Finite Group Databasebeta

← #20285442

Subgroups — #20285442

Order 6784.

370 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 115 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#16811155C212yesmaximal, complemented
2#16811023C211yesmaximal, characteristic, complemented
3#16811006C211yesmaximal, characteristic, complemented
4#16810976C212yesmaximal, complemented
5#16810882C211yesmaximal, characteristic, complemented
6#16810873C211yesmaximal, characteristic
7#16810868C212yesmaximal, complemented
8#16810731C211yesmaximal, characteristic, complemented
9#16810723C212yesmaximal, complemented
10#16810575C211yesmaximal, characteristic, complemented
11C53×#442C211yesmaximal, characteristic, complemented
12C2×Dic212C2211yescharacteristic
13C2×#11719923C2212yes—
14C2×#11719912C2211yescharacteristic
15C2×#11719912C2211yescharacteristic
16#13133210—22no—
17#13133208C2212yes—
18#13133204C2212yes—
19#13133199—22no—
20#13133197C2212yes—
21#13133188C2212yes—
22C4×Dic106C2212yes—
23C4×C22⋊D53—22no—
24#13133176—22no—
25C42×D53C2211yescharacteristic
26#13133150C2212yes—
27#13133149C2211yescharacteristic
28#13133142C2212yes—
29#13133140C2212yes—
30#13133138C2212yes—
31#13133128C2211yescharacteristic
32C8×Dic53C2211yescharacteristic
33C8×Dic53C2211yescharacteristic
34C2×D8×C53C2211yescharacteristic
35C4×D4×C53C2212yes—
36#106×C53C2211yescharacteristic
37#102×C53C2212yes—
38#97×C53C2211yescharacteristic
39C2×D4×C53C2312yes—
40C22×C4×C53—22no—
41C2×C22⋊D53—22no—
42C2×C4×D53D412yes—
43C2×C4×D53C2311yescharacteristic
44C2×Dic106C2312yes—
45D8×C53—22no—
46C2×C8×C53C2311yescharacteristic
47C2×C8×C53C2311yescharacteristic
48C4⋊C4×C53C2312yes—
49C22⋊C4×C53—22no—
50C42×C53C2311yescharacteristic
51C22⋊Dic53—22no—
52C22⋊Dic53—22no—
53C22⋊Dic53—22no—
54#11719923—24no—
55#11719921—22no—
56C4⋊Dic53—22no—
57C4⋊Dic53—22no—
58C4⋊Dic53C2312yes—
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—
69C23×C53—22no—
70C22×D53C2×D411yescharacteristic
71D4×C53—22no—
72D4×C53—22no—
73C2×C4×C53—22no—
74C2×C4×C53—44no—
75C2×C4×C53C2×D412yes—
76C2×C4×C53C2411yescharacteristic
77C22⋊D53—44no—
78C2×Dic53—22no—
79C2×Dic53—22no—
80C2×Dic53C2×D412yes—
81C2×Dic53C2×D411yescharacteristic
82C4×D53—21no—
83C4×D53—21no—
84C4×D53—42no—
85Dic106—22no—
86Dic106—22no—
87C8×C53—21no—
88C8×C53#4112yes—
89#88322—21no—
90#88322#4112yes—
91C22×C53—42no—
92C22×C53—42no—
93C22×C53—42no—
94C22×C53C22×D411yescharacteristic
95C2×D53—21no—
96C2×D53—22no—
97C4×C53—22no—
98C4×C53—42no—
99C4×C53C2×#4111yescharacteristic
100C4×C53C2×#4111yescharacteristic
101Dic53—21no—
102Dic53—22no—
103Dic53—42no—
104Dic53—42no—
105#3183—531nomaximal, Sylow
106C2×C53—82no—
107C2×C53#53211yescharacteristic
108C2×C53#57512yes—
109D53—41no—
110#529—532no—
111#499—531no—
112#495—531no—
113#488—532no—
114#470—531no—
115#469—531no—
116#468—532no—
117#442D5311yescharacteristic, complemented
118#441—531no—
119#440—532no—
120C4×M16—531no—
121C53#318311yescharacteristic, complemented, Sylow
122C2×Q16—531no—
123C2×SD16—532no—
124C2×D8C2×D5311yescharacteristic
125C2×M16—531no—
126C2×M16—531no—
127#126—1062no—
128#125—532no—
129#124—532no—
130#123—1062no—
131C22⋊Q8—532no—
132#121—532no—
133C4×Q8—532no—
134C4×D4—1062no—
135C4×D4C2×D5312yes—
136#117—1062no—
137C2×C42—531no—
138#107—532no—
139#106—531no—
140#106C2×D5311yescharacteristic
141#103—532no—
142#103—532no—
143#102—532no—
144#102C2×D5312yes—
145#97—531no—
146#97C2×D5311yescharacteristic
147C4×C8—531no—
148C4×C8—531no—
149C2×Q8—532no—
150C2×D4—1062no—
151C2×D4C22×D5312yes—
152C22×C4—22no—
153C22×C4—531no—
154C22×C4—532no—
155Q16—1062no—
156SD16—1064no—
157D8—22no—
158M16—1062no—
159M16—1062no—
160C2×C8—531no—
161C2×C8—531no—
162C2×C8C22×D5311yescharacteristic
163C2×C8C22×D5311yescharacteristic
164C4⋊C4—532no—
165C4⋊C4—1062no—
166C4⋊C4—1062no—
167C4⋊C4—1062no—
168C4⋊C4—1062no—
169C4⋊C4C22×D5312yes—
170C22⋊C4—22no—
171C22⋊C4—1062no—
172C22⋊C4—1062no—
173C22⋊C4—1062no—
174C22⋊C4—1062no—
175C42—532no—
176C42—531no—
177C42—1062no—
178C42C22×D5311yescharacteristic
179C23—22no—
180C23—531no—
181Q8—1062no—
182Q8—1062no—
183D4—22no—
184D4—22no—
185D4—2124no—
186C2×C4—22no—
187C2×C4—44no—
188C2×C4—532no—
189C2×C4—531no—
190C2×C4—1061no—
191C2×C4—1061no—
192C2×C4—1062no—
193C2×C4—1062no—
194C2×C4—2122no—
195C2×C4D4×D5312yes—
196C2×C4C23×D5311yescharacteristic
197C8—21no—
198C8—532no—
199C8—1061no—
200C8#1171994512yes—
201C22—42no—
202C22—42no—
203C22—42no—
204C22—1062no—
205C22—1061no—
206C22C2×D4×D5311yescharacteristic
207C4—22no—
208C4—42no—
209C4—1062no—
210C4—1061no—
211C4—2122no—
212C4—2122no—
213C4#41×D5311yescharacteristic
214C4C2×#1171994511yescharacteristic
215C2—82no—
216C2—2121no—
217C2#1681117011yescharacteristic
218C2#1681141912yes—
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#442C2
socleC22×C53C22×D4
hypercenter#442D53
hypocenterC53#3183
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. #16811155C2semidirect
2. #16811023C2semidirect
3. #16811006C2semidirect
4. #16810976C2semidirect
5. #16810882C2semidirect
7. #16810868C2semidirect
8. #16810731C2semidirect
9. #16810723C2semidirect
10. #16810575C2semidirect
11. C53×#442C2semidirect
117. #442D53semidirect
121. C53#3183semidirect