Finite Group Databasebeta

← #20284641

Subgroups — #20284641

Order 6784.

458 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 129 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#16811135C211yesmaximal, characteristic
2C2×C4⋊C4×D53C211yesmaximal, characteristic, complemented
3#16810960C212yesmaximal, complemented
4#16810955C212yesmaximal, complemented
5#16810943C212yesmaximal, complemented
6#16810938C212yesmaximal, complemented
7#16810666C211yesmaximal, characteristic, complemented
8#16810625C211yesmaximal, characteristic, complemented
9#16810622C211yesmaximal, characteristic, complemented
10#16810586C211yesmaximal, characteristic, complemented
11C53×#428C211yesmaximal, characteristic, complemented
12C2×#11719945C2211yescharacteristic
13C22×C4×D53—21no—
14C22×C4×D53C2211yescharacteristic
15C2×#11719917C2211yescharacteristic
16#13133201—21no—
17#13133197C2211yescharacteristic
18#13133197C2211yescharacteristic
19#13133192C2211yescharacteristic
20#13133192C2211yescharacteristic
21#13133187—21no—
22C4×D212—21no—
23C4×D212—21no—
24C2×C4⋊Dic53C2211yescharacteristic
25C2×#11719919—21no—
26C4⋊C4×D53—21no—
27C4⋊C4×D53—21no—
28C4⋊C4×D53C2212yes—
29C4⋊C4×D53C2212yes—
30#13133150C2212yescomplemented
31#13133148C2212yescomplemented
32#13133142C2212yescomplemented
33#13133140C2212yescomplemented
34#13133139C2212yescomplemented
35#13133137C2212yescomplemented
36#13133129C2212yes—
37#13133129C2212yes—
38C2×M16×C53C2211yescharacteristic
39#117×C53C2211yescharacteristic
40C2×C4⋊C4×C53C2211yescharacteristic
41#107×C53C2212yescomplemented
42#106×C53C2212yescomplemented
43C23×D53D411yescharacteristic
44C22×C4×C53—21no—
45C22×C4×C53C2311yescharacteristic
46C2×C22⋊D53—21no—
47C22×Dic53—21no—
48C22×Dic53D411yescharacteristic
49#11719945—22no—
50#11719945—22no—
51C2×D212—21no—
52C2×D212—21no—
53C2×D212C2311yescharacteristic
54C2×C4×D53—22no—
55C2×C4×D53—22no—
56C2×C4×D53—22no—
57C2×C4×D53—22no—
58C2×C4×D53—21no—
59C2×C4×D53—44no—
60C2×C4×D53D411yescharacteristic
61C2×C4×D53D411yescharacteristic
62C2×C4×D53C2311yescharacteristic
63C2×C4×D53C2311yescharacteristic
64C2×Dic106C2311yescharacteristic
65M16×C53—22no—
66C2×C8×C53C2312yes—
67C4⋊C4×C53—21no—
68C4⋊C4×C53C2311yescharacteristic
69C4⋊C4×C53C2312yes—
70C4⋊C4×C53C2311yescharacteristic
71C22⋊C4×C53—21no—
72C42×C53—21no—
73#11719921—21no—
74#11719921—21no—
75#11719921—21no—
76#11719921—21no—
77C4⋊Dic53—21no—
78C4⋊Dic53—21no—
79C4⋊Dic53—21no—
80C4⋊Dic53C2312yes—
81#11719919—22no—
82#11719919—22no—
83#11719917—22no—
84C2×#88322C2312yes—
85C23×C53C2×D411yescharacteristic
86C22×D53—22no—
87C22×D53—22no—
88C22×D53—22no—
89C22×D53—21no—
90C22×D53C2×D411yescharacteristic
91C22×D53C2×D411yescharacteristic
92C2×C4×C53—22no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—42no—
96C2×C4×C53C2×D411yescharacteristic
97C2×C4×C53#4112yes—
98C2×C4×C53#4112yes—
99C2×C4×C53C2411yescharacteristic
100C22⋊D53—42no—
101C2×Dic53—22no—
102C2×Dic53—21no—
103C2×Dic53—22no—
104C2×Dic53—42no—
105C2×Dic53C2×D411yescharacteristic
106C2×Dic53C2×D411yescharacteristic
107D212—21no—
108D212—21no—
109D212—42no—
110D212—42no—
111C4×D53—22no—
112C4×D53—22no—
113C4×D53—21no—
114C4×D53—21no—
115C4×D53—21no—
116C4×D53—21no—
117C4×D53—44no—
118C4×D53—44no—
119C4×D53—42no—
120Dic106—21no—
121Dic106—21no—
122C8×C53—22no—
123#88322—22no—
124C22×C53—21no—
125C22×C53—21no—
126C22×C53#12312yes—
127C22×C53C22×D411yescharacteristic
128C2×D53—28no—
129C2×D53—21no—
130C2×D53—21no—
131C2×D53—21no—
132C2×D53—21no—
133C2×D53—41no—
134C2×D53—41no—
135C2×D53—41no—
136C2×D53AGL(1,ℤ/8)12yes—
137C2×D53#13712yes—
138C4×C53—42no—
139C4×C53—41no—
140C4×C53—41no—
141C4×C53#12312yes—
142C4×C53C2×#4111yescharacteristic
143C4×C53C2×#4111yescharacteristic
144Dic53—21no—
145Dic53—21no—
146Dic53—41no—
147Dic53—42no—
148#3092—531nomaximal, Sylow
149C2×C53—22no—
150C2×C53C2×#12311yescharacteristic
151C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
152C2×C53C2×#13711yescharacteristic
153D53—22no—
154D53—22no—
155D53—81no—
156#525—531no—
157C22×C4⋊C4—531no—
158#484—532no—
159#483—532no—
160#481—532no—
161#480—532no—
162#428—531no—
163#428D5311yescharacteristic, complemented
164#417—531no—
165#417—531no—
166#407—531no—
167C53#309211yescharacteristic, complemented, Sylow
168C2×#41—531no—
169C23×C4—531no—
170C23×C4—1061no—
171C2×M16—531no—
172C2×M16C2×D5311yescharacteristic
173#123—1061no—
174C22⋊Q8—531no—
175C22⋊Q8—531no—
176#121—531no—
177#121—531no—
178#120—1061no—
179C4×D4—1061no—
180C4×D4—1061no—
181#117C2×D5311yescharacteristic
182C2×C4⋊C4—531no—
183C2×C4⋊C4—532no—
184C2×C4⋊C4—532no—
185C2×C4⋊C4—1061no—
186C2×C4⋊C4—1061no—
187C2×C4⋊C4—1061no—
188C2×C4⋊C4C2×D5311yescharacteristic
189#107—532no—
190#107C2×D5312yescomplemented
191#106—532no—
192#106C2×D5312yescomplemented
193#103—532no—
194#103—532no—
195#102—532no—
196#102—532no—
197C22⋊C8—532no—
198C22⋊C8—532no—
199C24—531no—
200#41—1062no—
201#41—1062no—
202C2×Q8—531no—
203C2×D4—531no—
204C2×D4—1061no—
205C2×D4—1061no—
206C2×D4—1061no—
207C22×C4—21no—
208C22×C4—531no—
209C22×C4—531no—
210C22×C4—531no—
211C22×C4—531no—
212C22×C4—531no—
213C22×C4—1062no—
214C22×C4—1062no—
215C22×C4—1062no—
216C22×C4—1062no—
217C22×C4—1061no—
218C22×C4—1061no—
219C22×C4—2124no—
220C22×C4C22×D5311yescharacteristic
221M16—22no—
222M16—1062no—
223C2×C8—532no—
224C2×C8C22×D5312yes—
225C4⋊C4—21no—
226C4⋊C4—532no—
227C4⋊C4—1062no—
228C4⋊C4—1062no—
229C4⋊C4—1061no—
230C4⋊C4—1061no—
231C4⋊C4—1061no—
232C4⋊C4C22×D5312yes—
233C4⋊C4C22×D5311yescharacteristic
234C4⋊C4C22×D5311yescharacteristic
235C22⋊C4—21no—
236C22⋊C4—1061no—
237C22⋊C4—1061no—
238C22⋊C4—1061no—
239C22⋊C4—1061no—
240C42—21no—
241C23—531no—
242C23—531no—
243C23—1062no—
244C23—1062no—
245C23—1062no—
246C23—1061no—
247C23D4×D5311yescharacteristic
248Q8—1061no—
249Q8—1061no—
250D4—1061no—
251D4—1061no—
252D4—2122no—
253D4—2122no—
254D4—2122no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—42no—
259C2×C4—531no—
260C2×C4—531no—
261C2×C4—1062no—
262C2×C4—1062no—
263C2×C4—1062no—
264C2×C4—1061no—
265C2×C4—1061no—
266C2×C4—1061no—
267C2×C4—1061no—
268C2×C4—1061no—
269C2×C4—1062no—
270C2×C4—2124no—
271C2×C4—2124no—
272C2×C4—2122no—
273C2×C4—2122no—
274C2×C4#1171994512yes—
275C2×C4D4×D5311yescharacteristic
276C2×C4#1171994912yes—
277C2×C4C23×D5311yescharacteristic
278C8—22no—
279C8—1062no—
280C22—21no—
281C22—21no—
282C22—532no—
283C22—532no—
284C22—1068no—
285C22—1061no—
286C22—1061no—
287C22—1061no—
288C22—1061no—
289C22—2121no—
290C22—2121no—
291C22—2121no—
292C22#1313320212yes—
293C22C2×D4×D5311yescharacteristic
294C4—42no—
295C4—41no—
296C4—41no—
297C4—1061no—
298C4—1061no—
299C4—2121no—
300C4—2122no—
301C4#1313320212yes—
302C4C2×#1171994511yescharacteristic
303C4C2×#1171994911yescharacteristic
304C2—22no—
305C2—1062no—
306C2—1062no—
307C2—4241no—
308C2C2×#1313320211yescharacteristic
309C2AGL(1,ℤ/8)×D5311yescharacteristic
310C2#137×D5311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#428C2
socleC22×C53C22×D4
hypercenter#428D53
hypocenterC53#3092
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D53C2semidirect
3. #16810960C2semidirect
4. #16810955C2semidirect
5. #16810943C2semidirect
6. #16810938C2semidirect
7. #16810666C2semidirect
8. #16810625C2semidirect
9. #16810622C2semidirect
10. #16810586C2semidirect
11. C53×#428C2semidirect
30. #13133150C22semidirect
31. #13133148C22semidirect
32. #13133142C22semidirect
33. #13133140C22semidirect
34. #13133139C22semidirect
35. #13133137C22semidirect
41. #107×C53C22semidirect
42. #106×C53C22semidirect
163. #428D53semidirect
167. C53#3092semidirect
190. #107C2×D53semidirect
192. #106C2×D53semidirect