Finite Group Databasebeta

← #20285778

Subgroups — #20285778

Order 6784.

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

normal subgroups only

This group has 388 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#16811228C211yesmaximal, characteristic, complemented
2#16811199C211yesmaximal, characteristic, complemented
3#16811012C211yesmaximal, characteristic, complemented
4#16810996C211yesmaximal, characteristic, complemented
5#16810992C211yesmaximal, characteristic, complemented
6#16810978C211yesmaximal, characteristic, complemented
7#16810839C212yesmaximal, complemented
8#16810829C212yesmaximal, complemented
9#16810770C212yesmaximal, complemented
10#16810764C212yesmaximal, complemented
11C53×#433C211yesmaximal, characteristic, complemented
12C2×#11719945C2211yescharacteristic
13C22×Dic106C2211yescharacteristic
14C2×#11719913C2212yes—
15C2×#11719913C2212yes—
16C2×#11719913C2212yes—
17C2×D424C2212yes—
18#13133217C2211yescharacteristic
19#13133217C2211yescharacteristic
20#13133214C2211yescharacteristic
21#13133209C2211yescharacteristic
22#13133205C2211yescharacteristic
23#13133205C2211yescharacteristic
24#13133202—22no—
25#13133201—22no—
26#13133197—22no—
27#13133191C2212yes—
28#13133190—22no—
29C4×D212C2212yes—
30#13133142C2212yes—
31#13133142C2212yes—
32#13133142C2212yes—
33#13133139C2212yes—
34#117×C53C2211yescharacteristic
35C4⋊C8×C53C2212yes—
36C22⋊C8×C53C2212yes—
37#97×C53C2211yescharacteristic
38C4×C8×C53C2211yescharacteristic
39C22×C4×C53C2311yescharacteristic
40C2×C22⋊D53—21no—
41C2×C22⋊D53—22no—
42C22×Dic53—21no—
43#11719945—42no—
44C2×D212—22no—
45C2×D212C2311yescharacteristic
46C2×D212C2312yes—
47C2×C4×D53—21no—
48C2×C4×D53—22no—
49C2×Dic106—21no—
50C2×Dic106—42no—
51C2×Dic106C2311yescharacteristic
52C2×Dic106C2311yescharacteristic
53C2×Dic106C2311yescharacteristic
54C2×C8×C53C2312yes—
55C2×C8×C53C2312yes—
56C4⋊C4×C53D412yes—
57C22⋊C4×C53D412yes—
58C42×C53C2311yescharacteristic
59C42×C53C2311yescharacteristic
60#11719921—22no—
61#11719921—22no—
62#11719921—22no—
63#11719921—22no—
64C4⋊Dic53—22no—
65C4⋊Dic53—22no—
66C4⋊Dic53C2312yes—
67#11719919—22no—
68D424—42no—
69#11719913—42no—
70#11719913—42no—
71#11719913—42no—
72C23×C53C2×D411yescharacteristic
73C22×D53—22no—
74C22×D53—21no—
75C2×C4×C53—22no—
76C2×C4×C53C2×D411yescharacteristic
77C2×C4×C53C2×D412yes—
78C2×C4×C53C2×D412yes—
79C2×C4×C53C2411yescharacteristic
80C22⋊D53—42no—
81C22⋊D53—82no—
82C2×Dic53—21no—
83C2×Dic53—21no—
84C2×Dic53—21no—
85C2×Dic53—22no—
86C2×Dic53—42no—
87D212—22no—
88D212—22no—
89D212—22no—
90D212—82no—
91C4×D53—42no—
92C4×D53—82no—
93Dic106—22no—
94Dic106—22no—
95Dic106—22no—
96Dic106—42no—
97C8×C53—22no—
98C8×C53—22no—
99C22×C53—21no—
100C22×C53—22no—
101C22×C53C22×D411yescharacteristic
102C2×D53—41no—
103C2×D53—42no—
104C2×D53—42no—
105C2×D53—42no—
106C2×D53—42no—
107C4×C53—21no—
108C4×C53—22no—
109C4×C53—22no—
110C4×C53H(5,𝔽2)12yes—
111Dic53—41no—
112Dic53—41no—
113Dic53—41no—
114Dic53—42no—
115#3221—531nomaximal, Sylow
116C2×C53—41no—
117C2×C53#53511yescharacteristic
118C2×C53#57712yes—
119D53—81no—
120D53—82no—
121#540—531no—
122#536—531no—
123#496—531no—
124#492—531no—
125#491—531no—
126#488—531no—
127#463—532no—
128#461—532no—
129#449—532no—
130#448—532no—
131#433D5311yescharacteristic, complemented
132C53#322111yescharacteristic, complemented, Sylow
133C2×#41—531no—
134C22×Q8—531no—
135C2×SD16—532no—
136C2×SD16—532no—
137C2×SD16—532no—
138C2×D8—532no—
139C4⋊Q8—531no—
140C4⋊Q8—531no—
141#127—531no—
142#125—531no—
143#124—531no—
144#124—531no—
145#123—1062no—
146#123—1062no—
147C22⋊Q8—1062no—
148#121—532no—
149#121—1062no—
150C4×D4—532no—
151#117C2×D5311yescharacteristic
152C4⋊C8C2×D5312yes—
153#103—532no—
154#103—532no—
155#103—532no—
156#102—532no—
157C22⋊C8C2×D5312yes—
158#97C2×D5311yescharacteristic
159C4×C8C2×D5311yescharacteristic
160#41—2122no—
161C2×Q8—531no—
162C2×Q8—531no—
163C2×Q8—531no—
164C2×Q8—1061no—
165C2×Q8—2122no—
166C2×D4—531no—
167C2×D4—532no—
168C2×D4—1062no—
169C2×D4—1061no—
170C2×D4—1062no—
171C22×C4—1061no—
172C22×C4—1062no—
173C22×C4—1061no—
174C22×C4C22×D5311yescharacteristic
175SD16—2122no—
176SD16—2122no—
177SD16—2122no—
178D8—2122no—
179C2×C8C22×D5312yes—
180C2×C8C22×D5312yes—
181C4⋊C4—532no—
182C4⋊C4—1062no—
183C4⋊C4—1062no—
184C4⋊C4—1062no—
185C4⋊C4D21212yes—
186C22⋊C4—1062no—
187C22⋊C4—1062no—
188C22⋊C4—1062no—
189C22⋊C4—1062no—
190C22⋊C4D21212yes—
191C42C22×D5311yescharacteristic
192C42C22×D5311yescharacteristic
193C23—1061no—
194C23—1062no—
195C23C2×D21211yescharacteristic
196Q8—1062no—
197Q8—1062no—
198Q8—1062no—
199Q8—2122no—
200D4—1062no—
201D4—1062no—
202D4—1062no—
203D4—4242no—
204D4—4242no—
205D4—2122no—
206C2×C4—22no—
207C2×C4—1062no—
208C2×C4—1061no—
209C2×C4—1061no—
210C2×C4—1061no—
211C2×C4—4242no—
212C2×C4—2122no—
213C2×C4—2122no—
214C2×C4C2×D21212yes—
215C2×C4C2×D21212yes—
216C2×C4C2×D21211yescharacteristic
217C2×C4C23×D5311yescharacteristic
218C8—22no—
219C8—22no—
220C22—22no—
221C22—21no—
222C22—2122no—
223C22—2122no—
224C22—2122no—
225C22—2121no—
226C22—2122no—
227C22C22×D21211yescharacteristic
228C4—22no—
229C4—22no—
230C4—21no—
231C4—2122no—
232C4—2121no—
233C4—2121no—
234C4—2121no—
235C4#1313326312yes—
236C2—41no—
237C2—4242no—
238C2—4241no—
239C2#1681119211yescharacteristic
240C2#1681143512yes—
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#433C2
socleC22×C53C22×D4
hypercenter#433D53
hypocenterC53#3221
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ #117 ◁ #433

Complements

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

NH ≅ G/Ntype
1. #16811228C2semidirect
2. #16811199C2semidirect
3. #16811012C2semidirect
4. #16810996C2semidirect
5. #16810992C2semidirect
6. #16810978C2semidirect
7. #16810839C2semidirect
8. #16810829C2semidirect
9. #16810770C2semidirect
10. #16810764C2semidirect
11. C53×#433C2semidirect
131. #433D53semidirect
132. C53#3221semidirect