Finite Group Databasebeta

← #20285720

Subgroups — #20285720

Order 6784.

400 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 121 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#16811180C211yesmaximal, characteristic, complemented
2C2×#13133205C211yesmaximal, characteristic
3#16811017C212yesmaximal, complemented
4#16810983C212yesmaximal, complemented
5#16810846C214yesmaximal, complemented
6#16810781C212yesmaximal, complemented
7#16810776C212yesmaximal
8C53×#432C211yesmaximal, characteristic, complemented
9C22×Dic106C2211yescharacteristic
10C22×D212C2211yescharacteristic
11C2×Dic212C2214yescomplemented
12C2×#11719913C2214yes—
13#13133217C2211yescharacteristic, complemented
14#13133217C2211yescharacteristic, complemented
15#13133209—21no—
16#13133205—21no—
17#13133205—21no—
18#13133205C2212yes—
19#13133205C2212yes—
20#13133196—21no—
21#13133196C2212yes—
22C4×Dic106—21no—
23C4×Dic106C2212yescomplemented
24#13133176—21no—
25C2×#11719921—21no—
26C2×#11719921—21no—
27#13133142C2214yescomplemented
28#13133139C2214yes—
29C2×C42×C53C2211yescharacteristic
30C4⋊C8×C53C2212yescomplemented
31C22⋊C8×C53C2212yes—
32#97×C53C2212yescomplemented
33C23×D53—21no—
34C22×C4×C53D411yescharacteristic
35C22×C4×C53D411yescharacteristic
36C22×C4×C53C2311yescharacteristic
37C22×Dic53—21no—
38C2×D212—21no—
39C2×D212—42no—
40C2×D212C2312yes—
41C2×Dic106—21no—
42C2×Dic106—21no—
43C2×Dic106—42no—
44C2×Dic106C2312yes—
45C2×Dic106C2312yes—
46C2×C8×C53C2314yes—
47C42×C53D411yescharacteristic
48C42×C53D411yescharacteristic
49C42×C53C2311yescharacteristic
50C42×C53C2311yescharacteristic
51C22⋊Dic53—22no—
52#11719921—22no—
53#11719921—22no—
54#11719921—22no—
55#11719921—22no—
56C4⋊Dic53—21no—
57C4⋊Dic53—21no—
58C4⋊Dic53—21no—
59C4⋊Dic53C2312yes—
60#11719919—22no—
61#11719919—22no—
62C4×Dic53—22no—
63Dic212—44no—
64#11719913—44no—
65C23×C53C2×D411yescharacteristic
66C22×D53—22no—
67C22×D53—42no—
68C22×D53—42no—
69C22×D53—42no—
70C2×C4×C53—22no—
71C2×C4×C53—22no—
72C2×C4×C53—41no—
73C2×C4×C53C2×D411yescharacteristic
74C2×C4×C53C2×D411yescharacteristic
75C2×C4×C53C2×D411yescharacteristic
76C2×C4×C53C2×D411yescharacteristic
77C2×C4×C53C2×D411yescharacteristic
78C2×C4×C53C2411yescharacteristic
79C2×Dic53—22no—
80C2×Dic53—22no—
81C2×Dic53—22no—
82C2×Dic53—42no—
83D212—24no—
84D212—42no—
85Dic106—24no—
86Dic106—24no—
87Dic106—42no—
88Dic106—42no—
89C8×C53—24no—
90C22×C53—21no—
91C22×C53—21no—
92C22×C53#13712yes—
93C22×C53C22×D411yescharacteristic
94C2×D53—42no—
95C2×D53—42no—
96C2×D53—42no—
97C2×D53—84no—
98C4×C53—21no—
99C4×C53—21no—
100C4×C53—21no—
101C4×C53—21no—
102C4×C53—21no—
103C4×C53H(5,𝔽2)12yes—
104Dic53—42no—
105Dic53—42no—
106Dic53—42no—
107#3215—531nomaximal, Sylow
108C2×C53—22no—
109C2×C53#53511yescharacteristic
110C2×C53C2×#13711yescharacteristic
111C2×C53#57511yescharacteristic
112D53—82no—
113#533—531no—
114C2×#124—531no—
115#497—532no—
116#489—532no—
117#464—534no—
118#451—532no—
119#450—532no—
120#432D5311yescharacteristic, complemented
121C53#321511yescharacteristic, complemented, Sylow
122C22×Q8—531no—
123C22×D4—531no—
124C2×Q16—534no—
125C2×SD16—534no—
126C4⋊Q8—531no—
127C4⋊Q8—531no—
128#125—1061no—
129#124—532no—
130#124—532no—
131#124—1061no—
132#124—1061no—
133C22⋊Q8—532no—
134C22⋊Q8—1061no—
135C4×Q8—532no—
136C4×Q8—1061no—
137#117—1061no—
138C2×C22⋊C4—1061no—
139C2×C22⋊C4—1061no—
140C2×C42C2×D5311yescharacteristic
141C4⋊C8C2×D5312yescomplemented
142#103—534no—
143#102—534no—
144C22⋊C8C2×D5312yes—
145#97C2×D5312yescomplemented
146C24—1061no—
147C2×Q8—532no—
148C2×Q8—532no—
149C2×Q8—1061no—
150C2×Q8—1061no—
151C2×Q8—2122no—
152C2×D4—532no—
153C2×D4—1061no—
154C2×D4—2122no—
155C22×C4—1061no—
156C22×C4D21211yescharacteristic
157C22×C4D21211yescharacteristic
158C22×C4C22×D5311yescharacteristic
159Q16—2124no—
160SD16—2124no—
161C2×C8C22×D5314yes—
162C4⋊C4—532no—
163C4⋊C4—1061no—
164C4⋊C4—1061no—
165C4⋊C4—1062no—
166C4⋊C4—1062no—
167C4⋊C4—1061no—
168C22⋊C4—1062no—
169C22⋊C4—1062no—
170C22⋊C4—1062no—
171C22⋊C4—1062no—
172C22⋊C4—1062no—
173C42—1062no—
174C42D21211yescharacteristic
175C42D21211yescharacteristic
176C42C22×D5311yescharacteristic
177C42C22×D5311yescharacteristic
178C23—1062no—
179C23—2122no—
180C23—2122no—
181C23—2122no—
182C23C2×D21211yescharacteristic
183Q8—1064no—
184Q8—1064no—
185Q8—2122no—
186Q8—2122no—
187D4—1064no—
188D4—2122no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—41no—
192C2×C4—1062no—
193C2×C4—1062no—
194C2×C4—1062no—
195C2×C4—2122no—
196C2×C4C2×D21211yescharacteristic
197C2×C4C2×D21211yescharacteristic
198C2×C4C2×D21211yescharacteristic
199C2×C4C2×D21211yescharacteristic
200C2×C4C2×D21211yescharacteristic
201C2×C4C23×D5311yescharacteristic
202C8—24no—
203C22—21no—
204C22—21no—
205C22—4244no—
206C22—2122no—
207C22—2122no—
208C22—2122no—
209C22#1313325012yes—
210C22C22×D21211yescharacteristic
211C4—21no—
212C4—21no—
213C4—21no—
214C4—21no—
215C4—21no—
216C4—2122no—
217C4—2122no—
218C4—2122no—
219C4#1313326312yes—
220C2—22no—
221C2—4242no—
222C2#1681119211yescharacteristic
223C2C2×#1313325011yescharacteristic
224C2#1681142311yescharacteristic
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#432C2
socleC22×C53C22×D4
hypercenter#432D53
hypocenterC53#3215
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. #16811180C2semidirect
3. #16811017C2semidirect
4. #16810983C2semidirect
5. #16810846C2semidirect
6. #16810781C2semidirect
8. C53×#432C2semidirect
11. C2×Dic212C22semidirect
13. #13133217C22semidirect
14. #13133217C22semidirect
23. C4×Dic106C22semidirect
27. #13133142C22semidirect
30. C4⋊C8×C53C22semidirect
32. #97×C53C22semidirect
120. #432D53semidirect
121. C53#3215semidirect
141. C4⋊C8C2×D53semidirect
145. #97C2×D53semidirect