Finite Group Databasebeta

← #20285372

Subgroups — #20285372

Order 6784.

338 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 115 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#16811231C212yesmaximal
2#16811023C211yesmaximal, characteristic, complemented
3#16811001C211yesmaximal, characteristic, complemented
4#16810990C212yesmaximal, complemented
5#16810882C211yesmaximal, characteristic
6#16810878C211yesmaximal, characteristic
7#16810863C212yesmaximal
8#16810731C211yesmaximal, characteristic, complemented
9C4×#11719925C212yesmaximal, complemented
10#16810580C211yesmaximal, characteristic
11C53×#441C211yesmaximal, characteristic, complemented
12C2×Dic212C2211yescharacteristic
13C2×#11719925C2212yes—
14C2×#11719912C2211yescharacteristic
15C2×C8×D53C2211yescharacteristic
16#13133215C2212yes—
17#13133212—22no—
18#13133211—22no—
19#13133209—22no—
20#13133208C2212yes—
21#13133207—22no—
22#13133197C2212yes—
23#13133193C2212yes—
24C4×Dic106C2212yes—
25#13133174C2211yescharacteristic
26#13133150C2212yes—
27#13133149C2211yescharacteristic
28#13133142C2212yes—
29#13133141C2212yes—
30#13133140C2212yes—
31#13133127C2211yescharacteristic
32C4×#88322C2211yescharacteristic
33C8×Dic53C2211yescharacteristic
34C2×Q16×C53C2211yescharacteristic
35C4×Q8×C53C2212yes—
36#106×C53C2211yescharacteristic
37#103×C53C2212yes—
38#97×C53C2211yescharacteristic
39C2×Q8×C53C2312yes—
40C2×C4×D53C2311yescharacteristic
41C2×Dic106C2312yes—
42Q16×C53—22no—
43C2×C8×C53C2311yescharacteristic
44C2×C8×C53C2311yescharacteristic
45C4⋊C4×C53—22no—
46C4⋊C4×C53C2312yes—
47C42×C53—22no—
48C42×C53C2311yescharacteristic
49#11719925—24no—
50#11719921—22no—
51#11719921—22no—
52#11719921D412yes—
53C4⋊Dic53—22no—
54C4⋊Dic53—22no—
55C4⋊Dic53C2312yes—
56#11719919—22no—
57#11719919—22no—
58#11719919—22no—
59#11719919—22no—
60#11719919—22no—
61#11719919D412yes—
62C4×Dic53—22no—
63C4×Dic53C2311yescharacteristic
64C2×#88322C2311yescharacteristic
65C2×#88322C2311yescharacteristic
66Dic212—22no—
67#11719912—22no—
68C8×D53—22no—
69C22×D53C2×D411yescharacteristic
70Q8×C53—22no—
71Q8×C53—22no—
72C2×C4×C53—22no—
73C2×C4×C53—22no—
74C2×C4×C53C2×D412yes—
75C2×C4×C53C2411yescharacteristic
76C2×Dic53—22no—
77C2×Dic53—22no—
78C2×Dic53C2×D411yescharacteristic
79C2×Dic53C2×D412yes—
80C4×D53—21no—
81C4×D53—21no—
82Dic106—22no—
83Dic106—22no—
84C8×C53—21no—
85C8×C53#4112yes—
86#88322—21no—
87#88322#4112yes—
88C22×C53C22×D411yescharacteristic
89C2×D53—21no—
90C2×D53—22no—
91C4×C53—22no—
92C4×C53—42no—
93C4×C53—42no—
94C4×C53C2×#4111yescharacteristic
95C4×C53C2×#4111yescharacteristic
96Dic53—21no—
97Dic53—22no—
98Dic53—42no—
99Dic53—42no—
100#3177—531nomaximal, Sylow
101C2×C53#53211yescharacteristic
102C2×C53#57812yes—
103D53—41no—
104#541—532no—
105#499—531no—
106#494—531no—
107#491—532no—
108#470—531no—
109#470—531no—
110#467—532no—
111#441—531no—
112#441D5311yescharacteristic, complemented
113C4×Q16—532no—
114#405—531no—
115C53#317711yescharacteristic, complemented, Sylow
116C2×Q16—532no—
117C2×Q16—531no—
118C2×Q16C2×D5311yescharacteristic
119C2×M16—531no—
120C22×C8—531no—
121C4⋊Q8—532no—
122#126—1062no—
123#126—1062no—
124#125—532no—
125#125—1062no—
126#125—1062no—
127C22⋊Q8—532no—
128C22⋊Q8—532no—
129C4×Q8—532no—
130C4×Q8C2×D5312yes—
131#117—531no—
132#107—532no—
133#106—531no—
134#106C2×D5311yescharacteristic
135#103—532no—
136#103—532no—
137#103—532no—
138#103C2×D5312yes—
139#97—531no—
140#97C2×D5311yescharacteristic
141C4×C8—531no—
142C4×C8—531no—
143C2×Q8—532no—
144C2×Q8C22×D5312yes—
145C22×C4—531no—
146Q16—22no—
147Q16—1064no—
148Q16—1062no—
149M16—1062no—
150C2×C8—531no—
151C2×C8—531no—
152C2×C8—1062no—
153C2×C8C22×D5311yescharacteristic
154C2×C8C22×D5311yescharacteristic
155C4⋊C4—22no—
156C4⋊C4—532no—
157C4⋊C4—532no—
158C4⋊C4—1062no—
159C4⋊C4—1062no—
160C4⋊C4—1062no—
161C4⋊C4—1062no—
162C4⋊C4—1062no—
163C4⋊C4—1062no—
164C4⋊C4—1062no—
165C4⋊C4C22×D5312yes—
166C22⋊C4—532no—
167C22⋊C4—1062no—
168C22⋊C4—1062no—
169C42—22no—
170C42—531no—
171C42—1062no—
172C42C22×D5311yescharacteristic
173C23—531no—
174Q8—22no—
175Q8—22no—
176Q8—1062no—
177Q8—1062no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—532no—
181C2×C4—531no—
182C2×C4—1061no—
183C2×C4—1061no—
184C2×C4—1062no—
185C2×C4—1062no—
186C2×C4D4×D5312yes—
187C2×C4C23×D5311yescharacteristic
188C8—21no—
189C8—532no—
190C8—1061no—
191C8#1171994512yes—
192C22—1061no—
193C22—1062no—
194C22C2×D4×D5311yescharacteristic
195C4—22no—
196C4—42no—
197C4—42no—
198C4—1061no—
199C4—1062no—
200C4—2122no—
201C4—2122no—
202C4#41×D5311yescharacteristic
203C4C2×#1171994511yescharacteristic
204C2—2121no—
205C2#1681117011yescharacteristic
206C2#1681143912yes—
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#441C2
socleC22×C53C22×D4
hypercenter#441D53
hypocenterC53#3177
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #441

Complements

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

NH ≅ G/Ntype
2. #16811023C2semidirect
3. #16811001C2semidirect
4. #16810990C2semidirect
8. #16810731C2semidirect
9. C4×#11719925C2semidirect
11. C53×#441C2semidirect
112. #441D53semidirect
115. C53#3177semidirect