Finite Group Databasebeta

← #20285680

Subgroups — #20285680

Order 6784.

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

normal subgroups only

This group has 410 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#16811157C211yesmaximal, characteristic, complemented
2#16811005C212yesmaximal, complemented
3#16810977C212yesmaximal, complemented
4#16810842C212yesmaximal, complemented
5#16810835C212yesmaximal, complemented
6#16810779C211yesmaximal, characteristic, complemented
7#16810774C211yesmaximal, characteristic, complemented
8#16810762C211yesmaximal, characteristic, complemented
9#16810757C211yesmaximal, characteristic
10#16810676C211yesmaximal, characteristic, complemented
11C2×#124×C53C211yesmaximal, characteristic, complemented
12C2×#11719925C2212yescomplemented
13C2×#11719923C2212yes—
14C2×#11719924C2212yescomplemented
15C2×#11719922C2212yes—
16#13133211—21no—
17#13133209C2211yescharacteristic, complemented
18#13133205C2211yescharacteristic, complemented
19#13133200—21no—
20#13133196C2211yescharacteristic
21#13133191C2211yescharacteristic
22C4×Dic106C2211yescharacteristic, complemented
23C4×D212C2211yescharacteristic, complemented
24C4×C22⋊D53—21no—
25#13133174—21no—
26C4⋊#88322C2212yescomplemented
27#13133141C2212yescomplemented
28#13133141C2212yescomplemented
29#13133138C2212yes—
30#13133138C2212yes—
31C22⋊#88322C2211yescharacteristic
32C22⋊#88322C2211yescharacteristic
33C4×#88322C2212yescomplemented
34C22×Q8×C53C2211yescharacteristic
35C22×D4×C53C2211yescharacteristic
36#124×C53—21no—
37#124×C53—21no—
38#124×C53C2212yes—
39#124×C53C2212yes—
40C2×C22⋊C4×C53—21no—
41C2×C22⋊C4×C53—21no—
42C2×C42×C53C2211yescharacteristic
43C24×C53—21no—
44C2×Q8×C53—21no—
45C2×Q8×C53—42no—
46C2×Q8×C53C2312yes—
47C2×D4×C53—21no—
48C2×D4×C53—42no—
49C2×D4×C53C2312yes—
50C22×C4×C53—21no—
51C22×C4×C53D411yescharacteristic
52C22×C4×C53D411yescharacteristic
53C22×C4×C53C2311yescharacteristic
54C2×C22⋊D53—21no—
55C2×D212C2311yescharacteristic
56C2×C4×D53—21no—
57C2×Dic106C2311yescharacteristic
58C22⋊C4×C53—22no—
59C22⋊C4×C53—22no—
60C22⋊C4×C53—22no—
61C22⋊C4×C53—22no—
62C42×C53D411yescharacteristic
63C42×C53D411yescharacteristic
64C42×C53C2311yescharacteristic
65C42×C53C2311yescharacteristic
66C22⋊Dic53—21no—
67#11719925—42no—
68#11719924—42no—
69#11719923—42no—
70#11719922—42no—
71#11719921—21no—
72#11719921—21no—
73#11719921—21no—
74#11719921—21no—
75C4⋊Dic53C2311yescharacteristic
76C4⋊Dic53C2311yescharacteristic
77#11719919—21no—
78#11719919—21no—
79#11719919—21no—
80#11719919—21no—
81C4×Dic53—21no—
82C2×#88322C2312yes—
83C2×#88322C2312yes—
84C23×C53—22no—
85C23×C53—42no—
86C23×C53—42no—
87C23×C53—42no—
88C23×C53C2×D411yescharacteristic
89C22×D53—21no—
90Q8×C53—22no—
91Q8×C53—22no—
92Q8×C53—42no—
93D4×C53—22no—
94D4×C53—22no—
95D4×C53—42no—
96C2×C4×C53—22no—
97C2×C4×C53—22no—
98C2×C4×C53—22no—
99C2×C4×C53—41no—
100C2×C4×C53—42no—
101C2×C4×C53C2×D411yescharacteristic
102C2×C4×C53C2×D411yescharacteristic
103C2×C4×C53C2×D411yescharacteristic
104C2×C4×C53C2×D411yescharacteristic
105C2×C4×C53C2×D411yescharacteristic
106C2×C4×C53C2411yescharacteristic
107C22⋊D53—42no—
108C2×Dic53—21no—
109C2×Dic53—21no—
110C2×Dic53—21no—
111D212—22no—
112C4×D53—42no—
113Dic106—22no—
114#88322—22no—
115#88322—22no—
116C22×C53—21no—
117C22×C53—21no—
118C22×C53—42no—
119C22×C53—42no—
120C22×C53—42no—
121C22×C53—84no—
122C22×C53#13512yes—
123C22×C53C22×D411yescharacteristic
124C2×D53—41no—
125C2×D53—41no—
126C2×D53—41no—
127C4×C53—21no—
128C4×C53—21no—
129C4×C53—21no—
130C4×C53—21no—
131C4×C53—21no—
132C4×C53—42no—
133C4×C53H(5,𝔽2)12yes—
134Dic53—41no—
135Dic53—41no—
136Dic53—41no—
137#3211—531nomaximal, Sylow
138C2×C53—22no—
139C2×C53—82no—
140C2×C53#53511yescharacteristic
141C2×C53C2×#13511yescharacteristic
142C2×C53#57511yescharacteristic
143D53—81no—
144#529—531no—
145C2×#124D5311yescharacteristic, complemented
146#495—532no—
147#488—532no—
148#464—532no—
149#463—532no—
150#451—531no—
151#450—531no—
152#448—531no—
153#447—531no—
154#431—531no—
155C53#321111yescharacteristic, complemented, Sylow
156C22×Q8C2×D5311yescharacteristic
157C22×D4C2×D5311yescharacteristic
158C2×Q16—532no—
159C2×SD16—532no—
160C2×SD16—532no—
161C2×D8—532no—
162#126—1061no—
163#125—531no—
164#124—21no—
165#124—21no—
166#124—531no—
167#124C2×D5312yes—
168#124C2×D5312yes—
169#123—1061no—
170C22⋊Q8—531no—
171#121—531no—
172C4×Q8—531no—
173C4×D4—531no—
174C4×D4—1061no—
175#117—1061no—
176C2×C22⋊C4—21no—
177C2×C22⋊C4—21no—
178C2×C42C2×D5311yescharacteristic
179C4⋊C8—532no—
180#103—532no—
181#103—532no—
182#102—532no—
183#102—532no—
184C22⋊C8—531no—
185C22⋊C8—531no—
186C4×C8—532no—
187C24—21no—
188C2×Q8—21no—
189C2×Q8—42no—
190C2×Q8—531no—
191C2×Q8C22×D5312yes—
192C2×D4—21no—
193C2×D4—42no—
194C2×D4—531no—
195C2×D4—1061no—
196C2×D4C22×D5312yes—
197C22×C4—21no—
198C22×C4—1061no—
199C22×C4C22⋊D5311yescharacteristic
200C22×C4C22⋊D5311yescharacteristic
201C22×C4C22×D5311yescharacteristic
202Q16—2122no—
203SD16—2122no—
204SD16—2122no—
205D8—2122no—
206C2×C8—532no—
207C2×C8—532no—
208C4⋊C4—531no—
209C4⋊C4—531no—
210C4⋊C4—1061no—
211C4⋊C4—1061no—
212C4⋊C4—1061no—
213C4⋊C4—1061no—
214C22⋊C4—22no—
215C22⋊C4—22no—
216C22⋊C4—22no—
217C22⋊C4—22no—
218C22⋊C4—1061no—
219C22⋊C4—1061no—
220C22⋊C4—1061no—
221C22⋊C4—1061no—
222C22⋊C4—1061no—
223C42—1061no—
224C42C22⋊D5311yescharacteristic
225C42C22⋊D5311yescharacteristic
226C42C22×D5311yescharacteristic
227C42C22×D5311yescharacteristic
228C23—22no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23—1061no—
233C23C2×C22⋊D5311yescharacteristic
234Q8—22no—
235Q8—22no—
236Q8—42no—
237Q8—1062no—
238D4—22no—
239D4—22no—
240D4—42no—
241D4—1062no—
242D4—2122no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—1061no—
249C2×C4—1061no—
250C2×C4—1061no—
251C2×C4—2122no—
252C2×C4C2×C22⋊D5311yescharacteristic
253C2×C4C2×C22⋊D5311yescharacteristic
254C2×C4C2×C22⋊D5311yescharacteristic
255C2×C4C2×C22⋊D5311yescharacteristic
256C2×C4C2×C22⋊D5311yescharacteristic
257C2×C4C23×D5311yescharacteristic
258C8—1062no—
259C8—1062no—
260C22—21no—
261C22—21no—
262C22—42no—
263C22—42no—
264C22—42no—
265C22—84no—
266C22—2121no—
267C22—2121no—
268C22—2121no—
269C22#1313323712yes—
270C22C22×C22⋊D5311yescharacteristic
271C4—21no—
272C4—21no—
273C4—21no—
274C4—21no—
275C4—21no—
276C4—42no—
277C4—2121no—
278C4—2121no—
279C4—2121no—
280C4#1313326212yes—
281C2—22no—
282C2—82no—
283C2—4241no—
284C2#1681119011yescharacteristic
285C2C2×#1313323711yescharacteristic
286C2#1681142111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC2×#124×C53C2
socleC22×C53C22×D4
hypercenterC2×#124D53
hypocenterC53#3211
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. #16811157C2semidirect
2. #16811005C2semidirect
3. #16810977C2semidirect
4. #16810842C2semidirect
5. #16810835C2semidirect
6. #16810779C2semidirect
7. #16810774C2semidirect
8. #16810762C2semidirect
10. #16810676C2semidirect
11. C2×#124×C53C2semidirect
12. C2×#11719925C22semidirect
14. C2×#11719924C22semidirect
17. #13133209C22semidirect
18. #13133205C22semidirect
22. C4×Dic106C22semidirect
23. C4×D212C22semidirect
26. C4⋊#88322C22semidirect
27. #13133141C22semidirect
28. #13133141C22semidirect
33. C4×#88322C22semidirect
145. C2×#124D53semidirect
155. C53#3211semidirect