Finite Group Databasebeta

← #20286509

Subgroups — #20286509

Order 6784.

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

normal subgroups only

This group has 460 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#16811435C212yesmaximal, complemented
2#16811430C212yesmaximal, complemented
3#16810996C211yesmaximal, characteristic, complemented
4#16810887C212yesmaximal, complemented
5#16810886C212yesmaximal, complemented
6#16810807C212yesmaximal, complemented
7#16810792C212yesmaximal, complemented
8#16810741C211yesmaximal, characteristic, complemented
9C53×#443C211yesmaximal, characteristic, complemented
10#13133264C2212yes—
11#13133263C2212yescomplemented
12#13133250—22no—
13#13133250C2212yescomplemented
14#13133249—22no—
15#13133247C2212yescomplemented
16#13133245—22no—
17#13133245C2212yes—
18#13133242C2212yes—
19#13133241—22no—
20#13133239—22no—
21#13133239C2211yescharacteristic
22#13133238—22no—
23C2×#11719913—22no—
24C2×#11719913C2212yescomplemented
25C2×#11719913C2212yescomplemented
26SD16×D53—22no—
27#13133223C2212yescomplemented
28#13133217C2211yescharacteristic, complemented
29#13133214C2211yescharacteristic
30#13133153C2211yescharacteristic, complemented
31#13133145C2212yescomplemented
32#13133135C2212yescomplemented
33#13133133C2212yescomplemented
34#135×C53C2211yescharacteristic
35#131×C53C2212yescomplemented
36#108×C53C2211yescharacteristic, complemented
37#104×C53C2212yescomplemented
38C4×C8×C53C2211yescharacteristic, complemented
39#41×C53C2312yes—
40#11719949—22no—
41Q8×D53—22no—
42#11719947—22no—
43#11719947—42no—
44D4×D53—22no—
45D4×D53—42no—
46#11719945—22no—
47#11719945—22no—
48#11719945C2312yes—
49C2×D212—22no—
50C2×D212—41no—
51C2×D212C2312yes—
52C2×Dic106—22no—
53C2×Dic106C2312yes—
54Q16×C53D411yescharacteristic, complemented
55SD16×C53D412yescomplemented
56D8×C53D411yescharacteristic, complemented
57M16×C53—22no—
58M16×C53C2312yes—
59C2×C8×C53—22no—
60C2×C8×C53C2311yescharacteristic
61C2×C8×C53C2311yescharacteristic
62C42×C53C2311yescharacteristic
63#11719925—22no—
64#11719924—22no—
65#11719923—22no—
66#11719922—22no—
67C4⋊Dic53—41no—
68#11719917C2312yes—
69Dic212—22no—
70Dic212D411yescharacteristic, complemented
71D424—22no—
72D424D411yescharacteristic, complemented
73#11719913—22no—
74#11719913—22no—
75#11719913—22no—
76#11719913—22no—
77#11719913—22no—
78#11719913—22no—
79#11719913D412yescomplemented
80#11719912—22no—
81C8×D53—22no—
82C22×D53—22no—
83C22×D53—42no—
84Q8×C53C2×D412yes—
85D4×C53—22no—
86D4×C53C2×D412yes—
87C2×C4×C53—21no—
88C2×C4×C53—22no—
89C2×C4×C53C2411yescharacteristic
90C22⋊D53—22no—
91C22⋊D53—42no—
92C22⋊D53—42no—
93C2×Dic53—22no—
94C2×Dic53—42no—
95D212—22no—
96D212—22no—
97D212—42no—
98D212—82no—
99D212C2×D412yes—
100C4×D53—22no—
101C4×D53—42no—
102C4×D53—42no—
103Dic106—22no—
104Dic106—22no—
105Dic106—42no—
106Dic106C2×D412yes—
107C8×C53—22no—
108C8×C53C2×D411yescharacteristic
109C8×C53C2×D411yescharacteristic
110C8×C53C2×D411yescharacteristic
111C8×C53C2×D411yescharacteristic
112#88322—22no—
113C22×C53—22no—
114C22×C53H(5,𝔽2)11yescharacteristic
115C2×D53—22no—
116C2×D53—42no—
117C2×D53—82no—
118C2×D53—162no—
119C4×C53—22no—
120C4×C53—22no—
121C4×C53C22×D411yescharacteristic
122C4×C53C22×D411yescharacteristic
123Dic53—22no—
124Dic53—42no—
125#3291—531nomaximal, Sylow
126C2×C53—21no—
127C2×C53—42no—
128C2×C53D4211yescharacteristic
129D53—42no—
130D53—82no—
131#577—532no—
132#577—532no—
133#492—531no—
134#471—532no—
135#471—532no—
136#456—532no—
137#453—532no—
138#443—531no—
139#443D5311yescharacteristic, complemented
140C53#329111yescharacteristic, complemented, Sylow
1412−1+4—532no—
142H(5,𝔽2)—532no—
143#137—532no—
144#137—532no—
145#137—1062no—
146#137—1062no—
147AGL(1,ℤ/8)—532no—
148AGL(1,ℤ/8)—532no—
149AGL(1,ℤ/8)—1062no—
150AGL(1,ℤ/8)—1062no—
151#135—531no—
152#135—1062no—
153#135—1062no—
154#135C2×D5311yescharacteristic
155C2×SD16—532no—
156C2×SD16—532no—
157C2×SD16—1062no—
158C2×SD16—1062no—
159#131—532no—
160#131C2×D5312yescomplemented
161C4⋊Q8—531no—
162#127—531no—
163#108—531no—
164#108C2×D5311yescharacteristic, complemented
165#104—532no—
166#104C2×D5312yescomplemented
167#101—532no—
168#100—532no—
169C4×C8C2×D5311yescharacteristic, complemented
170#41—532no—
171#41—1062no—
172#41—1062no—
173#41—1062no—
174#41—1062no—
175#41—2122no—
176#41C22×D5312yes—
177C2×Q8—532no—
178C2×Q8—1062no—
179C2×Q8—1062no—
180C2×D4—532no—
181C2×D4—1062no—
182C2×D4—1062no—
183C2×D4—2121no—
184C2×D4—2122no—
185Q16—531no—
186Q16—1062no—
187Q16—1062no—
188Q16D21211yescharacteristic, complemented
189SD16—532no—
190SD16—1062no—
191SD16—1062no—
192SD16—1062no—
193SD16—1062no—
194SD16—1062no—
195SD16—1062no—
196SD16—1062no—
197SD16—1062no—
198SD16D21212yescomplemented
199D8—531no—
200D8—1062no—
201D8—1062no—
202D8D21211yescharacteristic, complemented
203M16—22no—
204M16—532no—
205M16—1062no—
206M16C22×D5312yes—
207C2×C8—22no—
208C2×C8—1062no—
209C2×C8C22×D5311yescharacteristic
210C2×C8C22×D5311yescharacteristic
211C4⋊C4—2121no—
212C42C22×D5311yescharacteristic
213C23—1062no—
214C23—2122no—
215Q8—532no—
216Q8—1062no—
217Q8—1062no—
218Q8—2122no—
219Q8C2×D21212yes—
220D4—22no—
221D4—532no—
222D4—1062no—
223D4—1062no—
224D4—1062no—
225D4—4242no—
226D4—2122no—
227D4—2122no—
228D4—2122no—
229D4C2×D21212yes—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—1062no—
233C2×C4—1062no—
234C2×C4—2122no—
235C2×C4—2122no—
236C2×C4—2122no—
237C2×C4C23×D5311yescharacteristic
238C8—22no—
239C8—1062no—
240C8C2×D21211yescharacteristic
241C8C2×D21211yescharacteristic
242C8D4×D5311yescharacteristic
243C8D4×D5311yescharacteristic
244C22—22no—
245C22—8482no—
246C22—1062no—
247C22—4242no—
248C22—2122no—
249C22#1313326311yescharacteristic
250C4—22no—
251C4—22no—
252C4—1062no—
253C4—2122no—
254C4C2×D4×D5311yescharacteristic
255C4C22×D21211yescharacteristic
256C2—21no—
257C2—42no—
258C2—4242no—
259C2—2122no—
260C2D4×D21211yescharacteristic
subgroupG/H
centerC2D4×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#443C2
socleC2×C53D42
hypercenter#443D53
hypocenterC53#3291
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ #443

Complements

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

NH ≅ G/Ntype
1. #16811435C2semidirect
2. #16811430C2semidirect
3. #16810996C2semidirect
4. #16810887C2semidirect
5. #16810886C2semidirect
6. #16810807C2semidirect
7. #16810792C2semidirect
8. #16810741C2semidirect
9. C53×#443C2semidirect
11. #13133263C22semidirect
13. #13133250C22semidirect
15. #13133247C22semidirect
24. C2×#11719913C22semidirect
25. C2×#11719913C22semidirect
27. #13133223C22semidirect
28. #13133217C22semidirect
30. #13133153C22semidirect
31. #13133145C22semidirect
32. #13133135C22semidirect
33. #13133133C22semidirect
35. #131×C53C22semidirect
36. #108×C53C22semidirect
37. #104×C53C22semidirect
38. C4×C8×C53C22semidirect
54. Q16×C53D4semidirect
55. SD16×C53D4semidirect
56. D8×C53D4semidirect
70. Dic212D4semidirect
72. D424D4semidirect
79. #11719913D4semidirect
139. #443D53semidirect
140. C53#3291semidirect
160. #131C2×D53semidirect
164. #108C2×D53semidirect
166. #104C2×D53semidirect
169. C4×C8C2×D53semidirect
188. Q16D212semidirect
198. SD16D212semidirect
202. D8D212semidirect