Finite Group Databasebeta

← #20284233

Subgroups — #20284233

Order 6784.

490 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 141 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×#13133248C211yesmaximal, characteristic, complemented
2#16811190C211yesmaximal, characteristic
3#16811101C211yesmaximal, characteristic, complemented
4#16810843C214yesmaximal, complemented
5#16810821C214yesmaximal, complemented
6#16810649C211yesmaximal, characteristic, complemented
7#16810628C211yesmaximal, characteristic, complemented
8#16810620C211yesmaximal, characteristic, complemented
9C22×SD16×C53C211yesmaximal, characteristic, complemented
10C2×#11719945C2211yescharacteristic
11#13133248—24no—
12C2×#11719925C2212yescomplemented
13C2×#11719924C2212yescomplemented
14C2×#11719917C2211yescharacteristic
15#13133213C2212yes—
16#13133204C2212yes—
17#13133188—22no—
18#13133186—22no—
19Q8×Dic53—22no—
20Q8×Dic53C2214yescomplemented
21#13133176—22no—
22#13133176C2211yescharacteristic
23#13133176C2211yescharacteristic
24#13133146C2214yescomplemented
25#13133142C2212yescomplemented
26#13133139C2212yescomplemented
27#13133138C2214yes—
28C22×Q8×C53C2211yescharacteristic
29C22×D4×C53C2211yescharacteristic
30C2×SD16×C53—24no—
31C2×SD16×C53C2214yes—
32C22×C8×C53C2211yescharacteristic
33C24×C53—21no—
34C2×Q8×C53—21no—
35C2×Q8×C53—22no—
36C2×Q8×C53D414yescomplemented
37C2×Q8×C53C2312yes—
38C2×D4×C53—22no—
39C2×D4×C53—21no—
40C2×D4×C53—22no—
41C2×D4×C53C2312yes—
42C22×C4×C53—21no—
43C22×C4×C53C2311yescharacteristic
44C2×C22⋊D53—21no—
45C2×C22⋊D53—24no—
46#11719945—22no—
47#11719945—22no—
48C2×D212C2311yescharacteristic
49C2×C4×D53—21no—
50C2×Dic106C2311yescharacteristic
51SD16×C53—28no—
52C2×C8×C53—22no—
53C2×C8×C53C2312yes—
54C22⋊Dic53—21no—
55C22⋊Dic53—24no—
56C22⋊Dic53D412yes—
57#11719925—24no—
58#11719924—24no—
59C4⋊Dic53—24no—
60C4⋊Dic53D412yes—
61C4⋊Dic53C2312yes—
62C4×Dic53—21no—
63C4×Dic53—24no—
64C4×Dic53C2312yes—
65#11719917—22no—
66C2×#88322C2312yes—
67C23×C53—22no—
68C23×C53—42no—
69C23×C53—82no—
70C23×C53C2×D411yescharacteristic
71C22×D53—21no—
72Q8×C53—24no—
73Q8×C53—22no—
74Q8×C53C2×D414yes—
75D4×C53—22no—
76D4×C53—22no—
77D4×C53—42no—
78C2×C4×C53—24no—
79C2×C4×C53—22no—
80C2×C4×C53C2×D411yescharacteristic
81C2×C4×C53C2×D412yes—
82C2×C4×C53#4112yes—
83C2×C4×C53C2411yescharacteristic
84C22⋊D53—42no—
85C22⋊D53—84no—
86C2×Dic53—21no—
87C2×Dic53—22no—
88C2×Dic53C2×D414yes—
89D212—21no—
90D212—21no—
91C4×D53—42no—
92Dic106—21no—
93Dic106—21no—
94C8×C53—22no—
95#88322—22no—
96C22×C53—22no—
97C22×C53—42no—
98C22×C53—44no—
99C22×C53—84no—
100C22×C53#12112yes—
101C22×C53C22×D411yescharacteristic
102C2×D53—41no—
103C2×D53—42no—
104C4×C53—24no—
105C4×C53#12112yes—
106C4×C53C22×D411yescharacteristic
107C4×C53C2×#4111yescharacteristic
108Dic53—24no—
109Dic53—41no—
110Dic53—42no—
111#3043—531nomaximal, Sylow
112C2×C53—22no—
113C2×C53—82no—
114C2×C53C2×#12111yescharacteristic
115C2×C53#57712yes—
116D53—81no—
117C2×#137—531no—
118C22×SD16D5311yescharacteristic, complemented
119#535—531no—
120#519—531no—
121#464—534no—
122#460—534no—
123#424—531no—
124#418—531no—
125#416—531no—
126C53#304311yescharacteristic, complemented, Sylow
127C2×#41—531no—
128C22×Q8C2×D5311yescharacteristic
129C22×D4C2×D5311yescharacteristic
130#137—1064no—
131C2×Q16—532no—
132C2×SD16—24no—
133C2×SD16—532no—
134C2×SD16C2×D5314yes—
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—531no—
145#117—1062no—
146C4⋊C8—534no—
147#103—532no—
148#102—534no—
149#102—532no—
150C24—21no—
151#41—1062no—
152#41—1062no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8—531no—
156C2×Q8C22⋊D5314yescomplemented
157C2×Q8C22×D5312yes—
158C2×D4—21no—
159C2×D4—22no—
160C2×D4—22no—
161C2×D4—531no—
162C2×D4—1064no—
163C2×D4—1061no—
164C2×D4C22×D5312yes—
165C22×C4—21no—
166C22×C4—1061no—
167C22×C4C22×D5311yescharacteristic
168Q16—1064no—
169SD16—28no—
170SD16—1064no—
171M16—1062no—
172C2×C8—22no—
173C2×C8—532no—
174C2×C8C22×D5312yes—
175C4⋊C4—532no—
176C4⋊C4—532no—
177C4⋊C4—1064no—
178C22⋊C4—532no—
179C22⋊C4—1064no—
180C22⋊C4—1061no—
181C42—532no—
182C42—1064no—
183C42—1061no—
184C23—22no—
185C23—42no—
186C23—82no—
187C23—1061no—
188C23D4×D5311yescharacteristic
189Q8—24no—
190Q8—22no—
191Q8—1061no—
192Q8—1061no—
193Q8C2×C22⋊D5314yes—
194D4—22no—
195D4—22no—
196D4—42no—
197D4—1061no—
198D4—1061no—
199D4—4244no—
200D4—2122no—
201C2×C4—24no—
202C2×C4—22no—
203C2×C4—534no—
204C2×C4—1062no—
205C2×C4—1061no—
206C2×C4—2122no—
207C2×C4D4×D5311yescharacteristic
208C2×C4#1171994712yes—
209C2×C4C2×C22⋊D5312yes—
210C2×C4C23×D5311yescharacteristic
211C8—22no—
212C8—1062no—
213C22—22no—
214C22—42no—
215C22—44no—
216C22—84no—
217C22—2121no—
218C22—2122no—
219C22#1313318912yes—
220C22C2×D4×D5311yescharacteristic
221C4—24no—
222C4—1064no—
223C4—2121no—
224C4—2122no—
225C4#1313318912yes—
226C4C22×C22⋊D5311yescharacteristic
227C4C2×#1171994711yescharacteristic
228C2—22no—
229C2—82no—
230C2—4241no—
231C2C2×#1313318911yescharacteristic
232C2#1681142912yes—
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC22×SD16×C53C2
socleC22×C53C22×D4
hypercenterC22×SD16D53
hypocenterC53#3043
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#13133248C2semidirect
3. #16811101C2semidirect
4. #16810843C2semidirect
5. #16810821C2semidirect
6. #16810649C2semidirect
7. #16810628C2semidirect
8. #16810620C2semidirect
9. C22×SD16×C53C2semidirect
12. C2×#11719925C22semidirect
13. C2×#11719924C22semidirect
20. Q8×Dic53C22semidirect
24. #13133146C22semidirect
25. #13133142C22semidirect
26. #13133139C22semidirect
36. C2×Q8×C53D4semidirect
118. C22×SD16D53semidirect
126. C53#3043semidirect
156. C2×Q8C22⋊D53semidirect