Finite Group Databasebeta

← #20284227

Subgroups — #20284227

Order 6784.

494 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 149 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×#13133243C211yesmaximal, characteristic
2C22×#11719923C211yesmaximal, characteristic, complemented
3#16811199C211yesmaximal, characteristic, complemented
4#16810838C214yesmaximal, complemented
5#16810828C214yesmaximal, complemented
6#16810650C211yesmaximal, characteristic, complemented
7#16810631C211yesmaximal, characteristic, complemented
8#16810617C211yesmaximal, characteristic, complemented
9C53×#518C211yesmaximal, characteristic, complemented
10C2×#11719945C2211yescharacteristic
11C22×Dic106C2211yescharacteristic
12#13133243—24no—
13C2×#11719923—24no—
14C2×#11719923C2212yes—
15C2×#11719923C2214yes—
16C2×#11719922C2212yes—
17C2×#11719917C2211yescharacteristic
18C22×#88322C2211yescharacteristic
19#13133217C2212yescomplemented
20#13133205C2212yescomplemented
21#13133201—22no—
22#13133197—22no—
23C4⋊#88322C2214yescomplemented
24#13133140C2212yescomplemented
25#13133140C2214yescomplemented
26#13133137C2212yescomplemented
27C22×D4×C53C2211yescharacteristic
28C4×D4×C53—22no—
29C4×D4×C53C2214yes—
30#117×C53C2211yescharacteristic
31#117×C53C2211yescharacteristic
32C2×C22⋊C4×C53—22no—
33C24×C53—21no—
34C2×D4×C53—21no—
35C2×D4×C53—22no—
36C2×D4×C53D414yes—
37C2×D4×C53C2312yes—
38C22×C4×C53—24no—
39C22×C4×C53C2311yescharacteristic
40C2×C22⋊D53—21no—
41C22×Dic53—21no—
42#11719945—22no—
43#11719945—22no—
44C2×D212C2311yescharacteristic
45C2×C4×D53—21no—
46C2×Dic106—22no—
47C2×Dic106—22no—
48C2×Dic106—21no—
49C2×Dic106C2312yes—
50C2×Dic106C2311yescharacteristic
51C4⋊C4×C53D412yes—
52C4⋊C4×C53C2312yes—
53C22⋊C4×C53—21no—
54C22⋊C4×C53—24no—
55C22⋊C4×C53D412yes—
56C42×C53—21no—
57C42×C53C2312yes—
58#11719923—28no—
59#11719923—24no—
60#11719922—24no—
61#11719921—24no—
62C4⋊Dic53—24no—
63#11719917—22no—
64C2×#88322—22no—
65C2×#88322C2312yes—
66C2×#88322C2312yes—
67C23×C53—22no—
68C23×C53—24no—
69C23×C53—82no—
70C23×C53C2×D411yescharacteristic
71C22×D53—21no—
72D4×C53—24no—
73D4×C53—22no—
74D4×C53C2×D414yes—
75C2×C4×C53—22no—
76C2×C4×C53—84no—
77C2×C4×C53C2×D412yes—
78C2×C4×C53C2×D414yes—
79C2×C4×C53C2×D411yescharacteristic
80C2×C4×C53#4112yes—
81C2×C4×C53C2411yescharacteristic
82C22⋊D53—42no—
83C2×Dic53—22no—
84C2×Dic53—21no—
85C2×Dic53—42no—
86D212—21no—
87D212—21no—
88C4×D53—42no—
89Dic106—22no—
90Dic106—21no—
91Dic106—22no—
92Dic106—21no—
93Dic106—42no—
94#88322—22no—
95#88322—22no—
96C22×C53—22no—
97C22×C53—24no—
98C22×C53—44no—
99C22×C53—84no—
100C22×C53#12112yes—
101C22×C53C22×D411yescharacteristic
102C2×D53—41no—
103C2×D53—42no—
104C4×C53—24no—
105C4×C53—42no—
106C4×C53#12112yes—
107C4×C53C22×D411yescharacteristic
108C4×C53C2×#4111yescharacteristic
109Dic53—42no—
110Dic53—41no—
111#3042—531nomaximal, Sylow
112C2×C53—22no—
113C2×C53—44no—
114C2×C53C2×#12111yescharacteristic
115C2×C53#57712yes—
116D53—81no—
117C2×AGL(1,ℤ/8)—531no—
118C22×SD16—531no—
119#536—531no—
120#518D5311yescharacteristic, complemented
121#463—534no—
122#461—534no—
123#424—531no—
124#419—531no—
125#416—531no—
126C53#304211yescharacteristic, complemented, Sylow
127C2×#41—531no—
128C22×Q8—531no—
129C22×D4C2×D5311yescharacteristic
130AGL(1,ℤ/8)—1064no—
131C2×SD16—534no—
132C2×SD16—532no—
133C2×SD16—1064no—
134C2×D8—532no—
135C2×M16—531no—
136C22×C8—531no—
137C4⋊Q8—532no—
138#124—532no—
139#123—1062no—
140C22⋊Q8—1062no—
141C4×D4—22no—
142C4×D4C2×D5314yes—
143#117C2×D5311yescharacteristic
144#117C2×D5311yescharacteristic
145C2×C22⋊C4—22no—
146C4⋊C8—534no—
147#103—532no—
148#103—534no—
149#102—532no—
150C24—21no—
151#41—1062no—
152#41—1062no—
153C2×Q8—532no—
154C2×Q8—531no—
155C2×Q8—1061no—
156C2×Q8—1062no—
157C2×Q8—1062no—
158C2×D4—22no—
159C2×D4—21no—
160C2×D4—531no—
161C2×D4—1061no—
162C2×D4D21214yes—
163C2×D4C22×D5312yes—
164C22×C4—24no—
165C22×C4—1061no—
166C22×C4—1061no—
167C22×C4C22×D5311yescharacteristic
168SD16—1068no—
169SD16—1064no—
170D8—1064no—
171M16—1062no—
172C2×C8—532no—
173C2×C8—532no—
174C2×C8—1062no—
175C4⋊C4—1064no—
176C4⋊C4C22⋊D5312yes—
177C4⋊C4C22×D5312yes—
178C22⋊C4—21no—
179C22⋊C4—24no—
180C22⋊C4—1064no—
181C22⋊C4C22⋊D5312yes—
182C42—21no—
183C42C22×D5312yes—
184C23—22no—
185C23—24no—
186C23—82no—
187C23—1061no—
188C23C2×C22⋊D5311yescharacteristic
189Q8—1062no—
190Q8—1061no—
191Q8—1062no—
192Q8—1061no—
193Q8—2122no—
194D4—24no—
195D4—22no—
196D4—1061no—
197D4—1061no—
198D4—2122no—
199D4C2×D21214yes—
200C2×C4—22no—
201C2×C4—84no—
202C2×C4—1062no—
203C2×C4—1061no—
204C2×C4—2122no—
205C2×C4—2122no—
206C2×C4C2×D21212yes—
207C2×C4#1171994512yes—
208C2×C4C2×C22⋊D5314yes—
209C2×C4C2×C22⋊D5311yescharacteristic
210C2×C4C23×D5311yescharacteristic
211C8—1062no—
212C8—1062no—
213C22—22no—
214C22—24no—
215C22—44no—
216C22—84no—
217C22—2121no—
218C22—2122no—
219C22#1313319112yes—
220C22C22×C22⋊D5311yescharacteristic
221C4—24no—
222C4—42no—
223C4—2121no—
224C4—2122no—
225C4#1313319112yes—
226C4C22×D21211yescharacteristic
227C4C2×#1171994511yescharacteristic
228C2—22no—
229C2—44no—
230C2—4241no—
231C2C2×#1313319111yescharacteristic
232C2#1681143412yes—
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#518C2
socleC22×C53C22×D4
hypercenter#518D53
hypocenterC53#3042
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
2. C22×#11719923C2semidirect
3. #16811199C2semidirect
4. #16810838C2semidirect
5. #16810828C2semidirect
6. #16810650C2semidirect
7. #16810631C2semidirect
8. #16810617C2semidirect
9. C53×#518C2semidirect
19. #13133217C22semidirect
20. #13133205C22semidirect
23. C4⋊#88322C22semidirect
24. #13133140C22semidirect
25. #13133140C22semidirect
26. #13133137C22semidirect
120. #518D53semidirect
126. C53#3042semidirect