Finite Group Databasebeta

← #20284228

Subgroups — #20284228

Order 6784.

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

normal subgroups only

This group has 494 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×#13133245C211yesmaximal, characteristic
2C22×#11719913C211yesmaximal, characteristic, complemented
3#16811199C211yesmaximal, characteristic, complemented
4#16811097C211yesmaximal, characteristic, complemented
5#16810839C214yesmaximal, complemented
6#16810829C214yesmaximal, complemented
7#16810633C211yesmaximal, characteristic, complemented
8#16810620C211yesmaximal, characteristic, complemented
9C53×#424C211yesmaximal, characteristic, complemented
10C2×#11719945C2211yescharacteristic
11C22×Dic106C2211yescharacteristic
12C22×D212C2211yescharacteristic
13#13133245—24no—
14C2×#11719913—24no—
15C2×#11719913C2212yes—
16C2×#11719913C2214yes—
17C2×D424C2212yes—
18#13133217C2212yescomplemented
19#13133205C2212yescomplemented
20#13133201—22no—
21#13133197—22no—
22C4×D212C2214yes—
23#13133179—22no—
24#13133176C2211yescharacteristic
25C22⋊C4×D53—22no—
26#13133142C2212yescomplemented
27#13133142C2214yescomplemented
28#13133139C2212yescomplemented
29C2×M16×C53C2211yescharacteristic
30C22×C8×C53C2211yescharacteristic
31#117×C53C2211yescharacteristic
32C4⋊C8×C53C2214yescomplemented
33C23×D53—21no—
34C22×C4×C53C2311yescharacteristic
35C2×C22⋊D53—21no—
36C22×Dic53—21no—
37#11719945—22no—
38#11719945—22no—
39C2×D212—21no—
40C2×D212—22no—
41C2×D212D414yes—
42C2×D212C2311yescharacteristic
43C2×D212C2312yes—
44C2×C4×D53—21no—
45C2×C4×D53—24no—
46C2×Dic106—22no—
47C2×Dic106—22no—
48C2×Dic106—21no—
49C2×Dic106C2312yes—
50C2×Dic106C2311yescharacteristic
51M16×C53—22no—
52C2×C8×C53—22no—
53C2×C8×C53C2312yes—
54C2×C8×C53C2312yes—
55C4⋊C4×C53D412yes—
56C22⋊C4×C53D412yes—
57C42×C53C2312yes—
58C22⋊Dic53—21no—
59#11719921—24no—
60#11719921—24no—
61C4⋊Dic53—24no—
62C4⋊Dic53C2312yes—
63C4×Dic53—21no—
64D424—24no—
65#11719913—28no—
66#11719913—24no—
67C23×C53C2×D411yescharacteristic
68C22×D53—22no—
69C22×D53—24no—
70C22×D53—21no—
71C22×D53—82no—
72C2×C4×C53C2×D414yes—
73C2×C4×C53C2×D411yescharacteristic
74C2×C4×C53C2×D412yes—
75C2×C4×C53#4112yes—
76C2×C4×C53C2411yescharacteristic
77C22⋊D53—42no—
78C2×Dic53—22no—
79C2×Dic53—21no—
80C2×Dic53—22no—
81C2×Dic53—42no—
82D212—24no—
83D212—21no—
84D212—22no—
85D212—21no—
86D212C2×D414yes—
87C4×D53—42no—
88C4×D53—84no—
89Dic106—22no—
90Dic106—21no—
91Dic106—22no—
92Dic106—21no—
93Dic106—42no—
94C8×C53—22no—
95C8×C53—22no—
96C22×C53—22no—
97C22×C53#12112yes—
98C22×C53C22×D411yescharacteristic
99C2×D53—24no—
100C2×D53—44no—
101C2×D53—41no—
102C2×D53—42no—
103C2×D53—84no—
104C4×C53—24no—
105C4×C53#12112yes—
106C4×C53C22×D411yescharacteristic
107C4×C53C2×#4111yescharacteristic
108Dic53—42no—
109Dic53—41no—
110Dic53—42no—
111#3042—531nomaximal, Sylow
112C2×C53—22no—
113C2×C53C2×#12111yescharacteristic
114C2×C53#57712yes—
115D53—44no—
116D53—81no—
117C2×AGL(1,ℤ/8)—531no—
118C22×SD16—531no—
119#536—531no—
120#518—531no—
121#463—534no—
122#461—534no—
123#424D5311yescharacteristic, complemented
124#419—531no—
125#416—531no—
126C53#304211yescharacteristic, complemented, Sylow
127C2×#41—531no—
128C22×Q8—531no—
129C22×D4—531no—
130AGL(1,ℤ/8)—1064no—
131C2×SD16—534no—
132C2×SD16—532no—
133C2×SD16—1064no—
134C2×D8—532no—
135C2×M16C2×D5311yescharacteristic
136C22×C8C2×D5311yescharacteristic
137C4⋊Q8—532no—
138#124—532no—
139#123—1062no—
140C22⋊Q8—1062no—
141C4×D4—534no—
142C4×D4—1062no—
143#117—531no—
144#117C2×D5311yescharacteristic
145C2×C22⋊C4—1062no—
146C4⋊C8C2×D5314yescomplemented
147#103—532no—
148#103—534no—
149#102—532no—
150C24—1061no—
151#41—1062no—
152#41—1062no—
153C2×Q8—532no—
154C2×Q8—531no—
155C2×Q8—1061no—
156C2×Q8—1062no—
157C2×Q8—1062no—
158C2×D4—534no—
159C2×D4—532no—
160C2×D4—531no—
161C2×D4—1061no—
162C2×D4—1061no—
163C2×D4—1062no—
164C22×C4—1061no—
165C22×C4—1064no—
166C22×C4—1061no—
167C22×C4C22×D5311yescharacteristic
168SD16—1068no—
169SD16—1064no—
170D8—1064no—
171M16—22no—
172C2×C8—22no—
173C2×C8C22×D5312yes—
174C2×C8C22×D5312yes—
175C4⋊C4—532no—
176C4⋊C4—1064no—
177C4⋊C4D21212yes—
178C22⋊C4—1061no—
179C22⋊C4—1064no—
180C22⋊C4—1064no—
181C22⋊C4D21212yes—
182C42—1061no—
183C42C22×D5312yes—
184C23—1064no—
185C23—1062no—
186C23—1061no—
187C23—4242no—
188C23C2×D21211yescharacteristic
189Q8—1062no—
190Q8—1061no—
191Q8—1062no—
192Q8—1061no—
193Q8—2122no—
194D4—534no—
195D4—1064no—
196D4—1061no—
197D4—1062no—
198D4—1061no—
199D4—2122no—
200C2×C4—1062no—
201C2×C4—1062no—
202C2×C4—1061no—
203C2×C4—4244no—
204C2×C4—2122no—
205C2×C4—2122no—
206C2×C4C2×D21214yes—
207C2×C4C2×D21211yescharacteristic
208C2×C4D4×D5312yes—
209C2×C4#1171994912yes—
210C2×C4C23×D5311yescharacteristic
211C8—22no—
212C8—22no—
213C22—22no—
214C22—1064no—
215C22—4244no—
216C22—2121no—
217C22—2122no—
218C22—2124no—
219C22#1313319212yes—
220C22C22×D21211yescharacteristic
221C4—24no—
222C4—2121no—
223C4—2122no—
224C4—2122no—
225C4#1313319212yes—
226C4C2×D4×D5311yescharacteristic
227C4C2×#1171994911yescharacteristic
228C2—22no—
229C2—4241no—
230C2—2124no—
231C2C2×#1313319211yescharacteristic
232C2#1681143512yes—
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#424C2
socleC22×C53C22×D4
hypercenter#424D53
hypocenterC53#3042
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
2. C22×#11719913C2semidirect
3. #16811199C2semidirect
4. #16811097C2semidirect
5. #16810839C2semidirect
6. #16810829C2semidirect
7. #16810633C2semidirect
8. #16810620C2semidirect
9. C53×#424C2semidirect
18. #13133217C22semidirect
19. #13133205C22semidirect
26. #13133142C22semidirect
27. #13133142C22semidirect
28. #13133139C22semidirect
32. C4⋊C8×C53C22semidirect
123. #424D53semidirect
126. C53#3042semidirect
146. C4⋊C8C2×D53semidirect