Finite Group Databasebeta

← #20285712

Subgroups — #20285712

Order 6784.

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

normal subgroups only

This group has 434 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#16811173C211yesmaximal, characteristic, complemented
2#16811173C211yesmaximal, characteristic, complemented
3#16811012C212yesmaximal, complemented
4#16810983C212yesmaximal, complemented
5#16810829C212yesmaximal, complemented
6#16810819C212yesmaximal, complemented
7#16810770C214yesmaximal, complemented
8C53×#432C211yesmaximal, characteristic, complemented
9C2×#11719945—21no—
10C2×#11719945C2212yes—
11C2×#11719913C2214yes—
12C2×D424C2214yes—
13#13133217C2211yescharacteristic
14#13133217C2211yescharacteristic
15#13133214C2211yescharacteristic
16#13133214C2211yescharacteristic
17#13133205—21no—
18#13133205C2212yes—
19#13133191—21no—
20#13133191—21no—
21#13133191—21no—
22#13133191C2212yes—
23C4×D212—21no—
24C4×D212—21no—
25C4×D212—21no—
26C4×D212C2212yes—
27#13133142C2214yes—
28#13133139C2214yes—
29C2×C42×C53C2211yescharacteristic
30C4⋊C8×C53C2212yes—
31C22⋊C8×C53C2212yes—
32#97×C53C2212yes—
33C22×C4×C53D411yescharacteristic
34C22×C4×C53D411yescharacteristic
35C22×C4×C53C2311yescharacteristic
36C2×C22⋊D53—22no—
37C2×C22⋊D53—22no—
38#11719945—44no—
39#11719945—44no—
40C2×D212—21no—
41C2×D212—21no—
42C2×D212—21no—
43C2×D212—21no—
44C2×D212C2312yes—
45C2×D212C2312yes—
46C2×C4×D53—22no—
47C2×C4×D53—22no—
48C2×Dic106—21no—
49C2×Dic106C2312yes—
50C2×C8×C53C2314yes—
51C42×C53D411yescharacteristic
52C42×C53D411yescharacteristic
53C42×C53C2311yescharacteristic
54C42×C53C2311yescharacteristic
55#11719921—22no—
56#11719921—22no—
57#11719921—22no—
58#11719921—22no—
59C4⋊Dic53—21no—
60C4⋊Dic53—21no—
61C4⋊Dic53—21no—
62C4⋊Dic53C2312yes—
63D424—44no—
64#11719913—44no—
65C23×C53C2×D411yescharacteristic
66C22×D53—22no—
67C22×D53—22no—
68C2×C4×C53—22no—
69C2×C4×C53—22no—
70C2×C4×C53—41no—
71C2×C4×C53C2×D411yescharacteristic
72C2×C4×C53C2×D411yescharacteristic
73C2×C4×C53C2×D411yescharacteristic
74C2×C4×C53C2×D411yescharacteristic
75C2×C4×C53C2×D411yescharacteristic
76C2×C4×C53C2411yescharacteristic
77C22⋊D53—44no—
78C22⋊D53—44no—
79C2×Dic53—22no—
80C2×Dic53—22no—
81D212—24no—
82D212—22no—
83D212—22no—
84D212—42no—
85D212—42no—
86D212—42no—
87D212—42no—
88C4×D53—44no—
89C4×D53—44no—
90Dic106—24no—
91Dic106—42no—
92C8×C53—24no—
93C22×C53—21no—
94C22×C53—21no—
95C22×C53—21no—
96C22×C53C22×D411yescharacteristic
97C2×D53—42no—
98C2×D53—42no—
99C2×D53—42no—
100C2×D53—42no—
101C2×D53—42no—
102C2×D53—42no—
103C4×C53—21no—
104C4×C53—21no—
105C4×C53—21no—
106C4×C53—21no—
107C4×C53AGL(1,ℤ/8)12yes—
108C4×C53H(5,𝔽2)12yes—
109Dic53—42no—
110Dic53—42no—
111#3214—531nomaximal, Sylow
112C2×C53—41no—
113C2×C53#53511yescharacteristic
114C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
115C2×C53#57511yescharacteristic
116D53—82no—
117D53—82no—
118#532—531no—
119#532—531no—
120#496—532no—
121#489—532no—
122#461—532no—
123#459—532no—
124#449—534no—
125#432D5311yescharacteristic, complemented
126C53#321411yescharacteristic, complemented, Sylow
127C2×#41—532no—
128C2×#41—1061no—
129C2×SD16—534no—
130C2×D8—534no—
131C4⋊Q8—531no—
132C4⋊Q8—531no—
133#127—531no—
134#127—531no—
135#124—532no—
136#124—1061no—
137#121—532no—
138#121—1061no—
139#121—1061no—
140#121—1061no—
141C4×D4—532no—
142C4×D4—1061no—
143C4×D4—1061no—
144C4×D4—1061no—
145C2×C42C2×D5311yescharacteristic
146C4⋊C8C2×D5312yes—
147#103—534no—
148#102—534no—
149C22⋊C8C2×D5312yes—
150#97C2×D5312yes—
151#41—2124no—
152#41—2124no—
153C2×Q8—532no—
154C2×Q8—1061no—
155C2×D4—532no—
156C2×D4—532no—
157C2×D4—1062no—
158C2×D4—1062no—
159C2×D4—1061no—
160C2×D4—1061no—
161C2×D4—1061no—
162C2×D4—1061no—
163C22×C4—1062no—
164C22×C4—1062no—
165C22×C4D21211yescharacteristic
166C22×C4D21211yescharacteristic
167C22×C4C22×D5311yescharacteristic
168SD16—2124no—
169D8—2124no—
170C2×C8C22×D5314yes—
171C4⋊C4—532no—
172C4⋊C4—1061no—
173C4⋊C4—1061no—
174C4⋊C4—1061no—
175C22⋊C4—1062no—
176C22⋊C4—1062no—
177C22⋊C4—1062no—
178C22⋊C4—1062no—
179C42D21211yescharacteristic
180C42D21211yescharacteristic
181C42C22×D5311yescharacteristic
182C42C22×D5311yescharacteristic
183C23—1062no—
184C23—1062no—
185C23C2×D21211yescharacteristic
186Q8—1064no—
187Q8—2122no—
188D4—1062no—
189D4—1064no—
190D4—1062no—
191D4—2122no—
192D4—2122no—
193D4—2122no—
194D4—2124no—
195D4—2124no—
196D4—2122no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—41no—
200C2×C4—1062no—
201C2×C4—1062no—
202C2×C4—2124no—
203C2×C4—2124no—
204C2×C4C2×D21211yescharacteristic
205C2×C4C2×D21211yescharacteristic
206C2×C4C2×D21211yescharacteristic
207C2×C4C2×D21211yescharacteristic
208C2×C4C2×D21211yescharacteristic
209C2×C4C23×D5311yescharacteristic
210C8—24no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—2122no—
215C22—2122no—
216C22—2122no—
217C22—2122no—
218C22—2122no—
219C22—2122no—
220C22C22×D21211yescharacteristic
221C4—21no—
222C4—21no—
223C4—21no—
224C4—21no—
225C4—2122no—
226C4—2122no—
227C4#1313324512yes—
228C4#1313326312yes—
229C2—41no—
230C2—4242no—
231C2—4242no—
232C2#1681119211yescharacteristic
233C2C2×#1313324511yescharacteristic
234C2#1681142311yescharacteristic
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#432C2
socleC22×C53C22×D4
hypercenter#432D53
hypocenterC53#3214
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. #16811173C2semidirect
2. #16811173C2semidirect
3. #16811012C2semidirect
4. #16810983C2semidirect
5. #16810829C2semidirect
6. #16810819C2semidirect
7. #16810770C2semidirect
8. C53×#432C2semidirect
125. #432D53semidirect
126. C53#3214semidirect