Finite Group Databasebeta

← #20285179

Subgroups — #20285179

Order 6784.

572 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#13133248C212yesmaximal, complemented
2C2×#13133243C212yesmaximal, complemented
3#16811173C211yesmaximal, characteristic, complemented
4#16811018C212yesmaximal, complemented
5#16811013C212yesmaximal, complemented
6#16811005C214yesmaximal, complemented
7C4×#11719917C211yesmaximal, characteristic, complemented
8C2×#124×C53C211yesmaximal, characteristic, complemented
9C2×#11719945C2212yes—
10#13133248—28no—
11#13133243—28no—
12C2×#11719925C2214yescomplemented
13C2×#11719923C2214yes—
14C2×#11719924C2214yescomplemented
15C2×#11719922C2214yes—
16C2×#11719917C2212yes—
17#13133217C2211yescharacteristic, complemented
18#13133214C2211yescharacteristic, complemented
19#13133205C2212yescomplemented
20#13133191—22no—
21C4×D212—22no—
22#13133128C2212yescomplemented
23C4×#88322C2212yescomplemented
24C22×Q8×C53C2211yescharacteristic
25C22×D4×C53C2211yescharacteristic
26#124×C53—22no—
27#124×C53C2212yes—
28#124×C53C2212yes—
29C2×C22⋊C4×C53—22no—
30C2×C42×C53C2211yescharacteristic
31C24×C53—21no—
32C2×Q8×C53—21no—
33C2×Q8×C53—24no—
34C2×Q8×C53C2312yes—
35C2×D4×C53—24no—
36C2×D4×C53—21no—
37C2×D4×C53C2312yes—
38C22×C4×C53—21no—
39C22×C4×C53D412yes—
40C22×C4×C53C2311yescharacteristic
41C2×C22⋊D53—22no—
42#11719945—24no—
43#11719945—24no—
44C2×D212—22no—
45C2×D212C2312yes—
46C2×C4×D53—22no—
47C2×Dic106C2312yes—
48C22⋊C4×C53—24no—
49C22⋊C4×C53—24no—
50C42×C53D412yes—
51C42×C53C2311yescharacteristic
52C42×C53C2311yescharacteristic
53#11719925—28no—
54#11719924—28no—
55#11719923—28no—
56#11719922—28no—
57#11719921—24no—
58C4⋊Dic53—22no—
59#11719917D418yescomplemented
60C2×#88322C2314yes—
61C23×C53—22no—
62C23×C53—44no—
63C23×C53—42no—
64C23×C53C2×D411yescharacteristic
65C22×D53—22no—
66Q8×C53—24no—
67Q8×C53—42no—
68D4×C53—24no—
69D4×C53—42no—
70C2×C4×C53—22no—
71C2×C4×C53—42no—
72C2×C4×C53—42no—
73C2×C4×C53C2×D414yes—
74C2×C4×C53C2×D412yes—
75C2×C4×C53C2×D412yes—
76C2×C4×C53C2×D411yescharacteristic
77C2×C4×C53C2411yescharacteristic
78C22⋊D53—44no—
79C2×Dic53—22no—
80D212—22no—
81D212—22no—
82D212—44no—
83C4×D53—44no—
84Dic106—22no—
85Dic106—22no—
86#88322C2×D418yes—
87C22×C53—22no—
88C22×C53—42no—
89C22×C53—44no—
90C22×C53—84no—
91C22×C53#12712yes—
92C22×C53C22×D411yescharacteristic
93C2×D53—42no—
94C2×D53—44no—
95C4×C53—22no—
96C4×C53—22no—
97C4×C53—42no—
98C4×C53#12712yes—
99C4×C53C22×D412yes—
100Dic53—42no—
101#3153—531nomaximal, Sylow
102C2×C53—22no—
103C2×C53—82no—
104C2×C53C2×#12711yescharacteristic
105C2×C53#57512yes—
106D53—82no—
107C2×#137—532no—
108C2×AGL(1,ℤ/8)—532no—
109#532—531no—
110C2×#124D5311yescharacteristic, complemented
111#497—532no—
112#496—532no—
113#495—534no—
114C4×M16—531no—
115C53#315311yescharacteristic, complemented, Sylow
116C2×#41—532no—
117C22×Q8C2×D5311yescharacteristic
118C22×D4C2×D5311yescharacteristic
119#137—1068no—
120AGL(1,ℤ/8)—1068no—
121C2×Q16—534no—
122C2×SD16—534no—
123C2×SD16—534no—
124C2×D8—534no—
125C2×M16—532no—
126C4⋊Q8—531no—
127#127—531no—
128#124—22no—
129#124—532no—
130#124C2×D5312yes—
131#124C2×D5312yes—
132#121—1062no—
133C4×D4—1062no—
134C2×C22⋊C4—22no—
135C2×C42C2×D5311yescharacteristic
136#97—532no—
137C4×C8—532no—
138C24—21no—
139#41—1064no—
140#41—1064no—
141C2×Q8—21no—
142C2×Q8—24no—
143C2×Q8—532no—
144C2×Q8C22×D5312yes—
145C2×D4—24no—
146C2×D4—21no—
147C2×D4—532no—
148C2×D4—1062no—
149C2×D4—1062no—
150C2×D4C22×D5312yes—
151C22×C4—21no—
152C22×C4—1062no—
153C22×C4C22⋊D5312yes—
154C22×C4C22×D5311yescharacteristic
155Q16—1068no—
156SD16—1068no—
157SD16—1068no—
158D8—1068no—
159M16—538no—
160C2×C8—534no—
161C4⋊C4—1062no—
162C22⋊C4—24no—
163C22⋊C4—24no—
164C22⋊C4—1064no—
165C42C22⋊D5312yes—
166C42C22×D5311yescharacteristic
167C42C22×D5311yescharacteristic
168C23—22no—
169C23—44no—
170C23—42no—
171C23—1062no—
172C23C2×C22⋊D5311yescharacteristic
173Q8—24no—
174Q8—42no—
175Q8—1062no—
176Q8—1062no—
177D4—24no—
178D4—42no—
179D4—1062no—
180D4—1062no—
181D4—2124no—
182D4—2124no—
183C2×C4—22no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—1062no—
187C2×C4—2124no—
188C2×C4D4×D5314yes—
189C2×C4C2×C22⋊D5312yes—
190C2×C4C2×C22⋊D5312yes—
191C2×C4C2×C22⋊D5311yescharacteristic
192C2×C4C23×D5311yescharacteristic
193C8—538no—
194C22—22no—
195C22—42no—
196C22—44no—
197C22—84no—
198C22—2122no—
199C22—2124no—
200C22#1313321312yes—
201C22C22×C22⋊D5311yescharacteristic
202C4—22no—
203C4—22no—
204C4—42no—
205C4—2122no—
206C4#1313321312yes—
207C4C2×D4×D5312yes—
208C2—22no—
209C2—82no—
210C2—4242no—
211C2C2×#1313321311yescharacteristic
212C2#1681142112yes—
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC2×#124×C53C2
socleC22×C53C22×D4
hypercenterC2×#124D53
hypocenterC53#3153
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. C2×#13133248C2semidirect
2. C2×#13133243C2semidirect
3. #16811173C2semidirect
4. #16811018C2semidirect
5. #16811013C2semidirect
6. #16811005C2semidirect
7. C4×#11719917C2semidirect
8. C2×#124×C53C2semidirect
12. C2×#11719925C22semidirect
14. C2×#11719924C22semidirect
17. #13133217C22semidirect
18. #13133214C22semidirect
19. #13133205C22semidirect
22. #13133128C22semidirect
23. C4×#88322C22semidirect
59. #11719917D4semidirect
110. C2×#124D53semidirect
115. C53#3153semidirect