Finite Group Databasebeta

← #20284642

Subgroups — #20284642

Order 6784.

458 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 123 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#16811136C211yesmaximal, characteristic
2C2×C4⋊C4×D53C211yesmaximal, characteristic, complemented
3#16810956C211yesmaximal, characteristic, complemented
4#16810952C212yesmaximal, complemented
5#16810942C211yesmaximal, characteristic, complemented
6#16810941C211yesmaximal, characteristic, complemented
7#16810938C212yesmaximal, complemented
8#16810664C212yesmaximal, complemented
9#16810623C212yesmaximal, complemented
10#16810585C211yesmaximal, characteristic, complemented
11C53×#484C211yesmaximal, characteristic, complemented
12C2×#11719949C2211yescharacteristic
13C22×C4×D53—21no—
14C22×C4×D53C2211yescharacteristic
15C2×#11719912C2212yes—
16#13133202—21no—
17#13133193C2211yescharacteristic
18#13133192C2211yescharacteristic
19#13133191C2211yescharacteristic
20#13133187—21no—
21#13133179—21no—
22#13133177—21no—
23#13133175C2211yescharacteristic
24C2×C4⋊Dic53C2211yescharacteristic
25C2×#11719919—21no—
26C4⋊C4×D53—21no—
27C4⋊C4×D53—21no—
28C4⋊C4×D53C2212yes—
29C4⋊C4×D53C2212yes—
30#13133150C2212yescomplemented
31#13133149C2212yescomplemented
32#13133148C2212yescomplemented
33#13133141C2212yescomplemented
34#13133139C2212yescomplemented
35#13133137C2212yescomplemented
36C22⋊#88322C2211yescharacteristic
37#13133129C2212yes—
38C22⋊Q8×C53C2211yescharacteristic
39C2×C4⋊C4×C53C2211yescharacteristic
40#107×C53C2212yescomplemented
41#103×C53C2212yescomplemented
42C22⋊C8×C53C2211yescharacteristic
43C23×D53D411yescharacteristic
44C2×Q8×C53C2311yescharacteristic
45C22×C4×C53—21no—
46C22×C4×C53C2311yescharacteristic
47C2×C22⋊D53—21no—
48C22×Dic53—21no—
49C22×Dic53D411yescharacteristic
50#11719949—24no—
51C2×D212—21no—
52C2×D212—21no—
53C2×D212C2311yescharacteristic
54C2×C4×D53—24no—
55C2×C4×D53—22no—
56C2×C4×D53—22no—
57C2×C4×D53—21no—
58C2×C4×D53—44no—
59C2×C4×D53D411yescharacteristic
60C2×C4×D53D411yescharacteristic
61C2×C4×D53C2311yescharacteristic
62C2×C4×D53C2311yescharacteristic
63C2×C8×C53C2312yes—
64C4⋊C4×C53—21no—
65C4⋊C4×C53—21no—
66C4⋊C4×C53C2312yes—
67C4⋊C4×C53C2311yescharacteristic
68C22⋊C4×C53—21no—
69#11719921—21no—
70#11719921—21no—
71#11719921—21no—
72#11719921—21no—
73C4⋊Dic53—21no—
74C4⋊Dic53C2312yes—
75C4⋊Dic53C2311yescharacteristic
76#11719919—22no—
77#11719919—22no—
78#11719919—21no—
79C4×Dic53—21no—
80C2×#88322C2312yes—
81#11719912—24no—
82C23×C53C2×D411yescharacteristic
83C22×D53—22no—
84C22×D53—22no—
85C22×D53—22no—
86C22×D53—21no—
87C22×D53C2×D411yescharacteristic
88C22×D53C2×D411yescharacteristic
89Q8×C53—22no—
90C2×C4×C53—22no—
91C2×C4×C53—22no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—42no—
95C2×C4×C53C2×D411yescharacteristic
96C2×C4×C53C2411yescharacteristic
97C22⋊D53—42no—
98C2×Dic53—22no—
99C2×Dic53—22no—
100C2×Dic53—21no—
101C2×Dic53—42no—
102C2×Dic53C2×D411yescharacteristic
103C2×Dic53C2×D411yescharacteristic
104D212—22no—
105D212—42no—
106D212—42no—
107C4×D53—22no—
108C4×D53—22no—
109C4×D53—22no—
110C4×D53—44no—
111C4×D53—44no—
112C4×D53—42no—
113C4×D53#4114yes—
114C8×C53—22no—
115#88322—22no—
116C22×C53—21no—
117C22×C53—21no—
118C22×C53#13712yes—
119C22×C53C22×D411yescharacteristic
120C2×D53—28no—
121C2×D53—21no—
122C2×D53—21no—
123C2×D53—21no—
124C2×D53—21no—
125C2×D53—41no—
126C2×D53—41no—
127C2×D53—41no—
128C2×D53#12312yes—
129C2×D53AGL(1,ℤ/8)12yes—
130C4×C53—21no—
131C4×C53—41no—
132C4×C53—42no—
133C4×C53—41no—
134C4×C53C2×#4112yes—
135Dic53—21no—
136Dic53—42no—
137Dic53—41no—
138Dic53#12312yes—
139#3092—531nomaximal, Sylow
140C2×C53—22no—
141C2×C53C2×#12311yescharacteristic
142C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
143C2×C53C2×#13711yescharacteristic
144D53—22no—
145D53—22no—
146D53—81no—
147#525—531no—
148C22×C4⋊C4—531no—
149#484—531no—
150#484D5311yescharacteristic, complemented
151#483—532no—
152#481—531no—
153#481—531no—
154#480—532no—
155#428—532no—
156#417—532no—
157#407—531no—
158C53#309211yescharacteristic, complemented, Sylow
159C2×#41—531no—
160C23×C4—531no—
161C23×C4—1061no—
162C2×M16—532no—
163#123—1061no—
164C22⋊Q8—531no—
165C22⋊Q8C2×D5311yescharacteristic
166#121—531no—
167#121—531no—
168#120—1061no—
169C4×D4—1061no—
170C4×D4—1061no—
171#117—531no—
172C2×C4⋊C4—531no—
173C2×C4⋊C4—532no—
174C2×C4⋊C4—532no—
175C2×C4⋊C4—1061no—
176C2×C4⋊C4—1061no—
177C2×C4⋊C4—1061no—
178C2×C4⋊C4C2×D5311yescharacteristic
179#107—532no—
180#107C2×D5312yescomplemented
181#106—532no—
182#106—532no—
183#103—532no—
184#103C2×D5312yescomplemented
185#102—532no—
186#102—532no—
187C22⋊C8—531no—
188C22⋊C8—532no—
189C22⋊C8C2×D5311yescharacteristic
190C24—531no—
191#41—1064no—
192C2×Q8C22×D5311yescharacteristic
193C2×D4—531no—
194C2×D4—1061no—
195C2×D4—1061no—
196C2×D4—1061no—
197C22×C4—21no—
198C22×C4—531no—
199C22×C4—531no—
200C22×C4—531no—
201C22×C4—531no—
202C22×C4—531no—
203C22×C4—1064no—
204C22×C4—1061no—
205C22×C4—1062no—
206C22×C4—1062no—
207C22×C4—1061no—
208C22×C4—2124no—
209C22×C4C22×D5311yescharacteristic
210M16—1064no—
211C2×C8—532no—
212C2×C8C22×D5312yes—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—532no—
216C4⋊C4—531no—
217C4⋊C4—1062no—
218C4⋊C4—1062no—
219C4⋊C4—1061no—
220C4⋊C4—1061no—
221C4⋊C4C22×D5312yes—
222C4⋊C4C22×D5311yescharacteristic
223C22⋊C4—21no—
224C22⋊C4—1061no—
225C22⋊C4—1061no—
226C22⋊C4—1061no—
227C22⋊C4—1061no—
228C42—1061no—
229C23—531no—
230C23—531no—
231C23—1062no—
232C23—1062no—
233C23—1062no—
234C23—1061no—
235C23D4×D5311yescharacteristic
236Q8—22no—
237D4—1062no—
238D4—2122no—
239D4—2122no—
240D4—2122no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—42no—
246C2×C4—531no—
247C2×C4—534no—
248C2×C4—531no—
249C2×C4—1062no—
250C2×C4—1062no—
251C2×C4—1062no—
252C2×C4—1062no—
253C2×C4—1061no—
254C2×C4—1062no—
255C2×C4—2122no—
256C2×C4—2124no—
257C2×C4—2124no—
258C2×C4—2122no—
259C2×C4D4×D5311yescharacteristic
260C2×C4C23×D5311yescharacteristic
261C8—22no—
262C8—1062no—
263C22—21no—
264C22—21no—
265C22—532no—
266C22—532no—
267C22—1068no—
268C22—1061no—
269C22—1061no—
270C22—1061no—
271C22—1061no—
272C22—2121no—
273C22—2121no—
274C22—2121no—
275C22#1313324912yes—
276C22C2×D4×D5311yescharacteristic
277C4—21no—
278C4—41no—
279C4—42no—
280C4—41no—
281C4—532no—
282C4—1061no—
283C4—2122no—
284C4—2121no—
285C4#41×D5312yes—
286C2—22no—
287C2—1062no—
288C2—1062no—
289C2—4241no—
290C2#123×D5311yescharacteristic
291C2AGL(1,ℤ/8)×D5311yescharacteristic
292C2C2×#1313324911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#484C2
socleC22×C53C22×D4
hypercenter#484D53
hypocenterC53#3092
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
2. C2×C4⋊C4×D53C2semidirect
3. #16810956C2semidirect
4. #16810952C2semidirect
5. #16810942C2semidirect
6. #16810941C2semidirect
7. #16810938C2semidirect
8. #16810664C2semidirect
9. #16810623C2semidirect
10. #16810585C2semidirect
11. C53×#484C2semidirect
30. #13133150C22semidirect
31. #13133149C22semidirect
32. #13133148C22semidirect
33. #13133141C22semidirect
34. #13133139C22semidirect
35. #13133137C22semidirect
40. #107×C53C22semidirect
41. #103×C53C22semidirect
150. #484D53semidirect
158. C53#3092semidirect
180. #107C2×D53semidirect
184. #103C2×D53semidirect