Finite Group Databasebeta

← #20285703

Subgroups — #20285703

Order 6784.

444 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 111 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#124×D53C211yesmaximal, characteristic, complemented
3#16811013C211yesmaximal, characteristic, complemented
4#16811012C211yesmaximal, characteristic, complemented
5#16810985C211yesmaximal, characteristic, complemented
6#16810841C212yesmaximal, complemented
7#16810837C212yesmaximal, complemented
8#16810763C212yesmaximal, complemented
9#16810758C212yesmaximal, complemented
10#16810678C211yesmaximal, characteristic, complemented
11C53×#489C211yesmaximal, characteristic, complemented
12C2×#11719949—21no—
13C2×Q8×D53C2211yescharacteristic
14C2×D4×D53C2211yescharacteristic
15C2×#11719913C2212yescomplemented
16C2×#11719924C2212yescomplemented
17C2×D424C2212yes—
18C2×#11719922C2212yes—
19#13133214C2211yescharacteristic, complemented
20#13133206—21no—
21#13133206C2211yescharacteristic
22#13133205C2211yescharacteristic
23#13133204C2211yescharacteristic
24#13133203—21no—
25#13133203—21no—
26#13133192—21no—
27#13133192C2212yes—
28#13133179—21no—
29#13133179C2212yescomplemented
30C22⋊C4×D53—21no—
31C22⋊C4×D53—21no—
32C42×D53C2211yescharacteristic
33#13133146C2212yescomplemented
34#13133140C2212yescomplemented
35#13133137C2212yes—
36#13133129C2212yes—
37#13133128C2211yescharacteristic, complemented
38C4⋊Q8×C53C2211yescharacteristic, complemented
39#124×C53C2211yescharacteristic
40#103×C53C2212yescomplemented
41#102×C53C2212yes—
42#97×C53C2211yescharacteristic, complemented
43C23×D53—21no—
44C2×Q8×C53—21no—
45C2×Q8×C53C2311yescharacteristic
46C2×D4×C53C2311yescharacteristic
47C2×C22⋊D53—21no—
48#11719949—44no—
49Q8×D53—42no—
50D4×D53—42no—
51C2×D212—22no—
52C2×D212—21no—
53C2×D212—21no—
54C2×D212C2311yescharacteristic
55C2×D212C2312yes—
56C2×C4×D53—21no—
57C2×C4×D53—22no—
58C2×C4×D53D411yescharacteristic
59C2×C4×D53D411yescharacteristic
60C2×C4×D53C2311yescharacteristic
61C2×Dic106—21no—
62C2×Dic106C2311yescharacteristic
63C2×C8×C53C2312yes—
64C4⋊C4×C53—21no—
65C4⋊C4×C53C2312yes—
66C22⋊C4×C53—21no—
67C22⋊C4×C53—21no—
68C42×C53C2311yescharacteristic
69C22⋊Dic53—21no—
70C22⋊Dic53—21no—
71#11719924—42no—
72#11719922—42no—
73#11719921—21no—
74#11719921—21no—
75#11719921—21no—
76#11719921—21no—
77#11719921—22no—
78#11719921—22no—
79C4×Dic53D411yescharacteristic
80C4×Dic53D411yescharacteristic
81C4×Dic53C2311yescharacteristic
82C2×#88322C2312yes—
83D424—42no—
84#11719913—42no—
85C23×C53—21no—
86C22×D53—22no—
87C22×D53—21no—
88C22×D53—42no—
89C22×D53—42no—
90C22×D53—42no—
91C22×D53C2×D411yescharacteristic
92Q8×C53—22no—
93Q8×C53—42no—
94D4×C53—22no—
95C2×C4×C53—21no—
96C2×C4×C53—22no—
97C2×C4×C53C2×D411yescharacteristic
98C2×C4×C53C2×D411yescharacteristic
99C2×C4×C53C2411yescharacteristic
100C22⋊D53—42no—
101C2×Dic53—21no—
102C2×Dic53C2×D411yescharacteristic
103C2×Dic53C2×D411yescharacteristic
104C2×Dic53C2×D411yescharacteristic
105D212—22no—
106D212—22no—
107D212—22no—
108D212—42no—
109D212—42no—
110D212—44no—
111C4×D53—22no—
112C4×D53—22no—
113C4×D53—41no—
114C4×D53—44no—
115C4×D53—42no—
116Dic106—22no—
117Dic106—42no—
118C8×C53—22no—
119#88322—22no—
120C22×C53—41no—
121C22×C53—41no—
122C22×C53—41no—
123C22×C53C22×D411yescharacteristic
124C2×D53—21no—
125C2×D53—21no—
126C2×D53—41no—
127C2×D53—42no—
128C2×D53—42no—
129C2×D53—41no—
130C2×D53—42no—
131C2×D53—41no—
132C2×D53—84no—
133C2×D53AGL(1,ℤ/8)12yes—
134C4×C53—21no—
135C4×C53—21no—
136C4×C53—41no—
137C4×C53—42no—
138C4×C53H(5,𝔽2)12yes—
139Dic53—21no—
140Dic53—21no—
141Dic53—21no—
142Dic53—41no—
143#3213—531nomaximal, Sylow
144C2×C53—81no—
145C2×C53#53511yescharacteristic
146C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
147C2×C53#57511yescharacteristic
148D53—22no—
149D53—82no—
150D53—81no—
151#532—531no—
152C2×#124—531no—
153#496—531no—
154#496—531no—
155#489—531no—
156#489D5311yescharacteristic, complemented
157#463—532no—
158#463—532no—
159#448—532no—
160#447—532no—
161#432—531no—
162C53#321311yescharacteristic, complemented, Sylow
163C2×#41—1061no—
164C22×Q8—531no—
165C22×D4—531no—
166C2×SD16—532no—
167C2×SD16—532no—
168C2×D8—532no—
169C2×D8—532no—
170C4⋊Q8C2×D5311yescharacteristic, complemented
171#127—531no—
172#124—531no—
173#124—531no—
174#124—531no—
175#124—1061no—
176#124—1061no—
177#124—1061no—
178#124C2×D5311yescharacteristic
179#121—532no—
180#121—1061no—
181C4×D4—532no—
182C4×D4—1061no—
183C2×C22⋊C4—1061no—
184C2×C22⋊C4—1061no—
185C2×C42—531no—
186C4⋊C8—532no—
187#103—532no—
188#103C2×D5312yescomplemented
189#102—532no—
190#102C2×D5312yes—
191C22⋊C8—532no—
192#97—531no—
193#97C2×D5311yescharacteristic, complemented
194C24—1061no—
195#41—2124no—
196C2×Q8—21no—
197C2×Q8—531no—
198C2×Q8—1061no—
199C2×Q8—2122no—
200C2×Q8C22×D5311yescharacteristic
201C2×D4—531no—
202C2×D4—532no—
203C2×D4—1062no—
204C2×D4—1061no—
205C2×D4—1061no—
206C2×D4—1061no—
207C2×D4—2122no—
208C2×D4C22×D5311yescharacteristic
209C22×C4—531no—
210C22×C4—531no—
211C22×C4—531no—
212C22×C4—1062no—
213C22×C4—1061no—
214SD16—2122no—
215SD16—2122no—
216D8—2122no—
217D8—2122no—
218C2×C8—532no—
219C2×C8C22×D5312yes—
220C4⋊C4—21no—
221C4⋊C4C22×D5312yes—
222C22⋊C4—21no—
223C22⋊C4—21no—
224C22⋊C4—1061no—
225C22⋊C4—1061no—
226C22⋊C4—1062no—
227C22⋊C4—1062no—
228C22⋊C4—1061no—
229C22⋊C4—1061no—
230C22⋊C4—1061no—
231C22⋊C4—1061no—
232C42—531no—
233C42—531no—
234C42—531no—
235C42C22×D5311yescharacteristic
236C23—21no—
237C23—531no—
238C23—1061no—
239C23—1062no—
240C23—2122no—
241C23—2122no—
242C23—2122no—
243Q8—22no—
244Q8—42no—
245Q8—1062no—
246Q8—2122no—
247D4—22no—
248D4—1062no—
249D4—1062no—
250D4—1062no—
251D4—2122no—
252D4—2122no—
253D4—2124no—
254D4—2122no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—531no—
258C2×C4—531no—
259C2×C4—531no—
260C2×C4—1062no—
261C2×C4—1062no—
262C2×C4—1061no—
263C2×C4—2121no—
264C2×C4—2124no—
265C2×C4—2122no—
266C2×C4D4×D5311yescharacteristic
267C2×C4D4×D5311yescharacteristic
268C2×C4C23×D5311yescharacteristic
269C8—22no—
270C8—1062no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—532no—
275C22—1061no—
276C22—1061no—
277C22—4244no—
278C22—2122no—
279C22—2121no—
280C22—2122no—
281C22—2121no—
282C22—2121no—
283C22—2122no—
284C22C2×D4×D5311yescharacteristic
285C4—21no—
286C4—21no—
287C4—42no—
288C4—41no—
289C4—1061no—
290C4—1061no—
291C4—1061no—
292C4—2121no—
293C4#1313326312yes—
294C2—81no—
295C2—1062no—
296C2—4241no—
297C2—4242no—
298C2#1681119111yescharacteristic
299C2AGL(1,ℤ/8)×D5311yescharacteristic
300C2#1681142211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#489C2
socleC22×C53C22×D4
hypercenter#489D53
hypocenterC53#3213
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. #124×D53C2semidirect
3. #16811013C2semidirect
4. #16811012C2semidirect
5. #16810985C2semidirect
6. #16810841C2semidirect
7. #16810837C2semidirect
8. #16810763C2semidirect
9. #16810758C2semidirect
10. #16810678C2semidirect
11. C53×#489C2semidirect
15. C2×#11719913C22semidirect
16. C2×#11719924C22semidirect
19. #13133214C22semidirect
29. #13133179C22semidirect
33. #13133146C22semidirect
34. #13133140C22semidirect
37. #13133128C22semidirect
38. C4⋊Q8×C53C22semidirect
40. #103×C53C22semidirect
42. #97×C53C22semidirect
156. #489D53semidirect
162. C53#3213semidirect
170. C4⋊Q8C2×D53semidirect
188. #103C2×D53semidirect
193. #97C2×D53semidirect