Finite Group Databasebeta

← #20285178

Subgroups — #20285178

Order 6784.

572 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 137 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×#13133249C212yesmaximal, complemented
2C2×#13133244C212yesmaximal, complemented
3#16811174C211yesmaximal, characteristic, complemented
4#124×D53C211yesmaximal, characteristic, complemented
5#16811015C211yesmaximal, characteristic, complemented
6#16811013C211yesmaximal, characteristic, complemented
7#16811012C211yesmaximal, characteristic, complemented
8#16811008C212yesmaximal, complemented
9#16811004C212yesmaximal, complemented
10#16810575C211yesmaximal, characteristic, complemented
11C53×#497C211yesmaximal, characteristic, complemented
12C2×#11719949C2212yes—
13C2×Q8×D53C2211yescharacteristic
14C2×D4×D53C2211yescharacteristic
15#13133249—28no—
16#13133244—28no—
17C2×#11719925C2212yescomplemented
18C2×#11719913C2212yescomplemented
19C2×#11719924C2212yes—
20C2×#11719924C2212yescomplemented
21C2×D424C2212yes—
22C2×#11719922C2212yes—
23C2×#11719912C2212yes—
24#13133214C2211yescharacteristic, complemented
25#13133206C2211yescharacteristic
26#13133206C2212yescomplemented
27#13133205C2211yescharacteristic
28#13133204C2211yescharacteristic
29#13133203—22no—
30#13133192—22no—
31#13133179—22no—
32C22⋊C4×D53—22no—
33C42×D53C2211yescharacteristic
34#13133128C2211yescharacteristic, complemented
35C8×Dic53C2212yescomplemented
36C2×Q16×C53C2212yescomplemented
37C2×SD16×C53C2212yes—
38C4⋊Q8×C53C2211yescharacteristic, complemented
39#124×C53C2211yescharacteristic
40#97×C53C2211yescharacteristic, complemented
41C23×D53—21no—
42C2×Q8×C53C2311yescharacteristic
43C2×Q8×C53C2312yes—
44C2×D4×C53C2311yescharacteristic
45C2×C22⋊D53—21no—
46#11719949—24no—
47#11719949—24no—
48Q8×D53—24no—
49D4×D53—24no—
50C2×D212—22no—
51C2×D212—22no—
52C2×D212C2311yescharacteristic
53C2×D212C2312yes—
54C2×C4×D53—21no—
55C2×C4×D53—22no—
56C2×C4×D53D412yes—
57C2×C4×D53C2311yescharacteristic
58C2×Dic106—21no—
59C2×Dic106C2311yescharacteristic
60Q16×C53—24no—
61SD16×C53—24no—
62C2×C8×C53C2312yes—
63C4⋊C4×C53—22no—
64C22⋊C4×C53—22no—
65C42×C53C2311yescharacteristic
66C22⋊Dic53—22no—
67#11719925—24no—
68#11719924—24no—
69#11719924—24no—
70#11719922—24no—
71#11719921—22no—
72#11719921—22no—
73#11719921—24no—
74C4×Dic53D412yes—
75C4×Dic53C2311yescharacteristic
76C2×#88322C2312yes—
77D424—24no—
78#11719913—24no—
79#11719912D418yescomplemented
80C23×C53—21no—
81C22×D53—22no—
82C22×D53—21no—
83C22×D53—44no—
84C22×D53—42no—
85C22×D53C2×D411yescharacteristic
86Q8×C53—22no—
87Q8×C53—22no—
88Q8×C53—22no—
89D4×C53—22no—
90C2×C4×C53—21no—
91C2×C4×C53—22no—
92C2×C4×C53C2×D412yes—
93C2×C4×C53C2411yescharacteristic
94C22⋊D53—42no—
95C2×Dic53—21no—
96C2×Dic53C2×D412yes—
97C2×Dic53C2×D411yescharacteristic
98D212—22no—
99D212—22no—
100D212—22no—
101D212—44no—
102D212—44no—
103C4×D53—42no—
104C4×D53—42no—
105C4×D53—44no—
106C4×D53C2×D414yes—
107Dic106—22no—
108Dic106—42no—
109C8×C53C2×D414yes—
110#88322C2×D414yes—
111C22×C53—41no—
112C22×C53—42no—
113C22×C53C22×D411yescharacteristic
114C2×D53—22no—
115C2×D53—42no—
116C2×D53—41no—
117C2×D53—44no—
118C2×D53—42no—
119C2×D53—84no—
120C2×D53#12712yes—
121C4×C53—22no—
122C4×C53—41no—
123C4×C53—42no—
124C4×C53C22×D412yes—
125Dic53—22no—
126Dic53—41no—
127Dic53#12712yes—
128#3153—531nomaximal, Sylow
129C2×C53—81no—
130C2×C53C2×#12711yescharacteristic
131C2×C53#57512yes—
132D53—22no—
133D53—82no—
134D53—81no—
135C2×#137—532no—
136C2×AGL(1,ℤ/8)—532no—
137#532—531no—
138C2×#124—531no—
139#497—531no—
140#497D5311yescharacteristic, complemented
141#496—531no—
142#496—531no—
143#495—532no—
144#495—532no—
145C4×M16—531no—
146C53#315311yescharacteristic, complemented, Sylow
147C2×#41—532no—
148C22×Q8—531no—
149C22×D4—531no—
150#137—1068no—
151AGL(1,ℤ/8)—1068no—
152C2×Q16—532no—
153C2×Q16C2×D5312yescomplemented
154C2×SD16—532no—
155C2×SD16—532no—
156C2×SD16—532no—
157C2×SD16C2×D5312yes—
158C2×D8—532no—
159C2×D8—532no—
160C2×M16—532no—
161C4⋊Q8C2×D5311yescharacteristic, complemented
162#127—531no—
163#124—531no—
164#124—532no—
165#124—531no—
166#124—531no—
167#124—1062no—
168#124C2×D5311yescharacteristic
169#121—1062no—
170C4×D4—1062no—
171C2×C22⋊C4—1062no—
172C2×C42—531no—
173#97—531no—
174#97C2×D5311yescharacteristic, complemented
175C4×C8—532no—
176C24—1061no—
177#41—1064no—
178#41—1064no—
179C2×Q8—531no—
180C2×Q8—1061no—
181C2×Q8—1064no—
182C2×Q8C22×D5311yescharacteristic
183C2×Q8C22×D5312yes—
184C2×D4—531no—
185C2×D4—532no—
186C2×D4—1062no—
187C2×D4—1064no—
188C2×D4—1061no—
189C2×D4—1062no—
190C2×D4C22×D5311yescharacteristic
191C22×C4—531no—
192C22×C4—532no—
193C22×C4—1062no—
194C22×C4—1061no—
195Q16—24no—
196Q16—1064no—
197SD16—24no—
198SD16—1064no—
199SD16—1064no—
200SD16—1064no—
201D8—1064no—
202D8—1064no—
203M16—538no—
204C2×C8—532no—
205C2×C8C22×D5312yes—
206C4⋊C4—22no—
207C22⋊C4—22no—
208C22⋊C4—1062no—
209C22⋊C4—1064no—
210C22⋊C4—1062no—
211C22⋊C4—1062no—
212C42—532no—
213C42—531no—
214C42C22×D5311yescharacteristic
215C23—21no—
216C23—531no—
217C23—1061no—
218C23—1062no—
219C23—2122no—
220C23—2124no—
221Q8—22no—
222Q8—22no—
223Q8—22no—
224Q8—1062no—
225Q8—2122no—
226D4—22no—
227D4—1062no—
228D4—1062no—
229D4—1062no—
230D4—2124no—
231D4—2124no—
232D4—2122no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—531no—
236C2×C4—534no—
237C2×C4—532no—
238C2×C4—1061no—
239C2×C4—2122no—
240C2×C4—2124no—
241C2×C4—2122no—
242C2×C4D4×D5312yes—
243C2×C4C23×D5311yescharacteristic
244C8—534no—
245C8D4×D5314yes—
246C22—41no—
247C22—42no—
248C22—532no—
249C22—1062no—
250C22—4244no—
251C22—2122no—
252C22—2121no—
253C22—2124no—
254C22—2122no—
255C22C2×D4×D5311yescharacteristic
256C4—22no—
257C4—42no—
258C4—41no—
259C4—532no—
260C4—1062no—
261C4—2121no—
262C4C2×D4×D5312yes—
263C2—81no—
264C2—1062no—
265C2—4241no—
266C2—4242no—
267C2#127×D5311yescharacteristic
268C2#1681142212yes—
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#497C2
socleC22×C53C22×D4
hypercenter#497D53
hypocenterC53#3153
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×#13133249C2semidirect
2. C2×#13133244C2semidirect
3. #16811174C2semidirect
4. #124×D53C2semidirect
5. #16811015C2semidirect
6. #16811013C2semidirect
7. #16811012C2semidirect
8. #16811008C2semidirect
9. #16811004C2semidirect
10. #16810575C2semidirect
11. C53×#497C2semidirect
17. C2×#11719925C22semidirect
18. C2×#11719913C22semidirect
20. C2×#11719924C22semidirect
24. #13133214C22semidirect
26. #13133206C22semidirect
34. #13133128C22semidirect
35. C8×Dic53C22semidirect
36. C2×Q16×C53C22semidirect
38. C4⋊Q8×C53C22semidirect
40. #97×C53C22semidirect
79. #11719912D4semidirect
140. #497D53semidirect
146. C53#3153semidirect
153. C2×Q16C2×D53semidirect
161. C4⋊Q8C2×D53semidirect
174. #97C2×D53semidirect