semidirect
6. #16810843C2semidirect
7. #16810821C2semidirect
8. #16810782C2semidirect
9. #16810772C2semidirect
10. #16810682C2semidirect
11. C53×#492C2semidirect
14. C2×#11719925C22semidirect
18. #13133209C22semidirect
25. Q8×Dic53C22semidirect
27. #13133146C22semidirect
28. #13133142C22semidirect
31. #13133138C22semidirect
33. #13133128C22semidirect
35. C2×SD16×C53C22semidirect
36. C4⋊Q8×C53C22semidirect
38. C4×C8×C53C22semidirect
126. #492D53semidirect
134. C53#3222semidirect
140. C2×SD16C2×D53semidirect
141. C4⋊Q8C2×D53semidirect
161. C4×C8C2×D53semidirect