Finite Group Databasebeta

← #20284674

Subgroups — #20284674

Order 6784.

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

normal subgroups only

This group has 418 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
1#16811187C211yesmaximal, characteristic
2#16811106C211yesmaximal, characteristic, complemented
3#16810948C212yesmaximal, complemented
4#16810948C212yesmaximal, complemented
5#16810942C212yesmaximal, complemented
6#16810937C212yesmaximal, complemented
7#16810665C211yesmaximal, characteristic, complemented
8#16810661C211yesmaximal, characteristic, complemented
9#16810629C211yesmaximal, characteristic, complemented
10C2×#13133139C211yesmaximal, characteristic, complemented
11C53×#408C211yesmaximal, characteristic, complemented
12C22×D212C2211yescharacteristic
13#13133213—21no—
14#13133206—21no—
15#13133191C2212yes—
16#13133191C2212yes—
17#13133185—22no—
18Q8×Dic53—21no—
19D4×Dic53—22no—
20D4×Dic53—21no—
21#13133176C2211yescharacteristic
22#13133176C2211yescharacteristic
23#13133176C2212yescomplemented
24C2×C4⋊Dic53C2211yescharacteristic
25C2×C4⋊Dic53C2212yescomplemented
26#13133151C2211yescharacteristic, complemented
27#13133151C2211yescharacteristic, complemented
28#13133151C2212yescomplemented
29#13133149C2211yescharacteristic, complemented
30#13133149C2211yescharacteristic, complemented
31#13133149C2212yescomplemented
32#13133139C2212yes—
33#13133139C2212yes—
34#13133139C2212yes—
35#13133139C2212yes—
36C2×#41×C53C2211yescharacteristic
37C2×M16×C53C2211yescharacteristic
38C22×C8×C53C2211yescharacteristic
39C22⋊C8×C53C2212yescomplemented
40C22⋊C8×C53C2212yescomplemented
41C23×D53—21no—
42#41×C53—22no—
43#41×C53—22no—
44C2×Q8×C53D411yescharacteristic
45C2×D4×C53D412yes—
46C2×D4×C53D411yescharacteristic
47C22×C4×C53C2312yes—
48C22×C4×C53C2311yescharacteristic
49C2×C22⋊D53—22no—
50C2×C22⋊D53—22no—
51C22×Dic53—21no—
52C22×Dic53—22no—
53C2×D212—22no—
54C2×D212—22no—
55C2×D212—21no—
56C2×D212C2312yes—
57M16×C53—22no—
58C2×C8×C53—22no—
59C2×C8×C53C2311yescharacteristic
60C2×C8×C53C2311yescharacteristic
61C2×C8×C53C2312yes—
62C22⋊Dic53—21no—
63C22⋊Dic53—21no—
64C22⋊Dic53—22no—
65#11719921—22no—
66#11719921—22no—
67C4⋊Dic53—21no—
68C4⋊Dic53—22no—
69C4⋊Dic53C2312yes—
70C4⋊Dic53C2311yescharacteristic
71C4⋊Dic53C2311yescharacteristic
72C4⋊Dic53C2311yescharacteristic
73C4⋊Dic53C2311yescharacteristic
74C4×Dic53—21no—
75C4×Dic53—21no—
76C4×Dic53—22no—
77C23×C53C2×D412yes—
78C23×C53C2×D411yescharacteristic
79C22×D53—22no—
80C22×D53—42no—
81C22×D53—81no—
82C22×D53—81no—
83Q8×C53—22no—
84D4×C53—24no—
85D4×C53—22no—
86C2×C4×C53—22no—
87C2×C4×C53—22no—
88C2×C4×C53C2×D412yes—
89C2×C4×C53C2×D411yescharacteristic
90C2×C4×C53#4112yes—
91C2×C4×C53#4112yes—
92C2×C4×C53C2411yescharacteristic
93C22⋊D53—82no—
94C22⋊D53—82no—
95C2×Dic53—22no—
96C2×Dic53—21no—
97C2×Dic53—21no—
98C2×Dic53—21no—
99C2×Dic53—21no—
100C2×Dic53—81no—
101C2×Dic53—82no—
102D212—22no—
103D212—22no—
104D212—42no—
105C8×C53—21no—
106C8×C53—21no—
107C8×C53—22no—
108C22×C53—21no—
109C22×C53—21no—
110C22×C53—22no—
111C22×C53—22no—
112C22×C53—22no—
113C22×C53#12312yes—
114C22×C53C22×D411yescharacteristic
115C2×D53—42no—
116C2×D53—42no—
117C2×D53—42no—
118C2×D53—84no—
119C4×C53—22no—
120C4×C53#12312yes—
121C4×C53C2×#4111yescharacteristic
122C4×C53C2×#4111yescharacteristic
123Dic53—42no—
124Dic53—41no—
125Dic53—41no—
126Dic53—41no—
127Dic53—41no—
128#3096—531nomaximal, Sylow
129C2×C53—22no—
130C2×C53—42no—
131C2×C53C2×#12311yescharacteristic
132C2×C53#57611yescharacteristic
133C2×C53#57711yescharacteristic
134D53—82no—
135#535—531no—
136#520—531no—
137#482—532no—
138#482—532no—
139#481—532no—
140#480—532no—
141#428—531no—
142#427—531no—
143#418—531no—
144C2×#102—531no—
145#408D5311yescharacteristic, complemented
146C53#309611yescharacteristic, complemented, Sylow
147C2×#41C2×D5311yescharacteristic
148C22×D4—531no—
149C2×M16C2×D5311yescharacteristic
150C22×C8C2×D5311yescharacteristic
151#127—1061no—
152#124—1061no—
153#121—532no—
154#121—532no—
155#120—1062no—
156C4×Q8—1061no—
157C4×D4—1062no—
158C4×D4—1061no—
159#117—531no—
160#117—532no—
161#117—531no—
162C2×C4⋊C4—531no—
163C2×C4⋊C4—532no—
164#107—531no—
165#107—531no—
166#107—532no—
167#106—531no—
168#106—531no—
169#106—532no—
170#102—532no—
171#102—532no—
172#102—532no—
173#102—532no—
174C22⋊C8C2×D5312yescomplemented
175C22⋊C8C2×D5312yescomplemented
176C24—1061no—
177#41—22no—
178#41—22no—
179C2×Q8D21211yescharacteristic
180C2×D4—532no—
181C2×D4—1062no—
182C2×D4—1062no—
183C2×D4—1061no—
184C2×D4—1062no—
185C2×D4—1062no—
186C2×D4D21212yes—
187C2×D4D21211yescharacteristic
188C22×C4—1062no—
189C22×C4—1061no—
190C22×C4C22×D5312yes—
191C22×C4C22×D5311yescharacteristic
192M16—22no—
193C2×C8—22no—
194C2×C8C22×D5311yescharacteristic
195C2×C8C22×D5312yes—
196C2×C8C22×D5311yescharacteristic
197C4⋊C4—532no—
198C4⋊C4—531no—
199C4⋊C4—531no—
200C4⋊C4—531no—
201C4⋊C4—531no—
202C4⋊C4—1062no—
203C4⋊C4—1061no—
204C22⋊C4—1062no—
205C22⋊C4—1061no—
206C22⋊C4—1061no—
207C22⋊C4—1062no—
208C22⋊C4—1062no—
209C42—1062no—
210C42—1061no—
211C42—1061no—
212C23—1062no—
213C23—4241no—
214C23—4241no—
215C23—2122no—
216C23C2×D21212yes—
217C23C2×D21211yescharacteristic
218Q8—22no—
219D4—24no—
220D4—22no—
221D4—1062no—
222D4—1062no—
223D4—4242no—
224D4—4242no—
225D4—2122no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—1061no—
229C2×C4—1061no—
230C2×C4—1061no—
231C2×C4—1062no—
232C2×C4—1061no—
233C2×C4—4241no—
234C2×C4—4242no—
235C2×C4C2×D21212yes—
236C2×C4C2×D21211yescharacteristic
237C2×C4#1171994712yes—
238C2×C4#1171994712yes—
239C2×C4C23×D5311yescharacteristic
240C8—21no—
241C8—21no—
242C8—22no—
243C22—21no—
244C22—21no—
245C22—22no—
246C22—22no—
247C22—22no—
248C22—4244no—
249C22—2122no—
250C22—2122no—
251C22—2122no—
252C22#1313320112yes—
253C22C22×D21211yescharacteristic
254C4—22no—
255C4—2122no—
256C4—2121no—
257C4—2121no—
258C4—2121no—
259C4—2121no—
260C4#1313320112yes—
261C4C2×#1171994711yescharacteristic
262C4C2×#1171994711yescharacteristic
263C2—22no—
264C2—42no—
265C2—4242no—
266C2C2×#1313320111yescharacteristic
267C2#1681142811yescharacteristic
268C2#1681143511yescharacteristic
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#408C2
socleC22×C53C22×D4
hypercenter#408D53
hypocenterC53#3096
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
2. #16811106C2semidirect
3. #16810948C2semidirect
4. #16810948C2semidirect
5. #16810942C2semidirect
6. #16810937C2semidirect
7. #16810665C2semidirect
8. #16810661C2semidirect
9. #16810629C2semidirect
10. C2×#13133139C2semidirect
11. C53×#408C2semidirect
23. #13133176C22semidirect
25. C2×C4⋊Dic53C22semidirect
26. #13133151C22semidirect
27. #13133151C22semidirect
28. #13133151C22semidirect
29. #13133149C22semidirect
30. #13133149C22semidirect
31. #13133149C22semidirect
39. C22⋊C8×C53C22semidirect
40. C22⋊C8×C53C22semidirect
145. #408D53semidirect
146. C53#3096semidirect
174. C22⋊C8C2×D53semidirect
175. C22⋊C8C2×D53semidirect