Finite Group Databasebeta

← #20284239

Subgroups — #20284239

Order 6784.

490 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C2×#13133247C211yesmaximal, characteristic, complemented
2C22×#11719913C211yesmaximal, characteristic, complemented
3#16811187C211yesmaximal, characteristic
4#16811100C211yesmaximal, characteristic, complemented
5#16810848C214yesmaximal, complemented
6#16810822C214yesmaximal, complemented
7#16810649C211yesmaximal, characteristic, complemented
8#16810627C211yesmaximal, characteristic, complemented
9C53×#416C211yesmaximal, characteristic, complemented
10C22×Dic106C2211yescharacteristic
11C22×D212C2211yescharacteristic
12#13133247—24no—
13C2×#11719925C2212yescomplemented
14C2×#11719913—24no—
15C2×#11719913C2214yes—
16C2×#11719923C2212yescomplemented
17C2×#11719917C2211yescharacteristic
18#13133213C2212yes—
19#13133206C2212yes—
20#13133191—22no—
21#13133185—22no—
22C4×Dic106—22no—
23#13133182C2214yescomplemented
24#13133176C2211yescharacteristic
25#13133173—22no—
26#13133146C2214yescomplemented
27#13133137C2214yes—
28C2×#41×C53C2211yescharacteristic
29C22×C8×C53C2211yescharacteristic
30#117×C53C2211yescharacteristic
31#103×C53C2212yescomplemented
32#102×C53C2212yescomplemented
33C23×D53—21no—
34#41×C53—22no—
35#41×C53—22no—
36C2×Q8×C53C2311yescharacteristic
37C2×D4×C53—21no—
38C2×D4×C53C2311yescharacteristic
39C22×C4×C53—21no—
40C22×C4×C53C2311yescharacteristic
41C2×C22⋊D53—24no—
42C22×Dic53—21no—
43C2×D212—22no—
44C2×D212—22no—
45C2×D212—21no—
46C2×D212C2312yes—
47C2×Dic106—21no—
48C2×Dic106—22no—
49C2×Dic106D414yescomplemented
50C2×Dic106C2312yes—
51C2×C8×C53—22no—
52C2×C8×C53C2312yes—
53C4⋊C4×C53C2312yes—
54C22⋊C4×C53—21no—
55C42×C53—21no—
56C22⋊Dic53D412yes—
57#11719925—24no—
58#11719923—24no—
59#11719921—24no—
60C4⋊Dic53D412yes—
61#11719919—24no—
62C4×Dic53—24no—
63C4×Dic53C2312yes—
64#11719917—22no—
65C2×#88322C2312yes—
66#11719913—28no—
67C23×C53—21no—
68C23×C53C2×D411yescharacteristic
69C22×D53—22no—
70C22×D53—42no—
71C22×D53—82no—
72Q8×C53—21no—
73Q8×C53—21no—
74D4×C53—21no—
75D4×C53—21no—
76D4×C53—42no—
77C2×C4×C53—21no—
78C2×C4×C53—22no—
79C2×C4×C53—42no—
80C2×C4×C53C2×D411yescharacteristic
81C2×C4×C53C2×D412yes—
82C2×C4×C53#4112yes—
83C2×C4×C53C2411yescharacteristic
84C22⋊D53—84no—
85C2×Dic53—24no—
86C2×Dic53—22no—
87C2×Dic53C2×D414yes—
88D212—22no—
89D212—22no—
90D212—42no—
91Dic106—24no—
92Dic106—22no—
93Dic106C2×D414yes—
94C8×C53—22no—
95#88322—22no—
96C22×C53—22no—
97C22×C53—41no—
98C22×C53—42no—
99C22×C53#12112yes—
100C22×C53C22×D411yescharacteristic
101C2×D53—42no—
102C2×D53—44no—
103C2×D53—84no—
104C4×C53—41no—
105C4×C53—42no—
106C4×C53#12112yes—
107C4×C53C22×D411yescharacteristic
108C4×C53C2×#4111yescharacteristic
109Dic53—24no—
110Dic53—24no—
111#3043—531nomaximal, Sylow
112C2×C53—22no—
113C2×C53—81no—
114C2×C53C2×#12111yescharacteristic
115C2×C53#57712yes—
116D53—82no—
117C2×#137—531no—
118C22×SD16—531no—
119#535—531no—
120#519—531no—
121#464—534no—
122#460—534no—
123#424—531no—
124#418—531no—
125#416D5311yescharacteristic, complemented
126C53#304311yescharacteristic, complemented, Sylow
127C2×#41C2×D5311yescharacteristic
128C22×Q8—531no—
129C22×D4—531no—
130#137—1064no—
131C2×Q16—532no—
132C2×SD16—532no—
133C2×SD16—534no—
134C2×SD16—1064no—
135C2×M16—531no—
136C22×C8C2×D5311yescharacteristic
137#127—532no—
138#124—532no—
139#121—1062no—
140#120—1062no—
141C4×Q8—534no—
142C4×Q8—1062no—
143#117—531no—
144#117—1062no—
145#117C2×D5311yescharacteristic
146C4⋊C8—534no—
147#103C2×D5312yescomplemented
148#102—534no—
149#102C2×D5312yescomplemented
150C24—1061no—
151#41—22no—
152#41—22no—
153C2×Q8—534no—
154C2×Q8—532no—
155C2×Q8—1062no—
156C2×Q8—1061no—
157C2×Q8C22×D5311yescharacteristic
158C2×D4—21no—
159C2×D4—532no—
160C2×D4—1064no—
161C2×D4—1062no—
162C2×D4—1062no—
163C2×D4—1061no—
164C2×D4C22×D5311yescharacteristic
165C22×C4—21no—
166C22×C4—1061no—
167C22×C4C22×D5311yescharacteristic
168Q16—1064no—
169SD16—1068no—
170SD16—1064no—
171M16—1062no—
172C2×C8—22no—
173C2×C8—532no—
174C2×C8C22×D5312yes—
175C4⋊C4—532no—
176C4⋊C4—1064no—
177C4⋊C4C22×D5312yes—
178C22⋊C4—21no—
179C22⋊C4—532no—
180C22⋊C4—1064no—
181C42—21no—
182C42—532no—
183C42—1064no—
184C23—21no—
185C23—1062no—
186C23—4242no—
187C23—2122no—
188C23D4×D5311yescharacteristic
189Q8—21no—
190Q8—21no—
191Q8—534no—
192Q8—1064no—
193Q8—1062no—
194D4—21no—
195D4—21no—
196D4—42no—
197D4—1062no—
198D4—1062no—
199D4—4244no—
200D4—2122no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—42no—
204C2×C4—534no—
205C2×C4—1064no—
206C2×C4—1062no—
207C2×C4#1171994512yes—
208C2×C4D4×D5311yescharacteristic
209C2×C4D4×D5312yes—
210C2×C4C23×D5311yescharacteristic
211C8—22no—
212C8—1062no—
213C22—22no—
214C22—41no—
215C22—42no—
216C22—4244no—
217C22—2122no—
218C22—2124no—
219C22#1313319012yes—
220C22C2×D4×D5311yescharacteristic
221C4—41no—
222C4—42no—
223C4—1064no—
224C4—1064no—
225C4#1313319012yes—
226C4C2×D4×D5311yescharacteristic
227C4C2×#1171994511yescharacteristic
228C2—22no—
229C2—81no—
230C2—4242no—
231C2C2×#1313319011yescharacteristic
232C2#1681143012yes—
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#416C2
socleC22×C53C22×D4
hypercenter#416D53
hypocenterC53#3043
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. C2×#13133247C2semidirect
2. C22×#11719913C2semidirect
4. #16811100C2semidirect
5. #16810848C2semidirect
6. #16810822C2semidirect
7. #16810649C2semidirect
8. #16810627C2semidirect
9. C53×#416C2semidirect
13. C2×#11719925C22semidirect
16. C2×#11719923C22semidirect
23. #13133182C22semidirect
26. #13133146C22semidirect
31. #103×C53C22semidirect
32. #102×C53C22semidirect
49. C2×Dic106D4semidirect
125. #416D53semidirect
126. C53#3043semidirect
147. #103C2×D53semidirect
149. #102C2×D53semidirect