Finite Group Databasebeta

← #20285117

Subgroups — #20285117

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#16811174C211yesmaximal, characteristic
2#125×D53C211yesmaximal, characteristic, complemented
3#16810991C211yesmaximal, characteristic, complemented
4#16810989C211yesmaximal, characteristic, complemented
5#16810988C211yesmaximal, characteristic, complemented
6#16810979C214yesmaximal, complemented
7#16810623C214yesmaximal, complemented
8#16810575C211yesmaximal, characteristic, complemented
9C53×#491C211yesmaximal, characteristic, complemented
10C2×#11719949C2212yes—
11C2×#11719912C2212yes—
12#13133214C2211yescharacteristic
13#13133209C2211yescharacteristic
14#13133208C2212yes—
15#13133207—22no—
16#13133206C2212yes—
17#13133192—22no—
18#13133179—22no—
19C4⋊C4×D53—22no—
20C4⋊C4×D53C2212yes—
21C42×D53C2211yescharacteristic
22#13133141C2214yes—
23#13133139C2214yes—
24#13133137C2214yes—
25#13133128C2211yescharacteristic
26C8×Dic53C2212yes—
27C4⋊Q8×C53C2211yescharacteristic
28#125×C53C2211yescharacteristic
29#103×C53C2214yes—
30#97×C53C2211yescharacteristic
31C2×Q8×C53C2312yes—
32#11719949—28no—
33C2×D212—22no—
34C2×D212—22no—
35C2×D212C2312yes—
36C2×C4×D53—22no—
37C2×C4×D53—22no—
38C2×C4×D53D412yes—
39C2×C4×D53C2311yescharacteristic
40C2×C8×C53C2312yes—
41C4⋊C4×C53—22no—
42C4⋊C4×C53—22no—
43C4⋊C4×C53C2312yes—
44C42×C53C2311yescharacteristic
45#11719921—24no—
46C4⋊Dic53—22no—
47C4⋊Dic53C2312yes—
48#11719919—22no—
49#11719919—22no—
50#11719919—22no—
51C4×Dic53D412yes—
52C4×Dic53C2311yescharacteristic
53C2×#88322C2312yes—
54#11719912—24no—
55C22×D53—22no—
56C22×D53C2×D411yescharacteristic
57Q8×C53—24no—
58C2×C4×C53—22no—
59C2×C4×C53—22no—
60C2×C4×C53C2×D412yes—
61C2×C4×C53C2411yescharacteristic
62C2×Dic53—22no—
63C2×Dic53C2×D412yes—
64C2×Dic53C2×D411yescharacteristic
65D212—24no—
66D212—44no—
67D212—44no—
68C4×D53—42no—
69C4×D53—44no—
70C4×D53—44no—
71C4×D53#4114yes—
72C8×C53—22no—
73#88322—22no—
74C22×C53C22×D411yescharacteristic
75C2×D53—22no—
76C2×D53—42no—
77C2×D53—44no—
78C2×D53#12412yes—
79C4×C53—22no—
80C4×C53—42no—
81C4×C53—42no—
82C4×C53C2×#4112yes—
83Dic53—22no—
84Dic53—42no—
85Dic53#12412yes—
86#3142—531nomaximal, Sylow
87C2×C53C2×#12411yescharacteristic
88C2×C53#57512yes—
89D53—22no—
90D53—82no—
91#532—531no—
92C2×#125—531no—
93#491—531no—
94#491D5311yescharacteristic, complemented
95#490—531no—
96#490—531no—
97#488—534no—
98#417—534no—
99C4×M16—531no—
100C53#314211yescharacteristic, complemented, Sylow
101C2×#41—532no—
102C2×M16—532no—
103C4⋊Q8C2×D5311yescharacteristic
104#127—531no—
105#125—532no—
106#125—531no—
107#125—1062no—
108#125C2×D5311yescharacteristic
109#124—532no—
110#121—1062no—
111C4×D4—1062no—
112C2×C4⋊C4—532no—
113C2×C4⋊C4—1062no—
114C2×C42—531no—
115#103—534no—
116#103C2×D5314yes—
117#102—534no—
118#102—534no—
119#97—531no—
120#97C2×D5311yescharacteristic
121C4×C8—532no—
122#41—1068no—
123C2×Q8C22×D5312yes—
124C2×D4—532no—
125C2×D4—1062no—
126C2×D4—1062no—
127C22×C4—531no—
128C22×C4—532no—
129C22×C4—1062no—
130C22×C4—1062no—
131M16—1064no—
132C2×C8—532no—
133C2×C8C22×D5312yes—
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—532no—
137C4⋊C4—1062no—
138C4⋊C4—1062no—
139C4⋊C4—1062no—
140C4⋊C4—1062no—
141C4⋊C4C22×D5312yes—
142C22⋊C4—1064no—
143C42—532no—
144C42—531no—
145C42C22×D5311yescharacteristic
146C23—531no—
147C23—1062no—
148Q8—24no—
149D4—1064no—
150D4—2124no—
151D4—2124no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—531no—
155C2×C4—534no—
156C2×C4—532no—
157C2×C4—1062no—
158C2×C4—2122no—
159C2×C4—2124no—
160C2×C4—2124no—
161C2×C4D4×D5312yes—
162C2×C4C23×D5311yescharacteristic
163C8—22no—
164C8—1062no—
165C22—532no—
166C22—1062no—
167C22—2122no—
168C22—2124no—
169C22C2×D4×D5311yescharacteristic
170C4—22no—
171C4—42no—
172C4—42no—
173C4—532no—
174C4—1062no—
175C4—2122no—
176C4#41×D5312yes—
177C2—1062no—
178C2—4242no—
179C2#124×D5311yescharacteristic
180C2#1681142212yes—
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#491C2
socleC22×C53C22×D4
hypercenter#491D53
hypocenterC53#3142
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
2. #125×D53C2semidirect
3. #16810991C2semidirect
4. #16810989C2semidirect
5. #16810988C2semidirect
6. #16810979C2semidirect
7. #16810623C2semidirect
8. #16810575C2semidirect
9. C53×#491C2semidirect
94. #491D53semidirect
100. C53#3142semidirect