Finite Group Databasebeta

← #20285735

Subgroups — #20285735

Order 6784.

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

normal subgroups only

This group has 490 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
1C2×#13133214C211yesmaximal, characteristic, complemented
2#16811157C211yesmaximal, characteristic, complemented
3#16811012C212yesmaximal, complemented
4#16810988C212yesmaximal, complemented
5#16810824C212yesmaximal, complemented
6#16810819C212yesmaximal, complemented
7#16810776C212yesmaximal, complemented
8#16810759C212yesmaximal, complemented
9C53×#432C211yesmaximal, characteristic, complemented
10C22×D212—21no—
11C22×D212—21no—
12C22×D212C2212yes—
13C2×#11719913C2214yescomplemented
14C2×D424C2214yescomplemented
15#13133214—21no—
16#13133214—21no—
17#13133214C2212yes—
18#13133214C2212yes—
19#13133211—21no—
20#13133209C2211yescharacteristic, complemented
21#13133205C2211yescharacteristic, complemented
22#13133200—21no—
23#13133196C2211yescharacteristic
24#13133191C2211yescharacteristic
25C4×Dic106C2211yescharacteristic, complemented
26C4×D212C2211yescharacteristic, complemented
27C4×C22⋊D53—21no—
28#13133174—21no—
29#13133139C2214yescomplemented
30#13133139C2214yescomplemented
31C2×C42×C53C2211yescharacteristic
32C4⋊C8×C53C2212yescomplemented
33C22⋊C8×C53C2212yes—
34#97×C53C2212yescomplemented
35C23×D53—22no—
36C22×C4×C53D411yescharacteristic
37C22×C4×C53D411yescharacteristic
38C22×C4×C53C2311yescharacteristic
39C2×C22⋊D53—21no—
40C2×D212—22no—
41C2×D212—22no—
42C2×D212—22no—
43C2×D212—22no—
44C2×D212—22no—
45C2×D212—44no—
46C2×D212—44no—
47C2×D212—44no—
48C2×D212C2314yes—
49C2×D212C2311yescharacteristic
50C2×C4×D53—21no—
51C2×Dic106C2311yescharacteristic
52C2×C8×C53C2314yes—
53C42×C53D411yescharacteristic
54C42×C53D411yescharacteristic
55C42×C53C2311yescharacteristic
56C42×C53C2311yescharacteristic
57C22⋊Dic53—21no—
58#11719921—21no—
59#11719921—21no—
60#11719921—21no—
61#11719921—21no—
62C4⋊Dic53C2311yescharacteristic
63C4⋊Dic53C2311yescharacteristic
64#11719919—21no—
65#11719919—21no—
66#11719919—21no—
67#11719919—21no—
68C4×Dic53—21no—
69D424—44no—
70#11719913—44no—
71C23×C53C2×D411yescharacteristic
72C22×D53—21no—
73C22×D53—24no—
74C22×D53—44no—
75C22×D53—44no—
76C22×D53—44no—
77C2×C4×C53—22no—
78C2×C4×C53—22no—
79C2×C4×C53—22no—
80C2×C4×C53C2×D411yescharacteristic
81C2×C4×C53C2×D411yescharacteristic
82C2×C4×C53C2×D411yescharacteristic
83C2×C4×C53C2×D411yescharacteristic
84C2×C4×C53C2×D411yescharacteristic
85C2×C4×C53C2411yescharacteristic
86C22⋊D53—42no—
87C2×Dic53—21no—
88C2×Dic53—21no—
89C2×Dic53—21no—
90D212—24no—
91D212—24no—
92D212—22no—
93D212—44no—
94D212—44no—
95D212—44no—
96D212—44no—
97D212—44no—
98C4×D53—42no—
99Dic106—22no—
100C8×C53—24no—
101C22×C53—21no—
102C22×C53—21no—
103C22×C53AGL(1,ℤ/8)12yes—
104C22×C53C22×D411yescharacteristic
105C2×D53—44no—
106C2×D53—41no—
107C2×D53—41no—
108C2×D53—44no—
109C2×D53—44no—
110C2×D53—41no—
111C2×D53—88no—
112C4×C53—21no—
113C4×C53—21no—
114C4×C53—21no—
115C4×C53—21no—
116C4×C53AGL(1,ℤ/8)12yes—
117C4×C53H(5,𝔽2)12yes—
118Dic53—41no—
119Dic53—41no—
120Dic53—41no—
121#3217—531nomaximal, Sylow
122C2×C53—22no—
123C2×C53#53511yescharacteristic
124C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
125C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
126D53—81no—
127D53—84no—
128C2×#127—531no—
129#529—531no—
130#496—532no—
131#490—532no—
132#460—532no—
133#459—532no—
134#450—532no—
135#447—532no—
136#432D5311yescharacteristic, complemented
137C53#321711yescharacteristic, complemented, Sylow
138C22×D4—532no—
139C22×D4—1061no—
140C22×D4—1061no—
141C2×SD16—534no—
142C2×D8—534no—
143#127—532no—
144#127—532no—
145#127—1061no—
146#127—1061no—
147#126—1061no—
148#125—531no—
149#124—531no—
150#123—1061no—
151C22⋊Q8—531no—
152#121—531no—
153C4×Q8—531no—
154C4×D4—531no—
155C4×D4—1061no—
156#117—1061no—
157C2×C42C2×D5311yescharacteristic
158C4⋊C8C2×D5312yescomplemented
159#102—534no—
160#102—534no—
161C22⋊C8C2×D5312yes—
162#97C2×D5312yescomplemented
163C24—1062no—
164C2×Q8—531no—
165C2×D4—534no—
166C2×D4—531no—
167C2×D4—1061no—
168C2×D4—1062no—
169C2×D4—1062no—
170C2×D4—1062no—
171C2×D4—1062no—
172C2×D4—1062no—
173C2×D4—2124no—
174C2×D4—2124no—
175C2×D4—2124no—
176C22×C4—1061no—
177C22×C4D21211yescharacteristic
178C22×C4D21211yescharacteristic
179C22×C4C22×D5311yescharacteristic
180SD16—2124no—
181D8—2124no—
182C2×C8C22×D5314yes—
183C4⋊C4—531no—
184C4⋊C4—531no—
185C4⋊C4—1061no—
186C4⋊C4—1061no—
187C4⋊C4—1061no—
188C4⋊C4—1061no—
189C22⋊C4—1061no—
190C22⋊C4—1061no—
191C22⋊C4—1061no—
192C22⋊C4—1061no—
193C22⋊C4—1061no—
194C42—1061no—
195C42D21211yescharacteristic
196C42D21211yescharacteristic
197C42C22×D5311yescharacteristic
198C42C22×D5311yescharacteristic
199C23—1061no—
200C23—1064no—
201C23—2124no—
202C23—2124no—
203C23—2124no—
204C23C2×D21211yescharacteristic
205Q8—1062no—
206D4—1062no—
207D4—1064no—
208D4—1064no—
209D4—2124no—
210D4—2124no—
211D4—2124no—
212D4—2124no—
213D4—2122no—
214D4—2124no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—1061no—
219C2×C4—1061no—
220C2×C4—1061no—
221C2×C4—2122no—
222C2×C4C2×D21211yescharacteristic
223C2×C4C2×D21211yescharacteristic
224C2×C4C2×D21211yescharacteristic
225C2×C4C2×D21211yescharacteristic
226C2×C4C2×D21211yescharacteristic
227C2×C4C23×D5311yescharacteristic
228C8—24no—
229C22—21no—
230C22—21no—
231C22—4248no—
232C22—2121no—
233C22—2124no—
234C22—2124no—
235C22—2121no—
236C22—2124no—
237C22—2121no—
238C22#1313324512yes—
239C22C22×D21211yescharacteristic
240C4—21no—
241C4—21no—
242C4—21no—
243C4—21no—
244C4—2121no—
245C4—2121no—
246C4—2121no—
247C4#1313324512yes—
248C4#1313326312yes—
249C2—22no—
250C2—4244no—
251C2—4241no—
252C2#1681119211yescharacteristic
253C2C2×#1313324511yescharacteristic
254C2C2×#1313324511yescharacteristic
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#432C2
socleC22×C53C22×D4
hypercenter#432D53
hypocenterC53#3217
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
1. C2×#13133214C2semidirect
2. #16811157C2semidirect
3. #16811012C2semidirect
4. #16810988C2semidirect
5. #16810824C2semidirect
6. #16810819C2semidirect
7. #16810776C2semidirect
8. #16810759C2semidirect
9. C53×#432C2semidirect
13. C2×#11719913C22semidirect
14. C2×D424C22semidirect
20. #13133209C22semidirect
21. #13133205C22semidirect
25. C4×Dic106C22semidirect
26. C4×D212C22semidirect
29. #13133139C22semidirect
30. #13133139C22semidirect
32. C4⋊C8×C53C22semidirect
34. #97×C53C22semidirect
136. #432D53semidirect
137. C53#3217semidirect
158. C4⋊C8C2×D53semidirect
162. #97C2×D53semidirect