Finite Group Databasebeta

← #20285034

Subgroups — #20285034

Order 6784.

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

normal subgroups only

This group has 400 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#16811178C211yesmaximal, characteristic, complemented
2#16811171C211yesmaximal, characteristic
3#16810976C214yesmaximal, complemented
4#16810975C211yesmaximal, characteristic, complemented
5#16810974C211yesmaximal, characteristic, complemented
6#16810969C211yesmaximal, characteristic, complemented
7#16810616C214yesmaximal, complemented
8C4×C8×D53C211yesmaximal, characteristic, complemented
9C53×#486C211yesmaximal, characteristic, complemented
10C2×#11719947C2212yes—
11C2×C8×D53C2212yes—
12#13133217C2211yescharacteristic
13#13133217C2211yescharacteristic
14#13133208C2212yes—
15#13133204C2212yes—
16#13133197—21no—
17#13133193—21no—
18#13133188—21no—
19#13133188—21no—
20Q8×Dic53—21no—
21#13133182—21no—
22D4×Dic53—21no—
23D4×Dic53—21no—
24#13133175C2212yes—
25C42×D53C2211yescharacteristic
26#13133142C2214yes—
27#13133140C2214yes—
28#13133138C2214yes—
29C4×#88322C2211yescharacteristic
30C8×Dic53C2212yes—
31C4⋊Q8×C53C2211yescharacteristic
32#127×C53C2211yescharacteristic
33#102×C53C2214yes—
34C4×C8×C53C2211yescharacteristic
35C2×Q8×C53—21no—
36C2×D4×C53—21no—
37C2×D4×C53—21no—
38C2×D4×C53C2312yes—
39C2×C22⋊D53—22no—
40C22×Dic53—22no—
41#11719947—28no—
42C2×C4×D53D411yescharacteristic
43C2×C4×D53D411yescharacteristic
44C2×C4×D53C2311yescharacteristic
45C2×Dic106—21no—
46C2×Dic106C2312yes—
47C2×C8×C53C2312yes—
48C4⋊C4×C53—21no—
49C4⋊C4×C53C2312yes—
50C42×C53C2311yescharacteristic
51C22⋊Dic53—22no—
52C22⋊Dic53—22no—
53#11719921—22no—
54C4⋊Dic53—21no—
55C4⋊Dic53—21no—
56C4⋊Dic53—21no—
57C4⋊Dic53—22no—
58C4⋊Dic53C2312yes—
59#11719919—22no—
60C4×Dic53—22no—
61C4×Dic53D411yescharacteristic
62C4×Dic53D411yescharacteristic
63C4×Dic53C2311yescharacteristic
64C2×#88322C2312yes—
65C8×D53—24no—
66C23×C53—22no—
67C22×D53C2×D411yescharacteristic
68Q8×C53—42no—
69D4×C53—24no—
70D4×C53—42no—
71D4×C53—42no—
72C2×C4×C53—22no—
73C2×C4×C53C2×D411yescharacteristic
74C2×C4×C53C2×D411yescharacteristic
75C2×C4×C53C2411yescharacteristic
76C22⋊D53—44no—
77C2×Dic53—22no—
78C2×Dic53—22no—
79C2×Dic53—44no—
80C2×Dic53C2×D411yescharacteristic
81C2×Dic53C2×D411yescharacteristic
82C2×Dic53C2×D411yescharacteristic
83C4×D53—22no—
84C4×D53—22no—
85C4×D53#4114yes—
86Dic106—24no—
87Dic106—42no—
88C8×C53—22no—
89#88322—22no—
90C22×C53—42no—
91C22×C53—42no—
92C22×C53—42no—
93C22×C53C22×D411yescharacteristic
94C2×D53—21no—
95C2×D53—21no—
96C2×D53#12412yes—
97C4×C53—42no—
98C4×C53#13512yes—
99C4×C53#13512yes—
100C4×C53C2×#4112yes—
101Dic53—21no—
102Dic53—21no—
103Dic53—42no—
104Dic53—42no—
105Dic53#12412yes—
106#3131—531nomaximal, Sylow
107C2×C53—82no—
108C2×C53C2×#12411yescharacteristic
109C2×C53C2×#13511yescharacteristic
110C2×C53C2×#13511yescharacteristic
111D53—22no—
112#533—531no—
113#532—531no—
114#488—534no—
115#487—531no—
116#487—531no—
117#486—531no—
118#486D5311yescharacteristic, complemented
119#416—534no—
120C2×C4×C8—531no—
121C53#313111yescharacteristic, complemented, Sylow
122C2×#41—532no—
123C22×C8—532no—
124C4⋊Q8—531no—
125C4⋊Q8—531no—
126C4⋊Q8C2×D5311yescharacteristic
127#127C2×D5311yescharacteristic
128#125—532no—
129#124—532no—
130C22⋊Q8—1061no—
131C22⋊Q8—1061no—
132#121—1061no—
133#121—1061no—
134C4×Q8—1061no—
135C4×Q8—1061no—
136C4×D4—1061no—
137C4×D4—1061no—
138#117—532no—
139C2×C42—531no—
140#103—534no—
141#103—534no—
142#102—534no—
143#102C2×D5314yes—
144C4×C8—532no—
145C4×C8—531no—
146C4×C8C2×D5311yescharacteristic
147#41—1068no—
148C2×Q8—21no—
149C2×Q8—532no—
150C2×Q8—1061no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—1062no—
154C2×D4C22×D5312yes—
155C22×C4—531no—
156C22×C4—531no—
157C22×C4—531no—
158C22×C4—1062no—
159C2×C8—532no—
160C2×C8—1064no—
161C2×C8C22×D5312yes—
162C4⋊C4—21no—
163C4⋊C4—532no—
164C4⋊C4—1062no—
165C4⋊C4—1062no—
166C4⋊C4—1061no—
167C4⋊C4—1061no—
168C4⋊C4—1061no—
169C4⋊C4C22×D5312yes—
170C22⋊C4—1062no—
171C22⋊C4—1062no—
172C22⋊C4—1062no—
173C42—531no—
174C42—531no—
175C42—531no—
176C42—1062no—
177C42C22×D5311yescharacteristic
178C23—22no—
179C23—531no—
180Q8—42no—
181Q8—1064no—
182Q8—2122no—
183D4—24no—
184D4—42no—
185D4—42no—
186D4—2124no—
187C2×C4—22no—
188C2×C4—531no—
189C2×C4—534no—
190C2×C4—531no—
191C2×C4—531no—
192C2×C4—1062no—
193C2×C4—1062no—
194C2×C4—1062no—
195C2×C4—1062no—
196C2×C4—2124no—
197C2×C4D4×D5311yescharacteristic
198C2×C4D4×D5311yescharacteristic
199C2×C4C23×D5311yescharacteristic
200C8—22no—
201C8—1062no—
202C22—42no—
203C22—42no—
204C22—42no—
205C22—532no—
206C22—1061no—
207C22—1061no—
208C22C2×D4×D5311yescharacteristic
209C4—42no—
210C4—532no—
211C4—1061no—
212C4—1061no—
213C4—2122no—
214C4—2122no—
215C4#1313323612yes—
216C4#1313323812yes—
217C4#41×D5312yes—
218C2—82no—
219C2—1062no—
220C2#124×D5311yescharacteristic
221C2C2×#1313323611yescharacteristic
222C2C2×#1313323811yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#486C2
socleC22×C53C22×D4
hypercenter#486D53
hypocenterC53#3131
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #16811178C2semidirect
3. #16810976C2semidirect
4. #16810975C2semidirect
5. #16810974C2semidirect
6. #16810969C2semidirect
7. #16810616C2semidirect
8. C4×C8×D53C2semidirect
9. C53×#486C2semidirect
118. #486D53semidirect
121. C53#3131semidirect