Finite Group Databasebeta

← #20285519

Subgroups — #20285519

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
1C2×#13133193C211yesmaximal, characteristic, complemented
2#121×D53C211yesmaximal, characteristic
3#16810957C212yesmaximal, complemented
4#16810937C211yesmaximal, characteristic, complemented
5#16810934C211yesmaximal, characteristic, complemented
6#16810872C212yesmaximal, complemented
7#16810866C212yesmaximal, complemented
8#16810772C212yesmaximal, complemented
9#16810762C211yesmaximal, characteristic, complemented
10#16810585C211yesmaximal, characteristic, complemented
11C53×#448C211yesmaximal, characteristic, complemented
12C2×D4×D53—21no—
13C2×D4×D53C2211yescharacteristic
14C22×C4×D53C2211yescharacteristic
15C2×#11719924C2212yescomplemented
16C2×#11719912C2212yescomplemented
17#13133193—21no—
18#13133193—21no—
19#13133193C2212yes—
20#13133193C2212yescomplemented
21#13133192C2211yescharacteristic
22#13133191C2211yescharacteristic
23#13133190—21no—
24#13133189—21no—
25#13133188C2211yescharacteristic
26C2×C4⋊Dic53C2211yescharacteristic, complemented
27C2×#11719919—21no—
28C4⋊C4×D53C2211yescharacteristic
29C2×#11719921—21no—
30C22⋊C4×D53—21no—
31#13133151C2212yescomplemented
32#13133150C2212yescomplemented
33#13133141C2212yescomplemented
34#13133139C2212yescomplemented
35#13133138C2212yescomplemented
36C22⋊#88322C2211yescharacteristic, complemented
37#13133129C2212yes—
38C22×Q8×C53C2211yescharacteristic, complemented
39C2×SD16×C53C2212yescomplemented
40#121×C53C2211yescharacteristic
41#103×C53C2212yescomplemented
42C22⋊C8×C53C2211yescharacteristic, complemented
43C23×D53—21no—
44C23×D53D411yescharacteristic
45C2×Q8×C53—21no—
46C2×Q8×C53—42no—
47C2×Q8×C53C2312yes—
48C2×D4×C53—21no—
49C2×D4×C53C2311yescharacteristic
50C22×C4×C53—21no—
51C22×C4×C53C2311yescharacteristic
52C2×C22⋊D53—21no—
53C2×C22⋊D53—21no—
54C2×C22⋊D53—21no—
55C22×Dic53—21no—
56C22×Dic53D411yescharacteristic
57D4×D53—42no—
58D4×D53—44no—
59C2×D212—21no—
60C2×D212C2311yescharacteristic
61C2×C4×D53—21no—
62C2×C4×D53—42no—
63C2×C4×D53D411yescharacteristic
64C2×C4×D53D411yescharacteristic
65C2×C4×D53C2311yescharacteristic
66C2×C4×D53C2311yescharacteristic
67SD16×C53—42no—
68C2×C8×C53C2312yes—
69C4⋊C4×C53C2311yescharacteristic
70C22⋊C4×C53—21no—
71C22⋊Dic53—21no—
72#11719924—42no—
73#11719921—22no—
74#11719921—22no—
75#11719921—21no—
76#11719921—21no—
77C4⋊Dic53—21no—
78C4⋊Dic53C2312yes—
79C4⋊Dic53C2311yescharacteristic
80#11719919—21no—
81#11719919—22no—
82#11719919—22no—
83C2×#88322C2312yes—
84#11719912—42no—
85C23×C53—21no—
86C23×C53C2×D411yescharacteristic
87C22×D53—22no—
88C22×D53—22no—
89C22×D53—21no—
90C22×D53—41no—
91C22×D53—42no—
92C22×D53—42no—
93C22×D53—42no—
94C22×D53C2×D411yescharacteristic
95C22×D53C2×D411yescharacteristic
96Q8×C53—22no—
97Q8×C53—22no—
98Q8×C53—42no—
99D4×C53—22no—
100D4×C53—42no—
101C2×C4×C53—22no—
102C2×C4×C53—21no—
103C2×C4×C53—22no—
104C2×C4×C53—42no—
105C2×C4×C53C2×D411yescharacteristic
106C2×C4×C53C2411yescharacteristic
107C22⋊D53—42no—
108C22⋊D53—42no—
109C22⋊D53—42no—
110C2×Dic53—21no—
111C2×Dic53—22no—
112C2×Dic53—41no—
113C2×Dic53—42no—
114C2×Dic53C2×D411yescharacteristic
115C2×Dic53C2×D411yescharacteristic
116D212—22no—
117D212—42no—
118C4×D53—22no—
119C4×D53—22no—
120C4×D53—41no—
121C4×D53—41no—
122C4×D53—42no—
123C8×C53—22no—
124#88322—22no—
125C22×C53—21no—
126C22×C53—21no—
127C22×C53—41no—
128C22×C53—41no—
129C22×C53—41no—
130C22×C53AGL(1,ℤ/8)12yes—
131C22×C53C22×D411yescharacteristic
132C2×D53—21no—
133C2×D53—21no—
134C2×D53—21no—
135C2×D53—21no—
136C2×D53—21no—
137C2×D53—44no—
138C2×D53—41no—
139C2×D53—41no—
140C2×D53—41no—
141C2×D53—84no—
142C2×D53#13712yes—
143C4×C53—21no—
144C4×C53—42no—
145C4×C53—41no—
146C4×C53H(5,𝔽2)12yes—
147Dic53—21no—
148Dic53—21no—
149Dic53—41no—
150Dic53—42no—
151#3193—531nomaximal, Sylow
152C2×C53—22no—
153C2×C53—81no—
154C2×C53#53411yescharacteristic
155C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
156C2×C53C2×#13711yescharacteristic
157D53—22no—
158D53—41no—
159D53—81no—
160C2×C22⋊Q8—531no—
161C2×#121—531no—
162#484—532no—
163#480—531no—
164#480—531no—
165#468—532no—
166#468—532no—
167#450—532no—
168#448—531no—
169#448D5311yescharacteristic, complemented
170#407—531no—
171C53#319311yescharacteristic, complemented, Sylow
172C22×Q8C2×D5311yescharacteristic, complemented
173C22×D4—531no—
174C22×D4—1061no—
175C23×C4—531no—
176C2×SD16—532no—
177C2×SD16C2×D5312yescomplemented
178C2×M16—532no—
179C22⋊Q8—532no—
180C22⋊Q8—532no—
181C22⋊Q8—1061no—
182C22⋊Q8—1061no—
183#121—531no—
184#121—531no—
185#121—531no—
186#121—1061no—
187#121—1061no—
188#121C2×D5311yescharacteristic
189C2×C4⋊C4—531no—
190C2×C4⋊C4—531no—
191C2×C4⋊C4—1061no—
192C2×C22⋊C4—1061no—
193C2×C22⋊C4—1061no—
194#107—532no—
195#107—532no—
196#103—532no—
197#103C2×D5312yescomplemented
198#102—532no—
199#102—532no—
200C22⋊C8—531no—
201C22⋊C8—532no—
202C22⋊C8C2×D5311yescharacteristic, complemented
203C24—531no—
204C24—1061no—
205C2×Q8—21no—
206C2×Q8—42no—
207C2×Q8C22×D5312yes—
208C2×D4—21no—
209C2×D4—531no—
210C2×D4—1061no—
211C2×D4—1061no—
212C2×D4—1061no—
213C2×D4—1061no—
214C2×D4—2122no—
215C2×D4—2124no—
216C2×D4C22×D5311yescharacteristic
217C22×C4—21no—
218C22×C4—531no—
219C22×C4—531no—
220C22×C4—531no—
221C22×C4—531no—
222C22×C4—531no—
223C22×C4—1061no—
224C22×C4—1061no—
225C22×C4—2122no—
226C22×C4C22×D5311yescharacteristic
227SD16—42no—
228SD16—2122no—
229M16—2122no—
230C2×C8—532no—
231C2×C8C22×D5312yes—
232C4⋊C4—532no—
233C4⋊C4—531no—
234C4⋊C4—1061no—
235C4⋊C4—1062no—
236C4⋊C4—1062no—
237C4⋊C4—1061no—
238C4⋊C4C22×D5311yescharacteristic
239C22⋊C4—21no—
240C22⋊C4—1061no—
241C22⋊C4—1062no—
242C22⋊C4—1062no—
243C22⋊C4—1061no—
244C22⋊C4—1061no—
245C23—21no—
246C23—531no—
247C23—531no—
248C23—1062no—
249C23—1062no—
250C23—1061no—
251C23—2121no—
252C23—2122no—
253C23—2122no—
254C23—2122no—
255C23D4×D5311yescharacteristic
256Q8—22no—
257Q8—22no—
258Q8—42no—
259D4—22no—
260D4—42no—
261D4—1062no—
262D4—2122no—
263D4—2122no—
264D4—2122no—
265D4—2122no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—42no—
270C2×C4—531no—
271C2×C4—531no—
272C2×C4—1062no—
273C2×C4—1062no—
274C2×C4—1062no—
275C2×C4—1061no—
276C2×C4—2121no—
277C2×C4—2121no—
278C2×C4—2122no—
279C2×C4—2121no—
280C2×C4—2122no—
281C2×C4D4×D5311yescharacteristic
282C2×C4C23×D5311yescharacteristic
283C8—22no—
284C8—1062no—
285C22—21no—
286C22—21no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—532no—
291C22—1061no—
292C22—1061no—
293C22—1061no—
294C22—1061no—
295C22—1061no—
296C22—4244no—
297C22—2124no—
298C22—2121no—
299C22—2121no—
300C22—2121no—
301C22#1313324412yes—
302C22C2×D4×D5311yescharacteristic
303C4—21no—
304C4—41no—
305C4—42no—
306C4—1061no—
307C4—1061no—
308C4—2122no—
309C4—2121no—
310C4#1313326212yes—
311C2—22no—
312C2—81no—
313C2—1062no—
314C2—4241no—
315C2—2121no—
316C2#1681118311yescharacteristic
317C2C2×#1313324411yescharacteristic
318C2#137×D5311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#448C2
socleC22×C53C22×D4
hypercenter#448D53
hypocenterC53#3193
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C2×#13133193C2semidirect
3. #16810957C2semidirect
4. #16810937C2semidirect
5. #16810934C2semidirect
6. #16810872C2semidirect
7. #16810866C2semidirect
8. #16810772C2semidirect
9. #16810762C2semidirect
10. #16810585C2semidirect
11. C53×#448C2semidirect
15. C2×#11719924C22semidirect
16. C2×#11719912C22semidirect
20. #13133193C22semidirect
26. C2×C4⋊Dic53C22semidirect
31. #13133151C22semidirect
32. #13133150C22semidirect
33. #13133141C22semidirect
34. #13133139C22semidirect
35. #13133138C22semidirect
36. C22⋊#88322C22semidirect
38. C22×Q8×C53C22semidirect
39. C2×SD16×C53C22semidirect
41. #103×C53C22semidirect
42. C22⋊C8×C53C22semidirect
169. #448D53semidirect
171. C53#3193semidirect
172. C22×Q8C2×D53semidirect
177. C2×SD16C2×D53semidirect
197. #103C2×D53semidirect
202. C22⋊C8C2×D53semidirect