Finite Group Databasebeta

← #393×D53 #20277913

Subgroups — #393×D53 #20277913

Order 6784.

1036 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 1036 conjugacy classes of subgroups — too many to draw. Only its 203 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
1C22×Q8×D53C211yesmaximal, characteristic, complemented
2C2×C22⋊C4×D53C213yesmaximal
3#16810536C213yesmaximal, complemented
4#16810535C213yesmaximal, complemented
5#16810534C211yesmaximal, characteristic, complemented
6#95×D53C213yesmaximal, complemented
7C53×#393C211yesmaximal, characteristic, complemented
8C24×D53C2211yescharacteristic
9C22×Dic106C2213yescomplemented
10C2×Q8×D53—212no—
11C22×C4×D53C2213yes—
12C22×C4×D53C2213yes—
13C2×C22⋊Dic53C2213yes—
14C2×#11719921C2213yes—
15C2×#11719921C2213yes—
16C22⋊C4×D53—212no—
17#13133124C2213yescomplemented
18#13133124C2216yescomplemented
19C22×Q8×C53C2211yescharacteristic, complemented
20C2×C22⋊C4×C53C2213yes—
21#95×C53C2213yescomplemented
22C24×C53C2311yescharacteristic
23C23×D53—26no—
24C23×D53—26no—
25C23×D53Q814yescomplemented
26C23×D53C2311yescharacteristic
27C23×D53C2311yescharacteristic
28C2×Q8×C53—26no—
29C22×C4×C53C2313yes—
30C22×C4×C53C2313yes—
31C22×Dic53C2313yes—
32C22×Dic53C2313yes—
33Q8×D53—232no—
34C2×C4×D53—212no—
35C2×C4×D53—26no—
36C2×C4×D53—212no—
37C2×C4×D53D4112yes—
38C2×Dic106—26no—
39C2×Dic106—212no—
40C22⋊C4×C53—26no—
41C22⋊Dic53—26no—
42#11719921—26no—
43#11719921—26no—
44C23×C53—23no—
45C23×C53—23no—
46C23×C53C2×Q812yescomplemented
47C23×C53C2411yescharacteristic
48C22×D53—23no—
49C22×D53—23no—
50C22×D53—224no—
51C22×D53—44no—
52C22×D53—412no—
53C22×D53C2×D416yes—
54C22×D53C2×Q812yes—
55C22×D53C2×Q812yescomplemented
56C22×D53#4116yes—
57Q8×C53—28no—
58C2×C4×C53—26no—
59C2×C4×C53—23no—
60C2×C4×C53—26no—
61C2×C4×C53C2×D416yes—
62C2×Dic53—26no—
63C2×Dic53—23no—
64C2×Dic53—26no—
65C2×Dic53C2×D416yes—
66C4×D53—224no—
67C4×D53—412no—
68Dic106—224no—
69C22×C53—26no—
70C22×C53—41no—
71C22×C53—43no—
72C22×C53C22×D413yes—
73C22×C53C22×Q811yescharacteristic
74C22×C53C2×#4113yes—
75C2×D53—26no—
76C2×D53—41no—
77C2×D53—43no—
78C2×D53—416no—
79C2×D53#12014yes—
80C2×D53C22⋊Q8112yes—
81C2×D53#124112yes—
82C4×C53—26no—
83C4×C53—43no—
84Dic53—26no—
85Dic53—43no—
86C2×#393—531nomaximal, Sylow
87C2×C53—42no—
88C2×C53C2×#12011yescharacteristic
89C2×C53C2×C22⋊Q813yes—
90C2×C53C2×#12413yes—
91D53—42no—
92D53#39318yesdirect product factor
93C23×Q8—531no—
94C22×C22⋊C4—533no—
95#393—531no—
96#393—533no—
97#393—533no—
98#393D5311yescharacteristic, direct product factor
99C2×#95—533no—
100C53C2×#39311yescharacteristic, complemented, Sylow
101C25—531no—
102C22×Q8—533no—
103C22×Q8—10612no—
104C22×Q8C2×D5311yescharacteristic, complemented
105C23×C4—533no—
106C23×C4—533no—
107C2×C22⋊C4—533no—
108C2×C22⋊C4—533no—
109C2×C22⋊C4—533no—
110C2×C22⋊C4—10612no—
111C2×C22⋊C4C2×D5313yes—
112#95—536no—
113#95—533no—
114#95C2×D5313yescomplemented
115C24—531no—
116C24—534no—
117C24—531no—
118C24—1066no—
119C24—1066no—
120C24C22×D5311yescharacteristic
121C2×Q8—26no—
122C2×Q8—10612no—
123C2×Q8—10632no—
124C2×Q8—1066no—
125C22×C4—533no—
126C22×C4—5312no—
127C22×C4—533no—
128C22×C4—10612no—
129C22×C4—1066no—
130C22×C4—10612no—
131C22×C4C22×D5313yes—
132C22×C4C22×D5313yes—
133C22⋊C4—26no—
134C22⋊C4—1066no—
135C22⋊C4—1066no—
136C22⋊C4—1066no—
137C23—23no—
138C23—23no—
139C23—536no—
140C23—532no—
141C23—536no—
142C23—532no—
143C23—10624no—
144C23—1063no—
145C23—1063no—
146C23—2124no—
147C23—21212no—
148C23Q8×D5312yescomplemented
149C23C23×D5311yescharacteristic
150Q8—28no—
151Q8—10624no—
152C2×C4—26no—
153C2×C4—23no—
154C2×C4—26no—
155C2×C4—536no—
156C2×C4—10624no—
157C2×C4—1066no—
158C2×C4—1063no—
159C2×C4—1066no—
160C2×C4—21212no—
161C2×C4D4×D5316yes—
162C22—26no—
163C22—41no—
164C22—43no—
165C22—534no—
166C22—5312no—
167C22—5312no—
168C22—1066no—
169C22—21216no—
170C22—2121no—
171C22—2123no—
172C22C2×D4×D5313yes—
173C22C2×Q8×D5311yescharacteristic
174C22#41×D5313yes—
175C4—26no—
176C4—43no—
177C4—1066no—
178C4—2123no—
179C2—42no—
180C2—538no—
181C2—2122no—
182C2#120×D5311yescharacteristic
183C2C22⋊Q8×D5313yes—
184C2#124×D5313yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#393C2
hypercenter#393D53
hypocenterC53C2×#393
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D53C2semidirect
3. #16810536C2semidirect
4. #16810535C2semidirect
5. #16810534C2semidirect
6. #95×D53C2semidirect
7. C53×#393C2semidirect
9. C22×Dic106C22semidirect
17. #13133124C22semidirect
18. #13133124C22semidirect
19. C22×Q8×C53C22semidirect
21. #95×C53C22semidirect
25. C23×D53Q8semidirect
46. C23×C53C2×Q8semidirect
55. C22×D53C2×Q8semidirect
92. D53#393direct (N × H)
98. #393D53direct (N × H)
100. C53C2×#393semidirect
104. C22×Q8C2×D53semidirect
114. #95C2×D53semidirect
148. C23Q8×D53semidirect