Finite Group Databasebeta

← #20286501

Subgroups — #20286501

Order 6784.

454 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 125 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#16811431C212yesmaximal, complemented
2#16811425C211yesmaximal, characteristic, complemented
3#16811008C211yesmaximal, characteristic, complemented
4#16810895C211yesmaximal, characteristic, complemented
5#16810893C211yesmaximal, characteristic, complemented
6#16810888C212yesmaximal, complemented
7#16810799C212yesmaximal, complemented
8#16810796C212yesmaximal, complemented
9#16810740C211yesmaximal, characteristic, complemented
10#16810738C211yesmaximal, characteristic, complemented
11C53×#578C211yesmaximal, characteristic, complemented
12#13133263C2212yes—
13#13133248—22no—
14#13133248C2212yes—
15#13133244—22no—
16#13133242—22no—
17#13133242C2212yes—
18#13133242C2212yescomplemented
19#13133240—21no—
20#13133237—22no—
21#13133237C2211yescharacteristic
22C2×#11719924—22no—
23C2×#11719924C2212yescomplemented
24C2×D424C2211yescharacteristic
25D8×D53—21no—
26#13133224C2212yescomplemented
27#13133223C2211yescharacteristic
28#13133206C2212yescomplemented
29#13133153C2211yescharacteristic, complemented
30#13133152C2211yescharacteristic, complemented
31#13133144C2211yescharacteristic, complemented
32#13133144C2212yescomplemented
33#13133143C2211yescharacteristic
34#13133136C2212yescomplemented
35#13133133C2212yescomplemented
36C8×Dic53C2211yescharacteristic, complemented
372−1+4×C53C2212yescomplemented
38#137×C53—22no—
39#137×C53C2212yescomplemented
40#135×C53—21no—
41#135×C53C2211yescharacteristic
42C2×Q16×C53—21no—
43C2×Q16×C53C2211yescharacteristic, complemented
44#131×C53C2211yescharacteristic, complemented
45#41×C53—22no—
46#41×C53—22no—
47#41×C53—42no—
48#41×C53C2311yescharacteristic
49#41×C53C2312yes—
50C2×Q8×C53—22no—
51C2×Q8×C53—22no—
52C2×Q8×C53C2312yes—
53#11719949—22no—
54D4×D53—22no—
55D4×D53—42no—
56#11719945—22no—
57#11719945C2311yescharacteristic
58C2×D212—22no—
59C2×D212C2312yes—
60Q16×C53—22no—
61Q16×C53—21no—
62Q16×C53—21no—
63Q16×C53D411yescharacteristic
64SD16×C53—22no—
65SD16×C53D412yes—
66D8×C53D411yescharacteristic
67M16×C53—21no—
68M16×C53C2311yescharacteristic
69C2×C8×C53—21no—
70C2×C8×C53C2311yescharacteristic
71#11719925—22no—
72#11719925D411yescharacteristic
73#11719924—22no—
74#11719924—22no—
75#11719924—22no—
76#11719924—22no—
77#11719924—22no—
78#11719924D411yescharacteristic
79#11719923D411yescharacteristic
80#11719922—22no—
81#11719922—22no—
82#11719922D411yescharacteristic
83#11719921—42no—
84C4×Dic53C2311yescharacteristic
85#11719917—22no—
86#11719917C2312yes—
87#11719917C2311yescharacteristic
88C2×#88322—22no—
89C2×#88322C2311yescharacteristic
90D424—21no—
91D424—21no—
92#11719912—21no—
93C8×D53—21no—
94C22×D53—22no—
95C22×D53—42no—
96Q8×C53—22no—
97Q8×C53—22no—
98Q8×C53—42no—
99Q8×C53C2×D411yescharacteristic
100Q8×C53C2×D412yes—
101D4×C53—22no—
102D4×C53—21no—
103D4×C53—42no—
104D4×C53C2×D411yescharacteristic
105D4×C53C2×D412yes—
106C2×C4×C53—22no—
107C2×C4×C53—21no—
108C2×C4×C53—22no—
109C2×C4×C53—42no—
110C2×C4×C53—42no—
111C2×C4×C53C2411yescharacteristic
112C22⋊D53—21no—
113C22⋊D53—42no—
114C2×Dic53—21no—
115D212—22no—
116D212—22no—
117D212—42no—
118D212C2×D411yescharacteristic
119C4×D53—21no—
120C4×D53—42no—
121Dic106C2×D411yescharacteristic
122C8×C53—21no—
123C8×C53C2×D411yescharacteristic
124C8×C53C2×D411yescharacteristic
125#88322—21no—
126#88322—22no—
127#88322C2×D411yescharacteristic
128#88322C2×D411yescharacteristic
129C22×C53—22no—
130C22×C53—21no—
131C22×C53H(5,𝔽2)11yescharacteristic
132C2×D53—21no—
133C2×D53—42no—
134C2×D53—82no—
135C2×D53—162no—
136C4×C53—22no—
137C4×C53—21no—
138C4×C53—42no—
139C4×C53C22×D411yescharacteristic
140C4×C53C22×D411yescharacteristic
141Dic53—21no—
142Dic53—41no—
143#3290—531nomaximal, Sylow
144C2×C53—21no—
145C2×C53—42no—
146C2×C53—41no—
147C2×C53D4211yescharacteristic
148D53—41no—
149D53—82no—
150#578D5311yescharacteristic, complemented
151#577—532no—
152#576—531no—
153#495—531no—
154#473—531no—
155#472—531no—
156#471—532no—
157#455—532no—
158#454—532no—
159#443—531no—
160#443—531no—
161C53#329011yescharacteristic, complemented, Sylow
1622−1+4C2×D5312yescomplemented
163H(5,𝔽2)—532no—
164#137—22no—
165#137—532no—
166#137—1062no—
167#137C2×D5312yescomplemented
168AGL(1,ℤ/8)—532no—
169AGL(1,ℤ/8)—532no—
170AGL(1,ℤ/8)—1062no—
171AGL(1,ℤ/8)—1062no—
172#135—21no—
173#135—531no—
174#135—1062no—
175#135—1061no—
176#135C2×D5311yescharacteristic
177C2×Q16—21no—
178C2×Q16C2×D5311yescharacteristic, complemented
179C2×SD16—532no—
180C2×SD16—1062no—
181C2×D8—531no—
182C2×D8—1061no—
183#131—532no—
184#131—531no—
185#131C2×D5311yescharacteristic, complemented
186#124—532no—
187#108—531no—
188#108—531no—
189#104—531no—
190#104—532no—
191#104—531no—
192#101—532no—
193#100—532no—
194C4×C8—531no—
195#41—22no—
196#41—22no—
197#41—42no—
198#41—531no—
199#41—1062no—
200#41—1062no—
201#41C22×D5311yescharacteristic
202#41C22×D5312yes—
203C2×Q8—22no—
204C2×Q8—22no—
205C2×Q8C22×D5312yes—
206C2×D4—532no—
207C2×D4—1062no—
208C2×D4—1062no—
209C2×D4—2122no—
210Q16—22no—
211Q16—21no—
212Q16—21no—
213Q16—531no—
214Q16—1062no—
215Q16C22⋊D5311yescharacteristic
216SD16—22no—
217SD16—531no—
218SD16—531no—
219SD16—1062no—
220SD16—1062no—
221SD16—1062no—
222SD16—1062no—
223SD16—1062no—
224SD16C22⋊D5312yes—
225D8—531no—
226D8—1062no—
227D8—1062no—
228D8—1061no—
229D8—1061no—
230D8C22⋊D5311yescharacteristic
231M16—21no—
232M16—532no—
233M16—531no—
234M16—1062no—
235M16—1061no—
236M16C22×D5311yescharacteristic
237C2×C8—21no—
238C2×C8—531no—
239C2×C8—1062no—
240C2×C8—1061no—
241C2×C8C22×D5311yescharacteristic
242C22⋊C4—2122no—
243C42—531no—
244C23—1062no—
245C23—2122no—
246Q8—22no—
247Q8—22no—
248Q8—42no—
249Q8—531no—
250Q8D4×D5311yescharacteristic
251Q8C2×C22⋊D5312yes—
252D4—22no—
253D4—21no—
254D4—42no—
255D4—531no—
256D4—1061no—
257D4—1062no—
258D4—1062no—
259D4—2122no—
260D4—2122no—
261D4D4×D5311yescharacteristic
262D4C2×C22⋊D5312yes—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—1061no—
269C2×C4—1061no—
270C2×C4—2122no—
271C2×C4C23×D5311yescharacteristic
272C8—21no—
273C8—531no—
274C8—531no—
275C8—1062no—
276C8—1061no—
277C8C2×C22⋊D5311yescharacteristic
278C8C2×C22⋊D5311yescharacteristic
279C22—21no—
280C22—22no—
281C22—8482no—
282C22—1061no—
283C22—4242no—
284C22—2122no—
285C22#1313326211yescharacteristic
286C4—21no—
287C4—22no—
288C4—42no—
289C4—1061no—
290C4—2121no—
291C4C2×D4×D5311yescharacteristic
292C4C22×C22⋊D5311yescharacteristic
293C2—21no—
294C2—42no—
295C2—41no—
296C2—4242no—
297C2—2121no—
298C2D4×C22⋊D5311yescharacteristic
subgroupG/H
centerC2D4×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#578C2
socleC2×C53D42
hypercenter#578D53
hypocenterC53#3290
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ #578

Complements

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

NH ≅ G/Ntype
1. #16811431C2semidirect
2. #16811425C2semidirect
3. #16811008C2semidirect
4. #16810895C2semidirect
5. #16810893C2semidirect
6. #16810888C2semidirect
7. #16810799C2semidirect
8. #16810796C2semidirect
9. #16810740C2semidirect
10. #16810738C2semidirect
11. C53×#578C2semidirect
18. #13133242C22semidirect
23. C2×#11719924C22semidirect
26. #13133224C22semidirect
28. #13133206C22semidirect
29. #13133153C22semidirect
30. #13133152C22semidirect
31. #13133144C22semidirect
32. #13133144C22semidirect
34. #13133136C22semidirect
35. #13133133C22semidirect
36. C8×Dic53C22semidirect
37. 2−1+4×C53C22semidirect
39. #137×C53C22semidirect
43. C2×Q16×C53C22semidirect
44. #131×C53C22semidirect
150. #578D53semidirect
161. C53#3290semidirect
162. 2−1+4C2×D53semidirect
167. #137C2×D53semidirect
178. C2×Q16C2×D53semidirect
185. #131C2×D53semidirect