Finite Group Databasebeta

← #20285457

Subgroups — #20285457

Order 6784.

358 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 119 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#16811177C212yesmaximal
2#16811020C211yesmaximal, characteristic, complemented
3#16811001C211yesmaximal, characteristic, complemented
4#16810972C212yesmaximal, complemented
5#16810882C211yesmaximal, characteristic
6#16810880C212yesmaximal
7#16810878C211yesmaximal, characteristic
8#16810731C211yesmaximal, characteristic, complemented
9#16810728C212yesmaximal, complemented
10#16810575C211yesmaximal, characteristic
11C53×#441C211yesmaximal, characteristic, complemented
12C2×Dic212C2211yescharacteristic
13C2×#11719925C2212yes—
14C2×#11719912C2211yescharacteristic
15C2×#11719912C2211yescharacteristic
16#13133216C2212yes—
17#13133215C2212yes—
18#13133207—22no—
19#13133197C2212yes—
20#13133195—22no—
21#13133193C2212yes—
22C4×Dic106—22no—
23C4×Dic106C2212yes—
24#13133174—22no—
25C42×D53C2211yescharacteristic
26#13133149C2211yescharacteristic
27#13133148C2212yes—
28#13133142C2212yes—
29#13133141C2212yes—
30#13133140C2212yes—
31#13133128C2211yescharacteristic
32C8×Dic53C2211yescharacteristic
33C8×Dic53C2211yescharacteristic
34C2×Q16×C53C2211yescharacteristic
35C4×Q8×C53C2212yes—
36#106×C53C2211yescharacteristic
37#103×C53C2212yes—
38#97×C53C2211yescharacteristic
39C2×Q8×C53C2312yes—
40C2×C4×D53D412yes—
41C2×C4×D53C2311yescharacteristic
42C2×Dic106—22no—
43C2×Dic106C2312yes—
44Q16×C53—22no—
45C2×C8×C53C2311yescharacteristic
46C2×C8×C53C2311yescharacteristic
47C4⋊C4×C53—22no—
48C4⋊C4×C53C2312yes—
49C42×C53—22no—
50C42×C53C2311yescharacteristic
51#11719925—24no—
52#11719921—22no—
53#11719921—22no—
54C4⋊Dic53—22no—
55C4⋊Dic53C2312yes—
56#11719919—22no—
57#11719919—22no—
58#11719919—22no—
59#11719919—22no—
60#11719919—22no—
61C4×Dic53—22no—
62C4×Dic53D412yes—
63C4×Dic53C2311yescharacteristic
64C2×#88322C2311yescharacteristic
65C2×#88322C2311yescharacteristic
66Dic212—22no—
67#11719912—22no—
68#11719912—22no—
69C22×D53C2×D411yescharacteristic
70Q8×C53—22no—
71Q8×C53—22no—
72C2×C4×C53—22no—
73C2×C4×C53—22no—
74C2×C4×C53C2×D412yes—
75C2×C4×C53C2411yescharacteristic
76C2×Dic53—22no—
77C2×Dic53—22no—
78C2×Dic53C2×D411yescharacteristic
79C2×Dic53C2×D412yes—
80C4×D53—21no—
81C4×D53—21no—
82C4×D53—24no—
83Dic106—22no—
84Dic106—22no—
85Dic106—44no—
86C8×C53—21no—
87C8×C53#4112yes—
88#88322—21no—
89#88322#4112yes—
90C22×C53C22×D411yescharacteristic
91C2×D53—21no—
92C2×D53—22no—
93C4×C53—22no—
94C4×C53—42no—
95C4×C53—42no—
96C4×C53C2×#4111yescharacteristic
97C4×C53C2×#4111yescharacteristic
98Dic53—21no—
99Dic53—42no—
100Dic53—42no—
101Dic53#13714yes—
102#3185—531nomaximal, Sylow
103C2×C53#53211yescharacteristic
104C2×C53C2×#13712yes—
105D53—41no—
106#533—532no—
107#498—531no—
108#494—531no—
109#487—532no—
110#470—531no—
111#470—531no—
112#470—532no—
113#441—532no—
114#441—531no—
115#441D5311yescharacteristic, complemented
116C4×M16—531no—
117C53#318511yescharacteristic, complemented, Sylow
118C2×Q16—532no—
119C2×Q16—531no—
120C2×Q16C2×D5311yescharacteristic
121C2×M16—531no—
122C2×M16—531no—
123C4⋊Q8—532no—
124C4⋊Q8—532no—
125#125—1062no—
126C22⋊Q8—532no—
127C22⋊Q8—532no—
128C22⋊Q8—1062no—
129C4×Q8—532no—
130C4×Q8—1062no—
131C4×Q8C2×D5312yes—
132#117—1062no—
133C2×C42—531no—
134#106—531no—
135#106—532no—
136#106C2×D5311yescharacteristic
137#103—532no—
138#103—532no—
139#103—532no—
140#103C2×D5312yes—
141#97—531no—
142#97C2×D5311yescharacteristic
143C4×C8—531no—
144C4×C8—531no—
145C2×Q8—532no—
146C2×Q8—1062no—
147C2×Q8C22×D5312yes—
148C22×C4—532no—
149C22×C4—531no—
150Q16—22no—
151Q16—1064no—
152Q16—1062no—
153M16—1062no—
154M16—1062no—
155C2×C8—531no—
156C2×C8—531no—
157C2×C8C22×D5311yescharacteristic
158C2×C8C22×D5311yescharacteristic
159C4⋊C4—22no—
160C4⋊C4—532no—
161C4⋊C4—1062no—
162C4⋊C4—1062no—
163C4⋊C4—1062no—
164C4⋊C4—1062no—
165C4⋊C4—1062no—
166C4⋊C4—1062no—
167C4⋊C4C22×D5312yes—
168C22⋊C4—1062no—
169C22⋊C4—1062no—
170C42—22no—
171C42—532no—
172C42—531no—
173C42—1062no—
174C42C22×D5311yescharacteristic
175C23—531no—
176Q8—22no—
177Q8—22no—
178Q8—1062no—
179Q8—1062no—
180Q8—2124no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—532no—
184C2×C4—531no—
185C2×C4—1064no—
186C2×C4—1061no—
187C2×C4—1061no—
188C2×C4—1062no—
189C2×C4—1062no—
190C2×C4D4×D5312yes—
191C2×C4C23×D5311yescharacteristic
192C8—21no—
193C8—532no—
194C8—1061no—
195C8#1171994512yes—
196C22—1062no—
197C22—1061no—
198C22C2×D4×D5311yescharacteristic
199C4—22no—
200C4—42no—
201C4—42no—
202C4—534no—
203C4—1061no—
204C4—2122no—
205C4—2122no—
206C4#41×D5311yescharacteristic
207C4C2×#1171994511yescharacteristic
208C2—2121no—
209C2#1681117011yescharacteristic
210C2#137×D5312yes—
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#441C2
socleC22×C53C22×D4
hypercenter#441D53
hypocenterC53#3185
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #441

Complements

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

NH ≅ G/Ntype
2. #16811020C2semidirect
3. #16811001C2semidirect
4. #16810972C2semidirect
8. #16810731C2semidirect
9. #16810728C2semidirect
11. C53×#441C2semidirect
115. #441D53semidirect
117. C53#3185semidirect