Finite Group Databasebeta

← #20285851

Subgroups — #20285851

Order 6784.

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

normal subgroups only

This group has 370 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
1#16811173C211yesmaximal, characteristic
2#16811026C211yesmaximal, characteristic, complemented
3#16811019C211yesmaximal, characteristic, complemented
4#16810971C211yesmaximal, characteristic, complemented
5#16810971C211yesmaximal, characteristic, complemented
6#16810947C214yesmaximal, complemented
7#16810920C214yesmaximal, complemented
8#16810676C211yesmaximal, characteristic, complemented
9C53×#533C211yesmaximal, characteristic, complemented
10C2×#11719945—21no—
11#13133217C2211yescharacteristic
12#13133214C2211yescharacteristic
13#13133205—21no—
14#13133191—21no—
15#13133191C2212yes—
16C4×D212—21no—
17C4×D212C2212yes—
18#13133150C2214yes—
19#13133148C2214yes—
20C4⋊#88322C2212yes—
21#13133137C2214yes—
22#13133137C2214yes—
23C22⋊#88322C2212yes—
24C4×#88322C2211yescharacteristic
25C4×#88322C2211yescharacteristic
26C4⋊Q8×C53C2211yescharacteristic
27C4⋊Q8×C53C2211yescharacteristic
28#125×C53C2212yes—
29C22⋊Q8×C53—21no—
30C22⋊Q8×C53—21no—
31C4×Q8×C53—21no—
32C4×Q8×C53—21no—
33#117×C53C2212yes—
34C2×C42×C53C2211yescharacteristic
35C2×Q8×C53—21no—
36C2×Q8×C53—21no—
37C22×C4×C53D411yescharacteristic
38C22×C4×C53D411yescharacteristic
39C22×C4×C53C2311yescharacteristic
40C2×C22⋊D53—22no—
41#11719945—44no—
42C2×D212—21no—
43C2×D212—21no—
44C2×D212C2312yes—
45C2×C4×D53—22no—
46C2×Dic106—21no—
47C4⋊C4×C53—22no—
48C4⋊C4×C53—22no—
49C4⋊C4×C53—21no—
50C4⋊C4×C53—21no—
51C4⋊C4×C53C2312yes—
52C4⋊C4×C53C2312yes—
53C22⋊C4×C53—22no—
54C42×C53—22no—
55C42×C53D411yescharacteristic
56C42×C53D411yescharacteristic
57C42×C53C2311yescharacteristic
58C42×C53C2311yescharacteristic
59#11719921—22no—
60#11719921—22no—
61C4⋊Dic53—21no—
62C4⋊Dic53C2312yes—
63C2×#88322C2312yes—
64C2×#88322C2312yes—
65C23×C53C2×D411yescharacteristic
66C22×D53—22no—
67Q8×C53—42no—
68Q8×C53—42no—
69C2×C4×C53—22no—
70C2×C4×C53—22no—
71C2×C4×C53—22no—
72C2×C4×C53—22no—
73C2×C4×C53—22no—
74C2×C4×C53C2×D411yescharacteristic
75C2×C4×C53C2×D411yescharacteristic
76C2×C4×C53C2×D411yescharacteristic
77C2×C4×C53C2×D411yescharacteristic
78C2×C4×C53C2×D411yescharacteristic
79C2×C4×C53C2411yescharacteristic
80C22⋊D53—44no—
81C2×Dic53—22no—
82D212—24no—
83D212—42no—
84D212—42no—
85C4×D53—44no—
86Dic106—42no—
87#88322—22no—
88#88322—22no—
89C22×C53—21no—
90C22×C53—21no—
91C22×C53—21no—
92C22×C53C22×D411yescharacteristic
93C2×D53—42no—
94C2×D53—42no—
95C2×D53—42no—
96C4×C53—21no—
97C4×C53—21no—
98C4×C53—21no—
99C4×C53—42no—
100C4×C53—42no—
101C4×C53#13512yes—
102C4×C53AGL(1,ℤ/8)12yes—
103C4×C532−1+412yes—
104Dic53—42no—
105#3228—531nomaximal, Sylow
106C2×C53—41no—
107C2×C53#53611yescharacteristic
108C2×C53C2×#13511yescharacteristic
109C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
110D53—82no—
111#533D5311yescharacteristic, complemented
112#532—531no—
113#500—531no—
114#498—531no—
115#486—531no—
116#486—531no—
117#482—534no—
118#478—534no—
119#431—531no—
120C53#322811yescharacteristic, complemented, Sylow
121C2×#41—1061no—
122C4⋊Q8—531no—
123C4⋊Q8C2×D5311yescharacteristic
124C4⋊Q8C2×D5311yescharacteristic
125#127—531no—
126#125C2×D5312yes—
127#124—1061no—
128C22⋊Q8—21no—
129C22⋊Q8—21no—
130#121—532no—
131#121—1061no—
132C4×Q8—21no—
133C4×Q8—21no—
134C4×D4—532no—
135C4×D4—1061no—
136#117C2×D5312yes—
137C2×C42C2×D5311yescharacteristic
138#107—534no—
139#106—534no—
140C4⋊C8—532no—
141#102—534no—
142#102—534no—
143C22⋊C8—532no—
144C4×C8—531no—
145C4×C8—531no—
146#41—2124no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—1061no—
150C2×D4—532no—
151C2×D4—1062no—
152C2×D4—1061no—
153C2×D4—1061no—
154C22×C4—1062no—
155C22×C4C22⋊D5311yescharacteristic
156C22×C4C22⋊D5311yescharacteristic
157C22×C4C22×D5311yescharacteristic
158C2×C8—532no—
159C2×C8—532no—
160C4⋊C4—22no—
161C4⋊C4—22no—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—532no—
165C4⋊C4—1061no—
166C4⋊C4C22×D5312yes—
167C4⋊C4C22×D5312yes—
168C22⋊C4—22no—
169C22⋊C4—1062no—
170C22⋊C4—1062no—
171C42—22no—
172C42C22⋊D5311yescharacteristic
173C42C22⋊D5311yescharacteristic
174C42C22×D5311yescharacteristic
175C42C22×D5311yescharacteristic
176C23—1062no—
177C23C2×C22⋊D5311yescharacteristic
178Q8—42no—
179Q8—42no—
180Q8—2122no—
181D4—1064no—
182D4—2124no—
183D4—2122no—
184D4—2122no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—1062no—
191C2×C4—2124no—
192C2×C4C2×C22⋊D5311yescharacteristic
193C2×C4C2×C22⋊D5311yescharacteristic
194C2×C4C2×C22⋊D5311yescharacteristic
195C2×C4C2×C22⋊D5311yescharacteristic
196C2×C4C2×C22⋊D5311yescharacteristic
197C2×C4C23×D5311yescharacteristic
198C8—1062no—
199C8—1062no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—2122no—
204C22—2122no—
205C22—2122no—
206C22C22×C22⋊D5311yescharacteristic
207C4—21no—
208C4—21no—
209C4—21no—
210C4—42no—
211C4—42no—
212C4—2122no—
213C4#1313323712yes—
214C4#1313324212yes—
215C4#1313326512yes—
216C2—41no—
217C2—4242no—
218C2#1681119711yescharacteristic
219C2C2×#1313323711yescharacteristic
220C2C2×#1313324211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#533C2
socleC22×C53C22×D4
hypercenter#533D53
hypocenterC53#3228
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
2. #16811026C2semidirect
3. #16811019C2semidirect
4. #16810971C2semidirect
5. #16810971C2semidirect
6. #16810947C2semidirect
7. #16810920C2semidirect
8. #16810676C2semidirect
9. C53×#533C2semidirect
111. #533D53semidirect
120. C53#3228semidirect