Finite Group Databasebeta

← #20285749

Subgroups — #20285749

Order 6784.

396 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 109 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#16811013C211yesmaximal, characteristic, complemented
4#16810991C211yesmaximal, characteristic, complemented
5#16810988C211yesmaximal, characteristic, complemented
6#16810847C212yesmaximal, complemented
7#16810840C212yesmaximal, complemented
8#16810784C212yesmaximal, complemented
9#16810767C212yesmaximal, complemented
10#16810678C211yesmaximal, characteristic, complemented
11C53×#497C211yesmaximal, characteristic, complemented
12C2×#11719949C2212yes—
13C2×#11719924C2212yes—
14C2×#11719922C2212yes—
15#13133214C2211yescharacteristic
16#13133210—21no—
17#13133209C2211yescharacteristic
18#13133206C2212yes—
19#13133199—21no—
20#13133193C2211yescharacteristic
21#13133192—21no—
22#13133192—21no—
23#13133188C2211yescharacteristic
24Q8×Dic53C2211yescharacteristic
25D4×Dic53C2211yescharacteristic
26#13133179—21no—
27#13133179—21no—
28#13133177—21no—
29#13133173—21no—
30C42×D53C2211yescharacteristic
31#13133146C2212yes—
32#13133141C2212yes—
33#13133141C2212yes—
34#13133139C2212yes—
35#13133139C2212yes—
36#13133129C2212yes—
37#13133128C2211yescharacteristic
38C2×Q16×C53C2212yes—
39C2×SD16×C53C2212yes—
40C4⋊Q8×C53C2211yescharacteristic
41#124×C53C2211yescharacteristic
42#97×C53C2211yescharacteristic
43C2×Q8×C53C2311yescharacteristic
44C2×Q8×C53C2312yes—
45C2×D4×C53C2311yescharacteristic
46C2×C22⋊D53—21no—
47C22×Dic53—21no—
48#11719949—42no—
49#11719949—42no—
50C2×D212—22no—
51C2×D212—21no—
52C2×D212—21no—
53C2×D212C2312yes—
54C2×C4×D53—22no—
55C2×C4×D53D411yescharacteristic
56C2×C4×D53D411yescharacteristic
57C2×C4×D53C2311yescharacteristic
58Q16×C53—42no—
59SD16×C53—42no—
60C2×C8×C53C2312yes—
61C4⋊C4×C53—21no—
62C4⋊C4×C53—21no—
63C22⋊C4×C53—21no—
64C22⋊C4×C53—21no—
65C42×C53C2311yescharacteristic
66C22⋊Dic53—21no—
67C22⋊Dic53—21no—
68#11719924—42no—
69#11719922—42no—
70#11719921—21no—
71#11719921—22no—
72#11719921—22no—
73C4⋊Dic53—21no—
74C4⋊Dic53C2311yescharacteristic
75C4⋊Dic53C2311yescharacteristic
76#11719919—21no—
77#11719919—21no—
78#11719919—21no—
79C4×Dic53—21no—
80C4×Dic53D411yescharacteristic
81C4×Dic53D411yescharacteristic
82C4×Dic53C2311yescharacteristic
83C2×#88322C2312yes—
84C23×C53—21no—
85C22×D53—22no—
86C22×D53C2×D411yescharacteristic
87Q8×C53—22no—
88Q8×C53—22no—
89Q8×C53—22no—
90D4×C53—22no—
91C2×C4×C53—22no—
92C2×C4×C53—21no—
93C2×C4×C53C2×D411yescharacteristic
94C2×C4×C53C2×D411yescharacteristic
95C2×C4×C53C2411yescharacteristic
96C22⋊D53—42no—
97C2×Dic53—21no—
98C2×Dic53—21no—
99C2×Dic53—42no—
100C2×Dic53C2×D411yescharacteristic
101C2×Dic53C2×D411yescharacteristic
102C2×Dic53C2×D411yescharacteristic
103D212—22no—
104D212—22no—
105D212—42no—
106D212—42no—
107D212—44no—
108C4×D53—22no—
109C4×D53—41no—
110C4×D53—41no—
111C4×D53—44no—
112C8×C53—22no—
113#88322—22no—
114C22×C53—41no—
115C22×C53—41no—
116C22×C53—41no—
117C22×C53C22×D411yescharacteristic
118C2×D53—21no—
119C2×D53—21no—
120C2×D53—21no—
121C2×D53—42no—
122C2×D53—42no—
123C2×D53—42no—
124C4×C53—21no—
125C4×C53—21no—
126C4×C53—42no—
127C4×C53—41no—
128C4×C53H(5,𝔽2)12yes—
129Dic53—21no—
130Dic53—21no—
131Dic53—21no—
132Dic53—41no—
133Dic53—41no—
134#3219—531nomaximal, Sylow
135C2×C53—81no—
136C2×C53#53511yescharacteristic
137C2×C53#57511yescharacteristic
138C2×C53#57511yescharacteristic
139D53—41no—
140D53—82no—
141#532—531no—
142#529—531no—
143#497D5311yescharacteristic, complemented
144#496—531no—
145#491—531no—
146#490—531no—
147#464—532no—
148#463—532no—
149#452—532no—
150#449—532no—
151#432—531no—
152C53#321911yescharacteristic, complemented, Sylow
153C2×#41—532no—
154C2×Q16C2×D5312yes—
155C2×SD16—532no—
156C2×SD16C2×D5312yes—
157C2×D8—532no—
158C4⋊Q8C2×D5311yescharacteristic
159#127—531no—
160#126—1061no—
161#125—531no—
162#124—532no—
163#124C2×D5311yescharacteristic
164#123—1061no—
165C22⋊Q8—531no—
166#121—531no—
167#121—1061no—
168#121—1061no—
169C4×Q8—531no—
170C4×D4—531no—
171C4×D4—1061no—
172C4×D4—1061no—
173C4×D4—1061no—
174#117—1061no—
175C2×C42—531no—
176C4⋊C8—532no—
177#103—532no—
178#103—532no—
179#102—532no—
180#102—532no—
181C22⋊C8—532no—
182#97—531no—
183#97C2×D5311yescharacteristic
184#41—2122no—
185#41—2122no—
186C2×Q8C22×D5312yes—
187C2×Q8C22×D5311yescharacteristic
188C2×D4—532no—
189C2×D4—1062no—
190C2×D4—1061no—
191C2×D4—1061no—
192C2×D4—1061no—
193C2×D4C22×D5311yescharacteristic
194C22×C4—531no—
195C22×C4—531no—
196C22×C4—531no—
197C22×C4—1062no—
198C22×C4—1061no—
199Q16—42no—
200SD16—42no—
201SD16—2122no—
202D8—2122no—
203C2×C8—532no—
204C2×C8C22×D5312yes—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—531no—
208C4⋊C4—531no—
209C4⋊C4—1061no—
210C4⋊C4—1061no—
211C4⋊C4—1061no—
212C4⋊C4—1061no—
213C22⋊C4—21no—
214C22⋊C4—21no—
215C22⋊C4—1062no—
216C22⋊C4—1062no—
217C22⋊C4—1061no—
218C22⋊C4—1061no—
219C22⋊C4—1061no—
220C42—531no—
221C42—531no—
222C42—531no—
223C42—1061no—
224C42C22×D5311yescharacteristic
225C23—21no—
226C23—531no—
227C23—1062no—
228Q8—22no—
229Q8—22no—
230Q8—22no—
231D4—22no—
232D4—1062no—
233D4—1062no—
234D4—2122no—
235D4—2122no—
236D4—2124no—
237D4—2122no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—531no—
241C2×C4—531no—
242C2×C4—531no—
243C2×C4—1062no—
244C2×C4—1061no—
245C2×C4—1061no—
246C2×C4—2121no—
247C2×C4—2121no—
248C2×C4—2124no—
249C2×C4—2122no—
250C2×C4D4×D5311yescharacteristic
251C2×C4D4×D5311yescharacteristic
252C2×C4C23×D5311yescharacteristic
253C8—22no—
254C8—1062no—
255C22—41no—
256C22—41no—
257C22—41no—
258C22—1061no—
259C22—1061no—
260C22—1061no—
261C22—2122no—
262C22—2122no—
263C22—2122no—
264C22C2×D4×D5311yescharacteristic
265C4—21no—
266C4—21no—
267C4—42no—
268C4—41no—
269C4—1061no—
270C4—1061no—
271C4—1061no—
272C4—2121no—
273C4—2121no—
274C4#1313326212yes—
275C2—81no—
276C2—4242no—
277C2—2121no—
278C2#1681118611yescharacteristic
279C2#1681142211yescharacteristic
280C2#1681142211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#497C2
socleC22×C53C22×D4
hypercenter#497D53
hypocenterC53#3219
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. #16811174C2semidirect
2. #16811152C2semidirect
3. #16811013C2semidirect
4. #16810991C2semidirect
5. #16810988C2semidirect
6. #16810847C2semidirect
7. #16810840C2semidirect
8. #16810784C2semidirect
9. #16810767C2semidirect
10. #16810678C2semidirect
11. C53×#497C2semidirect
143. #497D53semidirect
152. C53#3219semidirect