Finite Group Databasebeta

← #20285789

Subgroups — #20285789

Order 6784.

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

normal subgroups only

This group has 396 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#16811235C211yesmaximal, characteristic, complemented
2#16811192C211yesmaximal, characteristic
3#16811017C211yesmaximal, characteristic, complemented
4#16810996C211yesmaximal, characteristic, complemented
5#16810988C211yesmaximal, characteristic, complemented
6#16810978C211yesmaximal, characteristic, complemented
7#16810846C212yesmaximal, complemented
8#16810824C212yesmaximal, complemented
9#16810787C212yesmaximal, complemented
10#16810776C212yesmaximal, complemented
11C53×#433C211yesmaximal, characteristic, complemented
12C2×#11719945C2211yescharacteristic
13C22×D212C2211yescharacteristic
14C2×Dic212C2212yescomplemented
15C2×#11719913C2212yes—
16C2×#11719913C2212yes—
17C2×#11719913C2212yescomplemented
18#13133217C2211yescharacteristic, complemented
19#13133214C2211yescharacteristic
20#13133214C2211yescharacteristic
21#13133210—22no—
22#13133209C2211yescharacteristic, complemented
23#13133207—22no—
24#13133205C2211yescharacteristic
25#13133205C2211yescharacteristic
26#13133196C2212yes—
27#13133192—22no—
28#13133187—22no—
29C4×Dic106C2212yescomplemented
30#13133142C2212yescomplemented
31#13133139C2212yes—
32#13133139C2212yes—
33#13133139C2212yescomplemented
34#117×C53C2211yescharacteristic
35C4⋊C8×C53C2212yescomplemented
36C22⋊C8×C53C2212yes—
37#97×C53C2211yescharacteristic, complemented
38C4×C8×C53C2211yescharacteristic, complemented
39C23×D53—21no—
40C22×C4×C53C2311yescharacteristic
41C2×C22⋊D53—21no—
42#11719945—42no—
43C2×D212—21no—
44C2×D212—22no—
45C2×D212—22no—
46C2×D212—42no—
47C2×D212C2311yescharacteristic
48C2×D212C2311yescharacteristic
49C2×D212C2311yescharacteristic
50C2×C4×D53—21no—
51C2×Dic106C2311yescharacteristic
52C2×Dic106C2312yes—
53C2×C8×C53C2312yes—
54C2×C8×C53C2312yes—
55C4⋊C4×C53D412yes—
56C22⋊C4×C53D412yes—
57C42×C53C2311yescharacteristic
58C42×C53C2311yescharacteristic
59C22⋊Dic53—22no—
60#11719921—22no—
61#11719921—22no—
62C4⋊Dic53—22no—
63C4⋊Dic53C2312yes—
64#11719919—22no—
65#11719919—22no—
66#11719919—22no—
67C4×Dic53—22no—
68Dic212—42no—
69#11719913—42no—
70#11719913—42no—
71#11719913—42no—
72C23×C53C2×D411yescharacteristic
73C22×D53—21no—
74C22×D53—21no—
75C22×D53—21no—
76C22×D53—42no—
77C22×D53—82no—
78C2×C4×C53—22no—
79C2×C4×C53C2×D411yescharacteristic
80C2×C4×C53C2×D412yes—
81C2×C4×C53C2×D412yes—
82C2×C4×C53C2411yescharacteristic
83C22⋊D53—42no—
84C2×Dic53—21no—
85C2×Dic53—22no—
86C2×Dic53—22no—
87D212—22no—
88D212—22no—
89D212—22no—
90D212—42no—
91D212—82no—
92D212—82no—
93C4×D53—42no—
94Dic106—22no—
95Dic106—22no—
96Dic106—22no—
97C8×C53—22no—
98C8×C53—22no—
99C22×C53—21no—
100C22×C53—22no—
101C22×C53C22×D411yescharacteristic
102C2×D53—41no—
103C2×D53—41no—
104C2×D53—41no—
105C2×D53—42no—
106C2×D53—42no—
107C2×D53—42no—
108C2×D53—84no—
109C4×C53—21no—
110C4×C53—22no—
111C4×C53—22no—
112C4×C53H(5,𝔽2)12yes—
113Dic53—41no—
114Dic53—42no—
115Dic53—42no—
116#3222—531nomaximal, Sylow
117C2×C53—41no—
118C2×C53#53511yescharacteristic
119C2×C53#57712yes—
120D53—81no—
121D53—81no—
122D53—81no—
123#541—531no—
124#535—531no—
125#497—531no—
126#492—531no—
127#490—531no—
128#488—531no—
129#464—532no—
130#460—532no—
131#452—532no—
132#450—532no—
133#433D5311yescharacteristic, complemented
134C53#322211yescharacteristic, complemented, Sylow
135C2×#41—531no—
136C22×D4—531no—
137C2×Q16—532no—
138C2×SD16—532no—
139C2×SD16—532no—
140C2×SD16—532no—
141C4⋊Q8—531no—
142#127—531no—
143#127—531no—
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#117C2×D5311yescharacteristic
154C4⋊C8C2×D5312yescomplemented
155#103—532no—
156#102—532no—
157#102—532no—
158#102—532no—
159C22⋊C8C2×D5312yes—
160#97C2×D5311yescharacteristic, complemented
161C4×C8C2×D5311yescharacteristic, complemented
162C24—1061no—
163#41—2122no—
164C2×Q8—531no—
165C2×Q8—532no—
166C2×D4—531no—
167C2×D4—531no—
168C2×D4—531no—
169C2×D4—1061no—
170C2×D4—1062no—
171C2×D4—1062no—
172C2×D4—1061no—
173C2×D4—2122no—
174C22×C4—1061no—
175C22×C4C22×D5311yescharacteristic
176Q16—2122no—
177SD16—2122no—
178SD16—2122no—
179SD16—2122no—
180C2×C8C22×D5312yes—
181C2×C8C22×D5312yes—
182C4⋊C4—532no—
183C4⋊C4—1062no—
184C4⋊C4—1062no—
185C4⋊C4—1062no—
186C4⋊C4—1062no—
187C4⋊C4D21212yes—
188C22⋊C4—1062no—
189C22⋊C4—1062no—
190C22⋊C4—1062no—
191C22⋊C4D21212yes—
192C42—1062no—
193C42C22×D5311yescharacteristic
194C42C22×D5311yescharacteristic
195C23—1061no—
196C23—1061no—
197C23—1061no—
198C23—4242no—
199C23—2122no—
200C23C2×D21211yescharacteristic
201Q8—1062no—
202Q8—1062no—
203Q8—1062no—
204D4—1062no—
205D4—1062no—
206D4—1062no—
207D4—4242no—
208D4—4242no—
209D4—2122no—
210D4—2122no—
211C2×C4—22no—
212C2×C4—1062no—
213C2×C4—1062no—
214C2×C4—1061no—
215C2×C4—2122no—
216C2×C4C2×D21212yes—
217C2×C4C2×D21212yes—
218C2×C4C2×D21211yescharacteristic
219C2×C4C23×D5311yescharacteristic
220C8—22no—
221C8—22no—
222C22—22no—
223C22—21no—
224C22—4244no—
225C22—2122no—
226C22—2122no—
227C22—2122no—
228C22—2121no—
229C22—2121no—
230C22—2121no—
231C22C22×D21211yescharacteristic
232C4—22no—
233C4—22no—
234C4—21no—
235C4—2122no—
236C4—2121no—
237C4—2122no—
238C4#1313326312yes—
239C2—41no—
240C2—4241no—
241C2—4241no—
242C2—4241no—
243C2#1681119211yescharacteristic
244C2#1681143512yes—
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#433C2
socleC22×C53C22×D4
hypercenter#433D53
hypocenterC53#3222
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ #117 ◁ #433

Complements

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

NH ≅ G/Ntype
1. #16811235C2semidirect
3. #16811017C2semidirect
4. #16810996C2semidirect
5. #16810988C2semidirect
6. #16810978C2semidirect
7. #16810846C2semidirect
8. #16810824C2semidirect
9. #16810787C2semidirect
10. #16810776C2semidirect
11. C53×#433C2semidirect
14. C2×Dic212C22semidirect
17. C2×#11719913C22semidirect
18. #13133217C22semidirect
22. #13133209C22semidirect
29. C4×Dic106C22semidirect
30. #13133142C22semidirect
33. #13133139C22semidirect
35. C4⋊C8×C53C22semidirect
37. #97×C53C22semidirect
38. C4×C8×C53C22semidirect
133. #433D53semidirect
134. C53#3222semidirect
154. C4⋊C8C2×D53semidirect
160. #97C2×D53semidirect
161. C4×C8C2×D53semidirect