Finite Group Databasebeta

← #20285518

Subgroups — #20285518

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×#13133188C211yesmaximal, characteristic
3#16810959C211yesmaximal, characteristic, complemented
4#16810956C211yesmaximal, characteristic, complemented
5#16810935C212yesmaximal, complemented
6#16810868C212yesmaximal, complemented
7#16810866C212yesmaximal, complemented
8#16810774C211yesmaximal, characteristic, complemented
9#16810760C212yesmaximal, complemented
10#16810585C211yesmaximal, characteristic, complemented
11C53×#450C211yesmaximal, characteristic, complemented
12C2×Q8×D53C2211yescharacteristic, complemented
13C22×C22⋊D53—21no—
14C22×C4×D53C2211yescharacteristic
15C2×#11719923C2212yescomplemented
16C2×#11719912C2212yescomplemented
17#13133197C2211yescharacteristic, complemented
18#13133196C2211yescharacteristic
19#13133195—21no—
20#13133194—21no—
21#13133193C2211yescharacteristic, complemented
22#13133188—21no—
23#13133188—21no—
24#13133188C2212yes—
25#13133188C2212yes—
26C2×C4⋊Dic53C2211yescharacteristic
27C4⋊C4×D53—21no—
28C4⋊C4×D53C2211yescharacteristic, complemented
29C2×C22⋊Dic53—21no—
30C22⋊C4×D53—21no—
31#13133151C2212yescomplemented
32#13133150C2212yescomplemented
33#13133142C2212yescomplemented
34#13133141C2212yescomplemented
35#13133138C2212yescomplemented
36C22⋊#88322C2211yescharacteristic
37#13133129C2212yescomplemented
38C22×D4×C53C2211yescharacteristic
39C2×SD16×C53C2212yescomplemented
40C22⋊Q8×C53C2211yescharacteristic
41#102×C53C2212yescomplemented
42C22⋊C8×C53C2211yescharacteristic
43C24×C53—21no—
44C23×D53D411yescharacteristic
45C2×Q8×C53C2311yescharacteristic
46C2×D4×C53—21no—
47C2×D4×C53—42no—
48C2×D4×C53C2312yes—
49C22×C4×C53C2311yescharacteristic
50C2×C22⋊D53—22no—
51C2×C22⋊D53—22no—
52C2×C22⋊D53—44no—
53C22×Dic53—21no—
54C22×Dic53D411yescharacteristic
55Q8×D53—42no—
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
65SD16×C53—42no—
66C2×C8×C53C2312yes—
67C4⋊C4×C53—21no—
68C4⋊C4×C53C2311yescharacteristic
69C22⋊C4×C53—21no—
70C22⋊Dic53—21no—
71C22⋊Dic53—22no—
72C22⋊Dic53—22no—
73#11719923—42no—
74#11719921—21no—
75#11719921—21no—
76C4⋊Dic53—21no—
77C4⋊Dic53—21no—
78C4⋊Dic53C2311yescharacteristic
79C4⋊Dic53C2312yes—
80#11719919—21no—
81#11719919—21no—
82#11719919—21no—
83C2×#88322C2312yes—
84#11719912—42no—
85C23×C53—22no—
86C23×C53—42no—
87C23×C53—42no—
88C23×C53—42no—
89C23×C53C2×D411yescharacteristic
90C22×D53—22no—
91C22×D53—22no—
92C22×D53—41no—
93C22×D53C2×D411yescharacteristic
94C22×D53C2×D411yescharacteristic
95Q8×C53—22no—
96D4×C53—22no—
97D4×C53—22no—
98D4×C53—42no—
99C2×C4×C53—22no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53C2×D411yescharacteristic
103C2×C4×C53C2411yescharacteristic
104C22⋊D53—44no—
105C22⋊D53—44no—
106C2×Dic53—21no—
107C2×Dic53—21no—
108C2×Dic53—22no—
109C2×Dic53—41no—
110C2×Dic53—42no—
111C2×Dic53C2×D411yescharacteristic
112C2×Dic53C2×D411yescharacteristic
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—42no—
126C22×C53—42no—
127C22×C53—42no—
128C22×C53—84no—
129C22×C53#13712yes—
130C22×C53C22×D411yescharacteristic
131C2×D53—21no—
132C2×D53—21no—
133C2×D53—21no—
134C2×D53—21no—
135C2×D53—21no—
136C2×D53—44no—
137C2×D53AGL(1,ℤ/8)12yes—
138C4×C53—21no—
139C4×C53—41no—
140C4×C53—41no—
141C4×C53H(5,𝔽2)12yes—
142Dic53—21no—
143Dic53—21no—
144Dic53—41no—
145Dic53—41no—
146Dic53—42no—
147#3193—531nomaximal, Sylow
148C2×C53—22no—
149C2×C53—82no—
150C2×C53#53411yescharacteristic
151C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
152C2×C53C2×#13711yescharacteristic
153D53—22no—
154D53—41no—
155C2×C22⋊Q8—531no—
156C2×#121—531no—
157#484—531no—
158#484—531no—
159#480—532no—
160#468—532no—
161#468—532no—
162#450—531no—
163#450D5311yescharacteristic, complemented
164#448—532no—
165#407—531no—
166C53#319311yescharacteristic, complemented, Sylow
167C22×Q8—531no—
168C22×D4—1061no—
169C22×D4C2×D5311yescharacteristic
170C23×C4—531no—
171C2×SD16—532no—
172C2×SD16C2×D5312yescomplemented
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—531no—
186C2×C4⋊C4—1061no—
187C2×C22⋊C4—1061no—
188C2×C22⋊C4—1061no—
189#107—532no—
190#107—532no—
191#103—532no—
192#103—532no—
193#102—532no—
194#102C2×D5312yescomplemented
195C22⋊C8—531no—
196C22⋊C8—532no—
197C22⋊C8C2×D5311yescharacteristic
198C24—21no—
199C24—531no—
200C2×Q8—531no—
201C2×Q8—1061no—
202C2×Q8—2122no—
203C2×Q8C22×D5311yescharacteristic
204C2×D4—21no—
205C2×D4—42no—
206C2×D4—1062no—
207C2×D4—1062no—
208C2×D4—2124no—
209C2×D4C22×D5312yes—
210C22×C4—531no—
211C22×C4—531no—
212C22×C4—531no—
213C22×C4—531no—
214C22×C4—531no—
215C22×C4—1061no—
216C22×C4—1061no—
217C22×C4—1061no—
218C22×C4—2122no—
219C22×C4C22×D5311yescharacteristic
220SD16—42no—
221SD16—2122no—
222M16—2122no—
223C2×C8—532no—
224C2×C8C22×D5312yes—
225C4⋊C4—21no—
226C4⋊C4—531no—
227C4⋊C4—532no—
228C4⋊C4—1061no—
229C4⋊C4—1061no—
230C4⋊C4—1061no—
231C4⋊C4—1061no—
232C4⋊C4—1061no—
233C4⋊C4C22×D5311yescharacteristic
234C22⋊C4—21no—
235C22⋊C4—1061no—
236C22⋊C4—1061no—
237C22⋊C4—1062no—
238C22⋊C4—1062no—
239C22⋊C4—1061no—
240C23—22no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—531no—
245C23—531no—
246C23—1062no—
247C23—1062no—
248C23—2121no—
249C23D4×D5311yescharacteristic
250Q8—22no—
251Q8—1062no—
252Q8—2122no—
253D4—22no—
254D4—22no—
255D4—42no—
256D4—2124no—
257D4—2124no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—531no—
262C2×C4—531no—
263C2×C4—1062no—
264C2×C4—1062no—
265C2×C4—1061no—
266C2×C4—1062no—
267C2×C4—1061no—
268C2×C4—2121no—
269C2×C4—2121no—
270C2×C4—2122no—
271C2×C4—2122no—
272C2×C4—2121no—
273C2×C4—2122no—
274C2×C4D4×D5311yescharacteristic
275C2×C4C23×D5311yescharacteristic
276C8—22no—
277C8—1062no—
278C22—21no—
279C22—21no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—84no—
284C22—532no—
285C22—1061no—
286C22—1061no—
287C22—1061no—
288C22—1061no—
289C22—1061no—
290C22—2124no—
291C22#1313324612yes—
292C22C2×D4×D5311yescharacteristic
293C4—21no—
294C4—41no—
295C4—41no—
296C4—1061no—
297C4—1061no—
298C4—2121no—
299C4—2122no—
300C4—2121no—
301C4#1313326212yes—
302C2—22no—
303C2—82no—
304C2—1062no—
305C2—2121no—
306C2#1681118311yescharacteristic
307C2AGL(1,ℤ/8)×D5311yescharacteristic
308C2C2×#1313324611yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#450C2
socleC22×C53C22×D4
hypercenter#450D53
hypocenterC53#3193
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C22⋊Q8×D53C2semidirect
3. #16810959C2semidirect
4. #16810956C2semidirect
5. #16810935C2semidirect
6. #16810868C2semidirect
7. #16810866C2semidirect
8. #16810774C2semidirect
9. #16810760C2semidirect
10. #16810585C2semidirect
11. C53×#450C2semidirect
12. C2×Q8×D53C22semidirect
15. C2×#11719923C22semidirect
16. C2×#11719912C22semidirect
17. #13133197C22semidirect
21. #13133193C22semidirect
28. C4⋊C4×D53C22semidirect
31. #13133151C22semidirect
32. #13133150C22semidirect
33. #13133142C22semidirect
34. #13133141C22semidirect
35. #13133138C22semidirect
37. #13133129C22semidirect
39. C2×SD16×C53C22semidirect
41. #102×C53C22semidirect
163. #450D53semidirect
166. C53#3193semidirect
172. C2×SD16C2×D53semidirect
194. #102C2×D53semidirect