Finite Group Databasebeta

← #20284665

Subgroups — #20284665

Order 6784.

410 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 127 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×#13133197C211yesmaximal, characteristic
2#117×D53C211yesmaximal, characteristic
3#16810967C212yesmaximal, complemented
4#16810966C212yesmaximal, complemented
5#16810960C212yesmaximal, complemented
6#16810955C212yesmaximal, complemented
7#16810663C211yesmaximal, characteristic, complemented
8#16810633C211yesmaximal, characteristic, complemented
9#16810631C211yesmaximal, characteristic, complemented
10#16810586C211yesmaximal, characteristic
11C53×#428C211yesmaximal, characteristic, complemented
12C22×Dic106C2211yescharacteristic
13C22×C4×D53C2211yescharacteristic
14C2×#11719917C2211yescharacteristic
15#13133197—21no—
16#13133197—21no—
17#13133197C2212yes—
18#13133197C2212yes—
19#13133176C2211yescharacteristic
20#13133175C2211yescharacteristic
21#13133175C2211yescharacteristic
22#13133174—21no—
23#13133173—21no—
24C2×C4⋊Dic53—21no—
25C4⋊C4×D53C2211yescharacteristic
26C4⋊C4×D53C2211yescharacteristic
27C2×#11719921—21no—
28C22⋊C4×D53—21no—
29C42×D53—21no—
30#13133150C2212yescomplemented
31#13133148C2212yescomplemented
32#13133142C2212yescomplemented
33#13133142C2212yescomplemented
34#13133140C2212yescomplemented
35#13133140C2212yescomplemented
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
46C22×Dic53—21no—
47C22×Dic53D411yescharacteristic
48C2×C4×D53—22no—
49C2×C4×D53—22no—
50C2×C4×D53—21no—
51C2×C4×D53—21no—
52C2×C4×D53D411yescharacteristic
53C2×C4×D53D411yescharacteristic
54C2×C4×D53C2311yescharacteristic
55C2×C4×D53C2311yescharacteristic
56C2×Dic106—22no—
57C2×Dic106—22no—
58C2×Dic106—21no—
59C2×Dic106C2312yes—
60M16×C53—22no—
61C2×C8×C53C2312yes—
62C4⋊C4×C53—21no—
63C4⋊C4×C53C2311yescharacteristic
64C4⋊C4×C53C2312yes—
65C4⋊C4×C53C2311yescharacteristic
66C22⋊C4×C53—21no—
67C42×C53—21no—
68C22⋊Dic53—21no—
69#11719921—21no—
70#11719921—21no—
71#11719921—22no—
72#11719921—22no—
73C4⋊Dic53—22no—
74C4⋊Dic53—22no—
75C4⋊Dic53C2311yescharacteristic
76C4⋊Dic53C2311yescharacteristic
77#11719919—21no—
78#11719919—21no—
79C4×Dic53—21no—
80C4×Dic53—21no—
81C4×Dic53—21no—
82#11719917—22no—
83C2×#88322C2312yes—
84C23×C53C2×D411yescharacteristic
85C22×D53—22no—
86C22×D53—22no—
87C22×D53—41no—
88C22×D53C2×D411yescharacteristic
89C22×D53C2×D411yescharacteristic
90C2×C4×C53—21no—
91C2×C4×C53—21no—
92C2×C4×C53—22no—
93C2×C4×C53—42no—
94C2×C4×C53C2×D411yescharacteristic
95C2×C4×C53#4112yes—
96C2×C4×C53#4112yes—
97C2×C4×C53C2411yescharacteristic
98C2×Dic53—22no—
99C2×Dic53—22no—
100C2×Dic53—21no—
101C2×Dic53—21no—
102C2×Dic53—42no—
103C2×Dic53C2×D411yescharacteristic
104C2×Dic53C2×D411yescharacteristic
105C4×D53—22no—
106C4×D53—22no—
107C4×D53—21no—
108C4×D53—21no—
109C4×D53—21no—
110C4×D53—21no—
111C4×D53—42no—
112C4×D53—42no—
113Dic106—22no—
114Dic106—22no—
115Dic106—42no—
116C8×C53—22no—
117#88322—22no—
118C22×C53—21no—
119C22×C53—21no—
120C22×C53#12312yes—
121C22×C53C22×D411yescharacteristic
122C2×D53—21no—
123C2×D53—21no—
124C2×D53—21no—
125C2×D53—21no—
126C2×D53—21no—
127C2×D53—44no—
128C2×D53#13712yes—
129C4×C53—41no—
130C4×C53—41no—
131C4×C53—42no—
132C4×C53#12312yes—
133C4×C53C2×#4111yescharacteristic
134C4×C53C2×#4111yescharacteristic
135Dic53—21no—
136Dic53—21no—
137Dic53—42no—
138Dic53—41no—
139Dic53—41no—
140#3095—531nomaximal, Sylow
141C2×C53—22no—
142C2×C53C2×#12311yescharacteristic
143C2×C53C2×#13711yescharacteristic
144C2×C53#57511yescharacteristic
145D53—22no—
146D53—41no—
147C2×C22⋊Q8—531no—
148C2×#117—531no—
149#485—532no—
150#485—532no—
151#484—532no—
152#483—532no—
153#428—531no—
154#428D5311yescharacteristic, complemented
155#419—531no—
156#419—531no—
157#407—531no—
158C53#309511yescharacteristic, complemented, Sylow
159C22×Q8—531no—
160C23×C4—531no—
161C2×M16—531no—
162C2×M16C2×D5311yescharacteristic
163C22⋊Q8—532no—
164C22⋊Q8—532no—
165C22⋊Q8—1061no—
166C22⋊Q8—1061no—
167#117—531no—
168#117—531no—
169#117—531no—
170#117—1061no—
171#117—1061no—
172#117C2×D5311yescharacteristic
173C2×C4⋊C4—531no—
174C2×C4⋊C4—531no—
175C2×C4⋊C4—1061no—
176C2×C4⋊C4C2×D5311yescharacteristic
177C2×C22⋊C4—1061no—
178C2×C22⋊C4—1061no—
179C2×C42—1061no—
180#107—532no—
181#107C2×D5312yescomplemented
182#106—532no—
183#106C2×D5312yescomplemented
184#103—532no—
185#103—532no—
186#103—532no—
187#103—532no—
188C22⋊C8—532no—
189C22⋊C8—532no—
190C24—531no—
191C2×Q8—532no—
192C2×Q8—1061no—
193C2×Q8—1062no—
194C2×Q8—1062no—
195C22×C4—21no—
196C22×C4—531no—
197C22×C4—531no—
198C22×C4—531no—
199C22×C4—531no—
200C22×C4—531no—
201C22×C4—1061no—
202C22×C4—1061no—
203C22×C4—1062no—
204C22×C4—1062no—
205C22×C4—1061no—
206C22×C4C22×D5311yescharacteristic
207M16—22no—
208M16—1062no—
209C2×C8—532no—
210C2×C8C22×D5312yes—
211C4⋊C4—21no—
212C4⋊C4—531no—
213C4⋊C4—531no—
214C4⋊C4—1061no—
215C4⋊C4—1061no—
216C4⋊C4—1062no—
217C4⋊C4—1062no—
218C4⋊C4C22×D5311yescharacteristic
219C4⋊C4C22×D5311yescharacteristic
220C4⋊C4C22×D5312yes—
221C22⋊C4—21no—
222C22⋊C4—1061no—
223C22⋊C4—1061no—
224C22⋊C4—1062no—
225C22⋊C4—1062no—
226C22⋊C4—1061no—
227C42—21no—
228C42—1061no—
229C42—1061no—
230C42—1061no—
231C23—531no—
232C23—531no—
233C23—1062no—
234C23—1062no—
235C23—2121no—
236C23D4×D5311yescharacteristic
237Q8—1062no—
238Q8—1062no—
239Q8—2122no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—531no—
245C2×C4—531no—
246C2×C4—1062no—
247C2×C4—1062no—
248C2×C4—1062no—
249C2×C4—1061no—
250C2×C4—1061no—
251C2×C4—1061no—
252C2×C4—1061no—
253C2×C4—1061no—
254C2×C4—1061no—
255C2×C4—1062no—
256C2×C4—2122no—
257C2×C4—2122no—
258C2×C4—2122no—
259C2×C4#1171994512yes—
260C2×C4D4×D5311yescharacteristic
261C2×C4#1171994912yes—
262C2×C4C23×D5311yescharacteristic
263C8—22no—
264C8—1062no—
265C22—21no—
266C22—21no—
267C22—532no—
268C22—1061no—
269C22—1061no—
270C22—1061no—
271C22—1061no—
272C22—1061no—
273C22—2124no—
274C22#1313320212yes—
275C22C2×D4×D5311yescharacteristic
276C4—41no—
277C4—42no—
278C4—41no—
279C4—1061no—
280C4—1061no—
281C4—2122no—
282C4—2121no—
283C4—2121no—
284C4#1313320212yes—
285C4C2×#1171994511yescharacteristic
286C4C2×#1171994911yescharacteristic
287C2—22no—
288C2—1062no—
289C2—2121no—
290C2C2×#1313320211yescharacteristic
291C2#137×D5311yescharacteristic
292C2#1681141911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#428C2
socleC22×C53C22×D4
hypercenter#428D53
hypocenterC53#3095
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
3. #16810967C2semidirect
4. #16810966C2semidirect
5. #16810960C2semidirect
6. #16810955C2semidirect
7. #16810663C2semidirect
8. #16810633C2semidirect
9. #16810631C2semidirect
11. C53×#428C2semidirect
30. #13133150C22semidirect
31. #13133148C22semidirect
32. #13133142C22semidirect
33. #13133142C22semidirect
34. #13133140C22semidirect
35. #13133140C22semidirect
41. #107×C53C22semidirect
42. #106×C53C22semidirect
154. #428D53semidirect
158. C53#3095semidirect
181. #107C2×D53semidirect
183. #106C2×D53semidirect