Finite Group Databasebeta

← #20285731

Subgroups — #20285731

Order 6784.

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

normal subgroups only

This group has 490 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
1C2×#13133213C211yesmaximal, characteristic, complemented
2#16811154C211yesmaximal, characteristic, complemented
3#16811011C212yesmaximal, complemented
4#16810986C212yesmaximal, complemented
5#16810822C212yesmaximal, complemented
6#16810816C212yesmaximal, complemented
7#16810776C211yesmaximal, characteristic, complemented
8#16810774C211yesmaximal, characteristic, complemented
9#16810756C211yesmaximal, characteristic, complemented
10#16810677C211yesmaximal, characteristic, complemented
11C53×#447C211yesmaximal, characteristic, complemented
12C22×D212C2211yescharacteristic
13C22×C22⋊D53—21no—
14C22×C22⋊D53—21no—
15C2×#11719913C2212yescomplemented
16C2×#11719923C2212yescomplemented
17C2×#11719922C2212yescomplemented
18#13133213—21no—
19#13133213—21no—
20#13133213C2212yes—
21#13133213C2212yes—
22#13133210—21no—
23#13133208C2211yescharacteristic, complemented
24#13133204C2211yescharacteristic, complemented
25#13133198—21no—
26#13133196C2211yescharacteristic
27#13133182C2211yescharacteristic, complemented
28D4×Dic53—21no—
29D4×Dic53C2211yescharacteristic, complemented
30#13133173—21no—
31C2×C4×Dic53C2211yescharacteristic
32#13133146C2212yescomplemented
33#13133139C2212yescomplemented
34#13133138C2212yescomplemented
35#13133137C2212yescomplemented
36C22⋊#88322C2211yescharacteristic
37#13133127C2212yescomplemented
38C22×D4×C53C2211yescharacteristic
39C2×D8×C53C2212yescomplemented
40#121×C53C2211yescharacteristic
41#102×C53C2212yescomplemented
42C22⋊C8×C53C2211yescharacteristic
43C24×C53—21no—
44C23×D53—21no—
45C2×D4×C53—21no—
46C2×D4×C53—21no—
47C2×D4×C53—42no—
48C2×D4×C53C2312yes—
49C2×D4×C53C2311yescharacteristic
50C22×C4×C53C2311yescharacteristic
51C2×C22⋊D53—22no—
52C2×C22⋊D53—22no—
53C2×C22⋊D53—22no—
54C2×C22⋊D53—22no—
55C2×C22⋊D53—44no—
56C2×C22⋊D53—44no—
57C22×Dic53—21no—
58C22×Dic53D411yescharacteristic
59C22×Dic53D411yescharacteristic
60C2×D212—21no—
61C2×D212—42no—
62C2×D212C2312yes—
63C2×Dic106C2311yescharacteristic
64D8×C53—42no—
65C2×C8×C53C2312yes—
66C4⋊C4×C53C2311yescharacteristic
67C22⋊C4×C53—21no—
68C22⋊Dic53—21no—
69C22⋊Dic53—21no—
70C22⋊Dic53—21no—
71C22⋊Dic53—21no—
72#11719923—42no—
73#11719922—42no—
74C4⋊Dic53—21no—
75C4⋊Dic53C2311yescharacteristic
76#11719919—21no—
77#11719919—21no—
78#11719919—21no—
79C4×Dic53—21no—
80C4×Dic53D411yescharacteristic
81C4×Dic53D411yescharacteristic
82C4×Dic53C2311yescharacteristic
83C4×Dic53C2311yescharacteristic
84C2×#88322C2312yes—
85#11719913—42no—
86C23×C53—21no—
87C23×C53—22no—
88C23×C53—42no—
89C23×C53—42no—
90C23×C53—42no—
91C23×C53C2×D411yescharacteristic
92C22×D53—22no—
93C22×D53—42no—
94C22×D53—42no—
95C22×D53—42no—
96D4×C53—22no—
97D4×C53—22no—
98D4×C53—22no—
99D4×C53—42no—
100D4×C53—42no—
101C2×C4×C53—22no—
102C2×C4×C53—21no—
103C2×C4×C53C2×D411yescharacteristic
104C2×C4×C53C2411yescharacteristic
105C22⋊D53—44no—
106C22⋊D53—44no—
107C22⋊D53—44no—
108C22⋊D53—44no—
109C2×Dic53—22no—
110C2×Dic53—22no—
111C2×Dic53—21no—
112C2×Dic53—21no—
113C2×Dic53—42no—
114C2×Dic53C2×D411yescharacteristic
115C2×Dic53C2×D411yescharacteristic
116C2×Dic53C2×D411yescharacteristic
117C2×Dic53C2×D411yescharacteristic
118D212—22no—
119D212—22no—
120D212—42no—
121Dic106—22no—
122C8×C53—22no—
123#88322—22no—
124C22×C53—21no—
125C22×C53—21no—
126C22×C53—42no—
127C22×C53—41no—
128C22×C53—41no—
129C22×C53—42no—
130C22×C53—42no—
131C22×C53—41no—
132C22×C53—84no—
133C22×C53AGL(1,ℤ/8)12yes—
134C22×C53C22×D411yescharacteristic
135C2×D53—42no—
136C2×D53—42no—
137C2×D53—42no—
138C2×D53—84no—
139C4×C53—21no—
140C4×C53—41no—
141C4×C53H(5,𝔽2)12yes—
142Dic53—21no—
143Dic53—21no—
144Dic53—21no—
145Dic53—41no—
146Dic53—41no—
147Dic53AGL(1,ℤ/8)12yes—
148#3217—531nomaximal, Sylow
149C2×C53—22no—
150C2×C53—81no—
151C2×C53—82no—
152C2×C53#53511yescharacteristic
153C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
154C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
155D53—82no—
156C2×#127—531no—
157#529—531no—
158#496—532no—
159#490—532no—
160#460—532no—
161#459—532no—
162#450—531no—
163#450—531no—
164#447—531no—
165#447D5311yescharacteristic, complemented
166#432—531no—
167C53#321711yescharacteristic, complemented, Sylow
168C22×D4—531no—
169C22×D4—1061no—
170C22×D4—1061no—
171C22×D4C2×D5311yescharacteristic
172C2×SD16—532no—
173C2×SD16—532no—
174C2×D8—532no—
175C2×D8C2×D5312yescomplemented
176#127—532no—
177#127—532no—
178#127—1061no—
179#127—1061no—
180#126—1061no—
181#125—531no—
182#124—531no—
183#123—1061no—
184C22⋊Q8—531no—
185#121C2×D5311yescharacteristic
186C4×Q8—531no—
187C4×D4—531no—
188C4×D4—1061no—
189#117—1061no—
190C2×C42—531no—
191C4⋊C8—532no—
192#102—532no—
193#102—532no—
194#102—532no—
195#102C2×D5312yescomplemented
196C22⋊C8—531no—
197C22⋊C8C2×D5311yescharacteristic
198#97—532no—
199C24—21no—
200C24—1061no—
201C2×Q8—531no—
202C2×D4—21no—
203C2×D4—21no—
204C2×D4—42no—
205C2×D4—532no—
206C2×D4—1062no—
207C2×D4—1062no—
208C2×D4—1062no—
209C2×D4—1062no—
210C2×D4—1061no—
211C2×D4—2122no—
212C2×D4—2124no—
213C2×D4—2124no—
214C2×D4C22×D5312yes—
215C2×D4C22×D5311yescharacteristic
216C22×C4—531no—
217C22×C4—531no—
218C22×C4—1061no—
219C22×C4C22×D5311yescharacteristic
220SD16—2122no—
221SD16—2122no—
222D8—42no—
223D8—2122no—
224C2×C8—532no—
225C2×C8C22×D5312yes—
226C4⋊C4—531no—
227C4⋊C4—1061no—
228C4⋊C4—1061no—
229C4⋊C4—1061no—
230C4⋊C4—1061no—
231C4⋊C4C22×D5311yescharacteristic
232C22⋊C4—21no—
233C22⋊C4—1061no—
234C22⋊C4—1061no—
235C22⋊C4—1061no—
236C22⋊C4—1061no—
237C42—531no—
238C42—531no—
239C42—531no—
240C42—531no—
241C42—1061no—
242C23—21no—
243C23—22no—
244C23—42no—
245C23—42no—
246C23—42no—
247C23—1062no—
248C23—2122no—
249C23—2122no—
250C23—2122no—
251C23D4×D5311yescharacteristic
252Q8—1062no—
253D4—22no—
254D4—22no—
255D4—22no—
256D4—42no—
257D4—42no—
258D4—1062no—
259D4—1062no—
260D4—2122no—
261D4—2124no—
262D4—2124no—
263D4—2124no—
264D4—2124no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—531no—
268C2×C4—531no—
269C2×C4—531no—
270C2×C4—531no—
271C2×C4—1062no—
272C2×C4—1062no—
273C2×C4—1061no—
274C2×C4—1061no—
275C2×C4—2122no—
276C2×C4D4×D5311yescharacteristic
277C2×C4C23×D5311yescharacteristic
278C8—22no—
279C8—1062no—
280C22—21no—
281C22—21no—
282C22—41no—
283C22—42no—
284C22—41no—
285C22—42no—
286C22—42no—
287C22—41no—
288C22—84no—
289C22—4244no—
290C22—2122no—
291C22—2122no—
292C22—2122no—
293C22#1313324112yes—
294C22C2×D4×D5311yescharacteristic
295C4—21no—
296C4—41no—
297C4—532no—
298C4—1061no—
299C4—1061no—
300C4—1061no—
301C4—2121no—
302C4—2121no—
303C4#1313326212yes—
304C2—22no—
305C2—82no—
306C2—81no—
307C2—4242no—
308C2#1681118811yescharacteristic
309C2AGL(1,ℤ/8)×D5311yescharacteristic
310C2C2×#1313324111yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#447C2
socleC22×C53C22×D4
hypercenter#447D53
hypocenterC53#3217
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. C2×#13133213C2semidirect
2. #16811154C2semidirect
3. #16811011C2semidirect
4. #16810986C2semidirect
5. #16810822C2semidirect
6. #16810816C2semidirect
7. #16810776C2semidirect
8. #16810774C2semidirect
9. #16810756C2semidirect
10. #16810677C2semidirect
11. C53×#447C2semidirect
15. C2×#11719913C22semidirect
16. C2×#11719923C22semidirect
17. C2×#11719922C22semidirect
23. #13133208C22semidirect
24. #13133204C22semidirect
27. #13133182C22semidirect
29. D4×Dic53C22semidirect
32. #13133146C22semidirect
33. #13133139C22semidirect
34. #13133138C22semidirect
35. #13133137C22semidirect
37. #13133127C22semidirect
39. C2×D8×C53C22semidirect
41. #102×C53C22semidirect
165. #447D53semidirect
167. C53#3217semidirect
175. C2×D8C2×D53semidirect
195. #102C2×D53semidirect