Finite Group Databasebeta

← #20285531

Subgroups — #20285531

Order 6784.

466 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 121 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#16811134C211yesmaximal, characteristic, complemented
2C2×#13133191C211yesmaximal, characteristic, complemented
3#16810954C211yesmaximal, characteristic, complemented
4#16810948C212yesmaximal, complemented
5#16810937C211yesmaximal, characteristic, complemented
6#16810876C212yesmaximal, complemented
7#16810869C212yesmaximal, complemented
8#16810776C211yesmaximal, characteristic, complemented
9#16810770C212yesmaximal, complemented
10#16810759C211yesmaximal, characteristic, complemented
11C53×#407C211yesmaximal, characteristic, complemented
12C2×#11719945C2211yescharacteristic, complemented
13C22×D212C2211yescharacteristic
14C22×C22⋊D53—21no—
15C2×#11719913C2212yescomplemented
16C2×D424C2212yescomplemented
17#13133200—21no—
18#13133196C2211yescharacteristic
19#13133196C2211yescharacteristic, complemented
20#13133191—21no—
21#13133191—21no—
22#13133191C2211yescharacteristic
23#13133191C2212yes—
24#13133191C2212yes—
25#13133191C2211yescharacteristic, complemented
26#13133186—21no—
27C4×C22⋊D53—21no—
28C4×C22⋊D53—21no—
29#13133176C2211yescharacteristic, complemented
30C2×C4⋊Dic53C2211yescharacteristic
31C2×#11719921—21no—
32#13133151C2212yescomplemented
33#13133149C2212yescomplemented
34#13133142C2212yescomplemented
35#13133139C2212yescomplemented
36#13133139C2212yescomplemented
37#13133139C2212yescomplemented
38C23×C4×C53C2211yescharacteristic
39C2×M16×C53C2212yescomplemented
40C22⋊C8×C53C2211yescharacteristic
41C22⋊C8×C53C2211yescharacteristic
42C22⋊C8×C53C2212yescomplemented
43C24×C53D411yescharacteristic
44C23×D53—21no—
45C22×C4×C53—42no—
46C22×C4×C53D411yescharacteristic
47C22×C4×C53D411yescharacteristic
48C22×C4×C53D411yescharacteristic
49C22×C4×C53C2311yescharacteristic
50C22×C4×C53C2311yescharacteristic
51C22×C4×C53C2311yescharacteristic
52C2×C22⋊D53—21no—
53C2×C22⋊D53—21no—
54C2×C22⋊D53—21no—
55C2×C22⋊D53—22no—
56C2×C22⋊D53—22no—
57C2×C22⋊D53—44no—
58C22×Dic53—21no—
59#11719945—42no—
60C2×D212—21no—
61C2×D212—42no—
62C2×D212C2312yes—
63C2×D212C2311yescharacteristic
64C2×C4×D53—21no—
65C2×Dic106C2311yescharacteristic
66M16×C53—42no—
67C2×C8×C53C2312yes—
68C2×C8×C53C2312yes—
69C22⋊Dic53—21no—
70C22⋊Dic53—21no—
71C22⋊Dic53—21no—
72#11719921—22no—
73#11719921—22no—
74#11719921—21no—
75#11719921—21no—
76C4⋊Dic53—21no—
77C4⋊Dic53C2311yescharacteristic
78C4⋊Dic53C2311yescharacteristic
79C4⋊Dic53C2312yes—
80#11719919—21no—
81#11719919—21no—
82C4×Dic53—21no—
83D424—42no—
84#11719913—42no—
85C23×C53—22no—
86C23×C53—41no—
87C23×C53—41no—
88C23×C53C2×D411yescharacteristic
89C23×C53C2×D411yescharacteristic
90C23×C53C2×D411yescharacteristic
91C22×D53—21no—
92C22×D53—22no—
93C22×D53—42no—
94C22×D53—42no—
95C22×D53—42no—
96C2×C4×C53—22no—
97C2×C4×C53—22no—
98C2×C4×C53—22no—
99C2×C4×C53—41no—
100C2×C4×C53—41no—
101C2×C4×C53—41no—
102C2×C4×C53C2×D411yescharacteristic
103C2×C4×C53C2×D411yescharacteristic
104C2×C4×C53C2×D411yescharacteristic
105C2×C4×C53C2411yescharacteristic
106C22⋊D53—42no—
107C22⋊D53—42no—
108C22⋊D53—44no—
109C22⋊D53—44no—
110C22⋊D53—42no—
111C2×Dic53—21no—
112C2×Dic53—22no—
113C2×Dic53—21no—
114C2×Dic53—21no—
115C2×Dic53—42no—
116D212—22no—
117D212—22no—
118D212—22no—
119D212—42no—
120C4×D53—42no—
121Dic106—22no—
122C8×C53—22no—
123C8×C53—22no—
124C22×C53—21no—
125C22×C53—21no—
126C22×C53—21no—
127C22×C53—21no—
128C22×C53—21no—
129C22×C53—21no—
130C22×C53—21no—
131C22×C53—21no—
132C22×C53—44no—
133C22×C53AGL(1,ℤ/8)12yes—
134C22×C53C22×D411yescharacteristic
135C2×D53—41no—
136C2×D53—42no—
137C2×D53—42no—
138C2×D53—41no—
139C2×D53—41no—
140C2×D53—42no—
141C2×D53—84no—
142C4×C53—21no—
143C4×C53—21no—
144C4×C53—21no—
145C4×C53H(5,𝔽2)12yes—
146Dic53—41no—
147Dic53—42no—
148Dic53—41no—
149Dic53—41no—
150#3194—531nomaximal, Sylow
151C2×C53—22no—
152C2×C53—41no—
153C2×C53—41no—
154C2×C53#53411yescharacteristic
155C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
156C2×C53#57511yescharacteristic
157D53—81no—
158D53—82no—
159#525—531no—
160C2×#121—531no—
161#483—531no—
162#482—532no—
163#480—531no—
164#469—532no—
165#468—532no—
166#450—531no—
167#449—532no—
168#447—531no—
169#407D5311yescharacteristic, complemented
170C53#319411yescharacteristic, complemented, Sylow
171C2×#41—531no—
172C22×D4—531no—
173C22×D4—1061no—
174C23×C4C2×D5311yescharacteristic
175C2×SD16—532no—
176C2×D8—532no—
177C2×M16C2×D5312yescomplemented
178#123—1061no—
179C22⋊Q8—531no—
180C22⋊Q8—531no—
181#121—532no—
182#121—532no—
183#121—531no—
184#121—531no—
185#121—1061no—
186#121—1061no—
187#120—1061no—
188C4×D4—1061no—
189C4×D4—1061no—
190#117—531no—
191C2×C4⋊C4—531no—
192C2×C22⋊C4—1061no—
193#107—532no—
194#106—532no—
195#103—532no—
196#102—532no—
197#102—532no—
198#102—532no—
199C22⋊C8C2×D5311yescharacteristic
200C22⋊C8C2×D5311yescharacteristic
201C22⋊C8C2×D5312yescomplemented
202C24—1061no—
203C24D21211yescharacteristic
204#41—2122no—
205C2×Q8—531no—
206C2×D4—531no—
207C2×D4—532no—
208C2×D4—1061no—
209C2×D4—1061no—
210C2×D4—1061no—
211C2×D4—1062no—
212C2×D4—1062no—
213C2×D4—1061no—
214C2×D4—2122no—
215C2×D4—2124no—
216C22×C4—42no—
217C22×C4—1061no—
218C22×C4—1061no—
219C22×C4D21211yescharacteristic
220C22×C4D21211yescharacteristic
221C22×C4D21211yescharacteristic
222C22×C4C22×D5311yescharacteristic
223C22×C4C22×D5311yescharacteristic
224C22×C4C22×D5311yescharacteristic
225SD16—2122no—
226D8—2122no—
227M16—42no—
228C2×C8C22×D5312yes—
229C2×C8C22×D5312yes—
230C4⋊C4—531no—
231C4⋊C4—531no—
232C4⋊C4—532no—
233C4⋊C4—1061no—
234C4⋊C4—1061no—
235C4⋊C4—1061no—
236C22⋊C4—1061no—
237C22⋊C4—1061no—
238C22⋊C4—1061no—
239C22⋊C4—1062no—
240C22⋊C4—1062no—
241C22⋊C4—1061no—
242C22⋊C4—1061no—
243C42—1061no—
244C23—22no—
245C23—41no—
246C23—41no—
247C23—1061no—
248C23—1062no—
249C23—2122no—
250C23—2122no—
251C23—2122no—
252C23C2×D21211yescharacteristic
253C23C2×D21211yescharacteristic
254C23C2×D21211yescharacteristic
255Q8—1062no—
256D4—1062no—
257D4—1062no—
258D4—1062no—
259D4—2122no—
260D4—2122no—
261D4—2124no—
262D4—2124no—
263D4—2122no—
264D4—2122no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—1061no—
272C2×C4—1061no—
273C2×C4—1062no—
274C2×C4—1061no—
275C2×C4—2122no—
276C2×C4—2122no—
277C2×C4C2×D21211yescharacteristic
278C2×C4C2×D21211yescharacteristic
279C2×C4C2×D21211yescharacteristic
280C2×C4C23×D5311yescharacteristic
281C8—22no—
282C8—22no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—44no—
292C22—4244no—
293C22—2122no—
294C22—2121no—
295C22—2121no—
296C22—2122no—
297C22—2121no—
298C22—2122no—
299C22#1313324512yes—
300C22C22×D21211yescharacteristic
301C4—21no—
302C4—21no—
303C4—21no—
304C4—2122no—
305C4—2121no—
306C4—2121no—
307C4—2121no—
308C4#1313326212yes—
309C2—22no—
310C2—41no—
311C2—41no—
312C2—4242no—
313C2—4241no—
314C2#1681118511yescharacteristic
315C2C2×#1313324511yescharacteristic
316C2#1681142311yescharacteristic
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#407C2
socleC22×C53C22×D4
hypercenter#407D53
hypocenterC53#3194
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. #16811134C2semidirect
2. C2×#13133191C2semidirect
3. #16810954C2semidirect
4. #16810948C2semidirect
5. #16810937C2semidirect
6. #16810876C2semidirect
7. #16810869C2semidirect
8. #16810776C2semidirect
9. #16810770C2semidirect
10. #16810759C2semidirect
11. C53×#407C2semidirect
12. C2×#11719945C22semidirect
15. C2×#11719913C22semidirect
16. C2×D424C22semidirect
19. #13133196C22semidirect
25. #13133191C22semidirect
29. #13133176C22semidirect
32. #13133151C22semidirect
33. #13133149C22semidirect
34. #13133142C22semidirect
35. #13133139C22semidirect
36. #13133139C22semidirect
37. #13133139C22semidirect
39. C2×M16×C53C22semidirect
42. C22⋊C8×C53C22semidirect
169. #407D53semidirect
170. C53#3194semidirect
177. C2×M16C2×D53semidirect
201. C22⋊C8C2×D53semidirect