Finite Group Databasebeta

← #20285520

Subgroups — #20285520

Order 6784.

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

normal subgroups only

This group has 462 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
1C22⋊Q8×D53C211yesmaximal, characteristic, complemented
2C2×#13133192C211yesmaximal, characteristic
3#16810958C211yesmaximal, characteristic, complemented
4#16810938C212yesmaximal, complemented
5#16810872C212yesmaximal, complemented
6#16810870C212yesmaximal, complemented
7#16810776C211yesmaximal, characteristic, complemented
8#16810773C211yesmaximal, characteristic, complemented
9#16810763C212yesmaximal, complemented
10#16810585C211yesmaximal, characteristic, complemented
11C53×#484C211yesmaximal, characteristic, complemented
12C2×Q8×D53C2211yescharacteristic, complemented
13C22×D212—21no—
14C22×D212C2211yescharacteristic
15C22×C4×D53C2211yescharacteristic
16C2×#11719913C2212yescomplemented
17C2×#11719924C2212yescomplemented
18C2×#11719912C2212yescomplemented
19#13133197C2211yescharacteristic, complemented
20#13133196C2211yescharacteristic
21#13133195—21no—
22#13133194—21no—
23#13133193C2211yescharacteristic, complemented
24#13133192—21no—
25#13133192—21no—
26#13133192C2212yes—
27#13133192C2212yes—
28C4⋊C4×D53—21no—
29C4⋊C4×D53C2211yescharacteristic, complemented
30C2×#11719921—21no—
31C22⋊C4×D53—21no—
32#13133150C2212yescomplemented
33#13133140C2212yescomplemented
34#13133139C2212yescomplemented
35#13133137C2212yescomplemented
36C22⋊#88322C2211yescharacteristic
37#13133129C2212yescomplemented
38C22⋊Q8×C53C2211yescharacteristic
39C2×C4⋊C4×C53C2211yescharacteristic
40#107×C53C2212yescomplemented
41#103×C53C2212yescomplemented
42C22⋊C8×C53C2211yescharacteristic
43C23×D53—21no—
44C23×D53D411yescharacteristic
45C2×Q8×C53C2311yescharacteristic
46C22×C4×C53—21no—
47C22×C4×C53C2311yescharacteristic
48C22×Dic53D411yescharacteristic
49Q8×D53—42no—
50C2×D212—22no—
51C2×D212—22no—
52C2×D212—21no—
53C2×D212—42no—
54C2×D212—44no—
55C2×D212C2312yes—
56C2×C4×D53—21no—
57C2×C4×D53—21no—
58C2×C4×D53—42no—
59C2×C4×D53D411yescharacteristic
60C2×C4×D53D411yescharacteristic
61C2×C4×D53C2311yescharacteristic
62C2×C4×D53C2311yescharacteristic
63C2×Dic106—21no—
64C2×Dic106C2311yescharacteristic
65C2×C8×C53C2312yes—
66C4⋊C4×C53—21no—
67C4⋊C4×C53—21no—
68C4⋊C4×C53C2312yes—
69C4⋊C4×C53C2311yescharacteristic
70C22⋊C4×C53—21no—
71C22⋊Dic53—21no—
72#11719924—42no—
73#11719921—21no—
74#11719921—21no—
75#11719921—22no—
76#11719921—22no—
77C4⋊Dic53—21no—
78C4⋊Dic53C2311yescharacteristic
79#11719919—21no—
80#11719919—21no—
81#11719919—21no—
82C2×#88322C2312yes—
83#11719913—42no—
84#11719912—42no—
85C23×C53C2×D411yescharacteristic
86C22×D53—22no—
87C22×D53—22no—
88C22×D53—22no—
89C22×D53—41no—
90C22×D53—42no—
91C22×D53—42no—
92C22×D53—42no—
93C22×D53C2×D411yescharacteristic
94C22×D53C2×D411yescharacteristic
95Q8×C53—22no—
96C2×C4×C53—22no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—22no—
100C2×C4×C53—42no—
101C2×C4×C53C2×D411yescharacteristic
102C2×C4×C53C2411yescharacteristic
103C2×Dic53—21no—
104C2×Dic53—21no—
105C2×Dic53—41no—
106C2×Dic53C2×D411yescharacteristic
107C2×Dic53C2×D411yescharacteristic
108D212—22no—
109D212—22no—
110D212—42no—
111D212—44no—
112D212—44no—
113C4×D53—22no—
114C4×D53—22no—
115C4×D53—41no—
116C4×D53—41no—
117C4×D53—42no—
118C4×D53—42no—
119Dic106—22no—
120Dic106—42no—
121C8×C53—22no—
122#88322—22no—
123C22×C53—21no—
124C22×C53—21no—
125C22×C53#13712yes—
126C22×C53C22×D411yescharacteristic
127C2×D53—21no—
128C2×D53—21no—
129C2×D53—21no—
130C2×D53—21no—
131C2×D53—21no—
132C2×D53—44no—
133C2×D53—42no—
134C2×D53—42no—
135C2×D53—42no—
136C2×D53—84no—
137C2×D53AGL(1,ℤ/8)12yes—
138C4×C53—21no—
139C4×C53—41no—
140C4×C53—41no—
141C4×C53—42no—
142C4×C53H(5,𝔽2)12yes—
143Dic53—21no—
144Dic53—21no—
145Dic53—41no—
146Dic53—41no—
147#3193—531nomaximal, Sylow
148C2×C53—22no—
149C2×C53#53411yescharacteristic
150C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
151C2×C53C2×#13711yescharacteristic
152D53—22no—
153D53—41no—
154D53—82no—
155C2×C22⋊Q8—531no—
156C2×#121—531no—
157#484—531no—
158#484D5311yescharacteristic, complemented
159#480—532no—
160#468—532no—
161#468—532no—
162#450—531no—
163#450—531no—
164#448—532no—
165#407—531no—
166C53#319311yescharacteristic, complemented, Sylow
167C22×Q8—531no—
168C22×D4—531no—
169C22×D4—1061no—
170C23×C4—531no—
171C2×SD16—532no—
172C2×SD16—532no—
173C2×M16—532no—
174C22⋊Q8—531no—
175C22⋊Q8—531no—
176C22⋊Q8—531no—
177C22⋊Q8—1061no—
178C22⋊Q8—1061no—
179C22⋊Q8C2×D5311yescharacteristic
180#121—532no—
181#121—532no—
182#121—1061no—
183#121—1061no—
184C2×C4⋊C4—531no—
185C2×C4⋊C4—1061no—
186C2×C4⋊C4C2×D5311yescharacteristic
187C2×C22⋊C4—1061no—
188C2×C22⋊C4—1061no—
189#107—532no—
190#107C2×D5312yescomplemented
191#103—532no—
192#103C2×D5312yescomplemented
193#102—532no—
194#102—532no—
195C22⋊C8—531no—
196C22⋊C8—532no—
197C22⋊C8C2×D5311yescharacteristic
198C24—531no—
199C24—1061no—
200C2×Q8—531no—
201C2×Q8—1061no—
202C2×Q8—2122no—
203C2×Q8C22×D5311yescharacteristic
204C2×D4—532no—
205C2×D4—1062no—
206C2×D4—1062no—
207C2×D4—1061no—
208C2×D4—2122no—
209C2×D4—2124no—
210C22×C4—21no—
211C22×C4—531no—
212C22×C4—531no—
213C22×C4—531no—
214C22×C4—531no—
215C22×C4—531no—
216C22×C4—1061no—
217C22×C4—1061no—
218C22×C4—2122no—
219C22×C4C22×D5311yescharacteristic
220SD16—2122no—
221SD16—2122no—
222M16—2122no—
223C2×C8—532no—
224C2×C8C22×D5312yes—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—531no—
228C4⋊C4—1061no—
229C4⋊C4—1061no—
230C4⋊C4—1061no—
231C4⋊C4—1061no—
232C4⋊C4C22×D5311yescharacteristic
233C4⋊C4C22×D5312yes—
234C22⋊C4—21no—
235C22⋊C4—1062no—
236C22⋊C4—1062no—
237C22⋊C4—1061no—
238C22⋊C4—1061no—
239C22⋊C4—1061no—
240C23—531no—
241C23—531no—
242C23—1062no—
243C23—1062no—
244C23—1062no—
245C23—2121no—
246C23—2122no—
247C23—2122no—
248C23—2122no—
249C23D4×D5311yescharacteristic
250Q8—22no—
251Q8—1062no—
252Q8—2122no—
253D4—1062no—
254D4—1062no—
255D4—2122no—
256D4—2124no—
257D4—2124no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—42no—
263C2×C4—531no—
264C2×C4—531no—
265C2×C4—1062no—
266C2×C4—1062no—
267C2×C4—1061no—
268C2×C4—1061no—
269C2×C4—2121no—
270C2×C4—2121no—
271C2×C4—2122no—
272C2×C4—2122no—
273C2×C4—2121no—
274C2×C4D4×D5311yescharacteristic
275C2×C4C23×D5311yescharacteristic
276C8—22no—
277C8—1062no—
278C22—21no—
279C22—21no—
280C22—532no—
281C22—1061no—
282C22—1061no—
283C22—1061no—
284C22—1061no—
285C22—1061no—
286C22—4244no—
287C22—2124no—
288C22—2122no—
289C22—2122no—
290C22—2122no—
291C22#1313324912yes—
292C22C2×D4×D5311yescharacteristic
293C4—21no—
294C4—42no—
295C4—41no—
296C4—41no—
297C4—1061no—
298C4—1061no—
299C4—2121no—
300C4—2121no—
301C4#1313326312yes—
302C2—22no—
303C2—1062no—
304C2—4242no—
305C2—2121no—
306C2#1681118411yescharacteristic
307C2AGL(1,ℤ/8)×D5311yescharacteristic
308C2C2×#1313324911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#484C2
socleC22×C53C22×D4
hypercenter#484D53
hypocenterC53#3193
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
1. C22⋊Q8×D53C2semidirect
3. #16810958C2semidirect
4. #16810938C2semidirect
5. #16810872C2semidirect
6. #16810870C2semidirect
7. #16810776C2semidirect
8. #16810773C2semidirect
9. #16810763C2semidirect
10. #16810585C2semidirect
11. C53×#484C2semidirect
12. C2×Q8×D53C22semidirect
16. C2×#11719913C22semidirect
17. C2×#11719924C22semidirect
18. C2×#11719912C22semidirect
19. #13133197C22semidirect
23. #13133193C22semidirect
29. C4⋊C4×D53C22semidirect
32. #13133150C22semidirect
33. #13133140C22semidirect
34. #13133139C22semidirect
35. #13133137C22semidirect
37. #13133129C22semidirect
40. #107×C53C22semidirect
41. #103×C53C22semidirect
158. #484D53semidirect
166. C53#3193semidirect
190. #107C2×D53semidirect
192. #103C2×D53semidirect