Finite Group Databasebeta

← #20285416

Subgroups — #20285416

Order 6784.

410 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 115 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#16811174C211yesmaximal, characteristic, complemented
2#16811152C211yesmaximal, characteristic, complemented
3#16811029C211yesmaximal, characteristic, complemented
4#16811013C211yesmaximal, characteristic, complemented
5#16810988C211yesmaximal, characteristic, complemented
6#16810858C212yesmaximal, complemented
7#16810855C212yesmaximal, complemented
8#16810732C212yesmaximal, complemented
9#16810726C212yesmaximal, complemented
10#97×D53C211yesmaximal, characteristic, complemented
11C53×#489C211yesmaximal, characteristic, complemented
12C2×#11719949—21no—
13C2×#11719924C2212yes—
14C2×#11719922C2212yes—
15C2×C8×D53C2212yes—
16#13133214C2211yescharacteristic
17#13133210—21no—
18#13133209C2211yescharacteristic
19#13133206—21no—
20#13133199—21no—
21#13133193C2211yescharacteristic
22#13133192—21no—
23#13133192C2212yes—
24#13133188C2211yescharacteristic
25Q8×Dic53C2211yescharacteristic
26D4×Dic53C2211yescharacteristic
27#13133179—21no—
28#13133179C2212yes—
29#13133177—21no—
30#13133173—21no—
31C42×D53C2211yescharacteristic
32#13133150C2212yes—
33#13133148C2212yes—
34#13133139C2212yes—
35#13133139C2212yes—
36#13133128C2211yescharacteristic
37#13133127C2212yes—
38C4⋊Q8×C53C2211yescharacteristic
39#124×C53C2211yescharacteristic
40#103×C53C2212yes—
41#102×C53C2212yes—
42#97×C53C2211yescharacteristic
43C2×Q8×C53—21no—
44C2×Q8×C53C2311yescharacteristic
45C2×D4×C53C2311yescharacteristic
46C2×C22⋊D53—21no—
47C22×Dic53—21no—
48#11719949—44no—
49C2×D212—22no—
50C2×D212—21no—
51C2×D212—21no—
52C2×D212C2312yes—
53C2×C4×D53—22no—
54C2×C4×D53D411yescharacteristic
55C2×C4×D53D411yescharacteristic
56C2×C4×D53C2311yescharacteristic
57C2×C8×C53C2312yes—
58C4⋊C4×C53—21no—
59C4⋊C4×C53C2312yes—
60C22⋊C4×C53—21no—
61C22⋊C4×C53—21no—
62C42×C53C2311yescharacteristic
63C22⋊Dic53—21no—
64C22⋊Dic53—21no—
65#11719924—24no—
66#11719922—24no—
67#11719921—21no—
68#11719921—22no—
69#11719921—22no—
70C4⋊Dic53—21no—
71C4⋊Dic53C2311yescharacteristic
72C4⋊Dic53C2311yescharacteristic
73#11719919—21no—
74#11719919—21no—
75#11719919—21no—
76C4×Dic53—21no—
77C4×Dic53D411yescharacteristic
78C4×Dic53D411yescharacteristic
79C4×Dic53C2311yescharacteristic
80C2×#88322C2312yes—
81C8×D53—24no—
82C23×C53—21no—
83C22×D53—22no—
84C22×D53C2×D411yescharacteristic
85Q8×C53—22no—
86Q8×C53—42no—
87D4×C53—22no—
88C2×C4×C53—22no—
89C2×C4×C53—21no—
90C2×C4×C53C2×D411yescharacteristic
91C2×C4×C53C2×D411yescharacteristic
92C2×C4×C53C2411yescharacteristic
93C22⋊D53—42no—
94C2×Dic53—21no—
95C2×Dic53—21no—
96C2×Dic53—42no—
97C2×Dic53C2×D411yescharacteristic
98C2×Dic53C2×D411yescharacteristic
99C2×Dic53C2×D411yescharacteristic
100D212—22no—
101D212—22no—
102D212—42no—
103D212—42no—
104D212—44no—
105C4×D53—22no—
106C4×D53—22no—
107C4×D53—41no—
108C4×D53—44no—
109C8×C53—22no—
110#88322#4114yescomplemented
111C22×C53—41no—
112C22×C53—41no—
113C22×C53—41no—
114C22×C53C22×D411yescharacteristic
115C2×D53—21no—
116C2×D53—21no—
117C2×D53—42no—
118C2×D53—42no—
119C2×D53—42no—
120C2×D53AGL(1,ℤ/8)12yes—
121C4×C53—21no—
122C4×C53—21no—
123C4×C53—41no—
124C4×C53—42no—
125C4×C53C2×#4112yes—
126Dic53—21no—
127Dic53—21no—
128Dic53—21no—
129Dic53—41no—
130Dic53—41no—
131#3180—531nomaximal, Sylow
132C2×C53—81no—
133C2×C53#53211yescharacteristic
134C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
135C2×C53#57511yescharacteristic
136D53—22no—
137D53—82no—
138#532—531no—
139#529—531no—
140#501—531no—
141#496—531no—
142#490—531no—
143#489D5311yescharacteristic, complemented
144#466—532no—
145#465—532no—
146#442—532no—
147#440—532no—
148C2×#97—531no—
149C53#318011yescharacteristic, complemented, Sylow
150C2×#41—1061no—
151C2×SD16—532no—
152C2×D8—532no—
153C22×C8—532no—
154C4⋊Q8C2×D5311yescharacteristic
155#127—531no—
156#126—1061no—
157#125—531no—
158#124—1061no—
159#124C2×D5311yescharacteristic
160#123—1061no—
161C22⋊Q8—531no—
162#121—531no—
163#121—532no—
164#121—1061no—
165C4×Q8—531no—
166C4×D4—531no—
167C4×D4—532no—
168C4×D4—1061no—
169C4×D4—1061no—
170#117—1061no—
171C2×C42—531no—
172#107—532no—
173#106—532no—
174#103C2×D5312yes—
175#102—532no—
176#102—532no—
177#102C2×D5312yes—
178#97—532no—
179#97—531no—
180#97C2×D5311yescharacteristic
181#41—2124no—
182C2×Q8—21no—
183C2×Q8C22×D5311yescharacteristic
184C2×D4—532no—
185C2×D4—1062no—
186C2×D4—1061no—
187C2×D4—1061no—
188C2×D4—1061no—
189C2×D4C22×D5311yescharacteristic
190C22×C4—531no—
191C22×C4—531no—
192C22×C4—531no—
193C22×C4—1062no—
194C22×C4—1061no—
195SD16—1064no—
196D8—1064no—
197C2×C8—532no—
198C2×C8—1064no—
199C2×C8C22×D5312yes—
200C4⋊C4—21no—
201C4⋊C4—531no—
202C4⋊C4—531no—
203C4⋊C4—1061no—
204C4⋊C4—1061no—
205C4⋊C4—1061no—
206C4⋊C4—1061no—
207C4⋊C4C22×D5312yes—
208C22⋊C4—21no—
209C22⋊C4—21no—
210C22⋊C4—1062no—
211C22⋊C4—1062no—
212C22⋊C4—1061no—
213C22⋊C4—1061no—
214C22⋊C4—1061no—
215C42—531no—
216C42—531no—
217C42—531no—
218C42—1061no—
219C42C22×D5311yescharacteristic
220C23—21no—
221C23—531no—
222C23—1062no—
223Q8—22no—
224Q8—42no—
225D4—22no—
226D4—1062no—
227D4—1062no—
228D4—2122no—
229D4—2122no—
230D4—2124no—
231D4—2122no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—531no—
235C2×C4—531no—
236C2×C4—531no—
237C2×C4—1062no—
238C2×C4—1062no—
239C2×C4—1061no—
240C2×C4—1061no—
241C2×C4—2121no—
242C2×C4—2124no—
243C2×C4—2122no—
244C2×C4D4×D5311yescharacteristic
245C2×C4D4×D5311yescharacteristic
246C2×C4C23×D5311yescharacteristic
247C8—22no—
248C8—534no—
249C22—41no—
250C22—41no—
251C22—41no—
252C22—532no—
253C22—1061no—
254C22—1061no—
255C22—2122no—
256C22—2122no—
257C22—2122no—
258C22C2×D4×D5311yescharacteristic
259C4—21no—
260C4—21no—
261C4—42no—
262C4—41no—
263C4—1061no—
264C4—1061no—
265C4—1061no—
266C4—2121no—
267C4—2121no—
268C4#41×D5312yes—
269C2—81no—
270C2—1062no—
271C2—4242no—
272C2#1681117211yescharacteristic
273C2AGL(1,ℤ/8)×D5311yescharacteristic
274C2#1681142211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#489C2
socleC22×C53C22×D4
hypercenter#489D53
hypocenterC53#3180
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #16811174C2semidirect
2. #16811152C2semidirect
3. #16811029C2semidirect
4. #16811013C2semidirect
5. #16810988C2semidirect
6. #16810858C2semidirect
7. #16810855C2semidirect
8. #16810732C2semidirect
9. #16810726C2semidirect
10. #97×D53C2semidirect
11. C53×#489C2semidirect
110. #88322#41semidirect
143. #489D53semidirect
149. C53#3180semidirect