Finite Group Databasebeta

← #20285112

Subgroups — #20285112

Order 6784.

392 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 117 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#16811172C211yesmaximal, characteristic
2#125×D53C211yesmaximal, characteristic, complemented
3#16810990C212yesmaximal, complemented
4#16810986C212yesmaximal, complemented
5#16810980C211yesmaximal, characteristic, complemented
6#16810978C211yesmaximal, characteristic, complemented
7#16810977C211yesmaximal, characteristic, complemented
8#16810623C212yesmaximal, complemented
9#16810621C212yesmaximal, complemented
10C4×#11719912C211yesmaximal, characteristic, complemented
11C53×#488C211yesmaximal, characteristic, complemented
12C2×#11719949C2211yescharacteristic
13C2×#11719947C2211yescharacteristic
14C2×#11719912C2212yes—
15#13133215C2211yescharacteristic
16#13133213C2211yescharacteristic
17#13133209C2211yescharacteristic
18#13133208C2212yes—
19#13133207—22no—
20#13133205C2211yescharacteristic
21#13133190—22no—
22#13133177—22no—
23C4⋊C4×D53—22no—
24C4⋊C4×D53C2212yes—
25C42×D53C2211yescharacteristic
26#13133142C2212yes—
27#13133141C2212yes—
28#13133140C2212yes—
29#13133139C2212yes—
30#13133138C2212yes—
31#13133137C2212yes—
32#13133127C2212yes—
33C4×#88322C2211yescharacteristic
34#125×C53C2211yescharacteristic
35#124×C53C2211yescharacteristic
36#103×C53C2212yes—
37#102×C53C2212yes—
38C4×C8×C53C2211yescharacteristic
39C2×Q8×C53C2311yescharacteristic
40C2×D4×C53C2311yescharacteristic
41C2×C22⋊D53—21no—
42C2×C22⋊D53—22no—
43C22×Dic53—21no—
44#11719949—24no—
45#11719947—24no—
46C2×D212—21no—
47C2×D212C2311yescharacteristic
48C2×C4×D53—22no—
49C2×C4×D53—21no—
50C2×C4×D53D412yes—
51C2×C4×D53C2311yescharacteristic
52C2×Dic106C2311yescharacteristic
53C2×C8×C53C2312yes—
54C4⋊C4×C53—22no—
55C4⋊C4×C53C2312yes—
56C22⋊C4×C53—22no—
57C42×C53C2311yescharacteristic
58#11719921—22no—
59C4⋊Dic53—22no—
60C4⋊Dic53C2312yes—
61#11719919—22no—
62#11719919—22no—
63#11719919—22no—
64#11719919—22no—
65C4×Dic53D412yes—
66C4×Dic53C2311yescharacteristic
67C2×#88322C2312yes—
68#11719912—24no—
69C23×C53—21no—
70C22×D53—21no—
71C22×D53C2×D411yescharacteristic
72Q8×C53—22no—
73D4×C53—22no—
74C2×C4×C53—21no—
75C2×C4×C53—22no—
76C2×C4×C53C2×D412yes—
77C2×C4×C53C2411yescharacteristic
78C22⋊D53—44no—
79C22⋊D53—42no—
80C2×Dic53—21no—
81C2×Dic53—22no—
82C2×Dic53—42no—
83C2×Dic53C2×D412yes—
84C2×Dic53C2×D411yescharacteristic
85D212—22no—
86D212—42no—
87C4×D53—42no—
88C4×D53—44no—
89C4×D53—42no—
90C4×D53#4114yes—
91Dic106—22no—
92C8×C53—22no—
93#88322—22no—
94C22×C53—41no—
95C22×C53—42no—
96C22×C53C22×D411yescharacteristic
97C2×D53—22no—
98C2×D53—41no—
99C2×D53—42no—
100C2×D53#12412yes—
101C4×C53—22no—
102C4×C53—41no—
103C4×C53—42no—
104C4×C53C2×#4112yes—
105Dic53—22no—
106Dic53—41no—
107Dic53—42no—
108Dic53#12412yes—
109#3142—531nomaximal, Sylow
110C2×C53—81no—
111C2×C53C2×#12411yescharacteristic
112C2×C53#57512yes—
113D53—22no—
114D53—81no—
115#532—531no—
116C2×#125—531no—
117#491—532no—
118#490—532no—
119#488—531no—
120#488—531no—
121#488—531no—
122#488D5311yescharacteristic, complemented
123#417—532no—
124#417—532no—
125C4×M16—531no—
126C53#314211yescharacteristic, complemented, Sylow
127C2×#41—531no—
128C2×#41—531no—
129C2×M16—532no—
130C4⋊Q8—531no—
131#127—531no—
132#125—532no—
133#125—531no—
134#125—1062no—
135#125C2×D5311yescharacteristic
136#124—531no—
137#124C2×D5311yescharacteristic
138#121—1062no—
139C4×D4—1062no—
140C2×C4⋊C4—532no—
141C2×C4⋊C4—1062no—
142C2×C42—531no—
143#103—532no—
144#103—532no—
145#103—532no—
146#103C2×D5312yes—
147#102—532no—
148#102—532no—
149#102—532no—
150#102C2×D5312yes—
151#97—532no—
152C4×C8—531no—
153C4×C8C2×D5311yescharacteristic
154#41—1064no—
155#41—1064no—
156C2×Q8—531no—
157C2×Q8C22×D5311yescharacteristic
158C2×D4—531no—
159C2×D4—1061no—
160C2×D4—1062no—
161C2×D4—1061no—
162C2×D4C22×D5311yescharacteristic
163C22×C4—531no—
164C22×C4—532no—
165C22×C4—1061no—
166C22×C4—1062no—
167C22×C4—1061no—
168M16—1064no—
169C2×C8—532no—
170C2×C8C22×D5312yes—
171C4⋊C4—22no—
172C4⋊C4—532no—
173C4⋊C4—1062no—
174C4⋊C4—1062no—
175C4⋊C4—1062no—
176C4⋊C4—1062no—
177C4⋊C4—1062no—
178C4⋊C4C22×D5312yes—
179C22⋊C4—22no—
180C22⋊C4—1062no—
181C42—532no—
182C42—531no—
183C42C22×D5311yescharacteristic
184C23—21no—
185C23—531no—
186C23—1061no—
187Q8—22no—
188Q8—1062no—
189D4—22no—
190D4—1062no—
191D4—2122no—
192D4—2124no—
193D4—2122no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—531no—
197C2×C4—534no—
198C2×C4—532no—
199C2×C4—1061no—
200C2×C4—1062no—
201C2×C4—2122no—
202C2×C4—2122no—
203C2×C4—2124no—
204C2×C4—2122no—
205C2×C4D4×D5312yes—
206C2×C4C23×D5311yescharacteristic
207C8—22no—
208C8—1062no—
209C22—41no—
210C22—42no—
211C22—532no—
212C22—1062no—
213C22—2121no—
214C22—2122no—
215C22C2×D4×D5311yescharacteristic
216C4—22no—
217C4—41no—
218C4—42no—
219C4—532no—
220C4—1062no—
221C4—2121no—
222C4—2122no—
223C4#41×D5312yes—
224C2—81no—
225C2—1062no—
226C2—4241no—
227C2#124×D5311yescharacteristic
228C2#1681142012yes—
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#488C2
socleC22×C53C22×D4
hypercenter#488D53
hypocenterC53#3142
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
2. #125×D53C2semidirect
3. #16810990C2semidirect
4. #16810986C2semidirect
5. #16810980C2semidirect
6. #16810978C2semidirect
7. #16810977C2semidirect
8. #16810623C2semidirect
9. #16810621C2semidirect
10. C4×#11719912C2semidirect
11. C53×#488C2semidirect
122. #488D53semidirect
126. C53#3142semidirect