Finite Group Databasebeta

← #20276385

Subgroups — #20276385

Order 6784.

270 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 55 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#16811186C211yesmaximal, characteristic
2#16810595C212yesmaximal, complemented
3#16810416C211yesmaximal, characteristic, complemented
4#16810415C212yesmaximal, complemented
5C53×#353C211yesmaximal, characteristic, complemented
6C2×#11719947C2211yescharacteristic
7C2×D4×D53C411yescharacteristic, complemented
8#13133213C2211yescharacteristic
9#13133205C411yescharacteristic, complemented
10#13133204C411yescharacteristic, complemented
11#13133189—22no—
12#13133185—22no—
13#13133174C411yescharacteristic, complemented
14#13133173—22no—
15#13133132C2212yescomplemented
16#13133131—22no—
17#127×C53C2211yescharacteristic
18#99×C53C2212yescomplemented
19C23×D53—21no—
20C2×D4×C53—22no—
21C2×D4×C53C2×C411yescharacteristic
22C2×D4×C53C2311yescharacteristic
23C2×C22⋊D53—22no—
24C2×C22⋊D53—21no—
25C2×C22⋊D53D412yes—
26C22×Dic53D412yes—
27#11719947—42no—
28D4×D53—42no—
29C2×D212C2×C411yescharacteristic
30C2×C4×D53C2×C411yescharacteristic
31C2×Dic106C2×C411yescharacteristic
32C22⋊C4×C53—22no—
33C42×C53C2×C411yescharacteristic
34C22⋊Dic53—22no—
35C22⋊Dic53—22no—
36#11719921—21no—
37#11719921—22no—
38#11719919—21no—
39#11719919—42no—
40C4×Dic53—42no—
41C4×Dic53C2×C411yescharacteristic
42C23×C53—21no—
43C23×C53C2×D412yes—
44C22×D53—21no—
45C22×D53—42no—
46C22×D53—82no—
47C22×D53C22⋊C411yescharacteristic
48D4×C53—41no—
49D4×C53—41no—
50D4×C53—82no—
51C2×C4×C53—21no—
52C2×C4×C53—42no—
53C2×C4×C53C22×C411yescharacteristic
54C22⋊D53—42no—
55C22⋊D53—42no—
56C22⋊D53—82no—
57C2×Dic53—22no—
58C2×Dic53—22no—
59C2×Dic53—21no—
60C2×Dic53—42no—
61C2×Dic53C22⋊C412yes—
62C2×Dic53C22⋊C411yescharacteristic
63D212—41no—
64C4×D53—22no—
65Dic106—41no—
66C22×C53—22no—
67C22×C53—42no—
68C22×C53—42no—
69C22×C53—41no—
70C22×C53C2×C22⋊C411yescharacteristic
71C2×D53—21no—
72C2×D53—41no—
73C2×D53—42no—
74C2×D53—41no—
75C2×D53—84no—
76C4×C53—21no—
77C4×C53—41no—
78C4×C53—82no—
79Dic53—22no—
80Dic53—41no—
81Dic53—82no—
82Dic53#9912yes—
83#2128—531nomaximal, Sylow
84C2×C53—21no—
85C2×C53—42no—
86C2×C53—81no—
87C2×C53C2×#9911yescharacteristic
88D53—41no—
89D53—81no—
90#535—531no—
91#410—532no—
92#353—531no—
93#353—532no—
94#353D5311yescharacteristic, complemented
95C53#212811yescharacteristic, complemented, Sylow
96C2×#41—531no—
97C22×D4—531no—
98#127—531no—
99#127C2×D5311yescharacteristic
100#124—531no—
101#124—531no—
102#121—1062no—
103#120—1062no—
104#117—531no—
105#117—1062no—
106#99—532no—
107#99—1062no—
108#99C2×D5312yescomplemented
109C24—1061no—
110#41—2122no—
111C2×Q8—531no—
112C2×D4—22no—
113C2×D4—532no—
114C2×D4—531no—
115C2×D4—1061no—
116C2×D4—1062no—
117C2×D4—2122no—
118C2×D4C4×D5311yescharacteristic
119C2×D4C22×D5311yescharacteristic
120C22×C4—532no—
121C22×C4—531no—
122C4⋊C4—1061no—
123C4⋊C4—2122no—
124C22⋊C4—22no—
125C22⋊C4—1062no—
126C22⋊C4—1061no—
127C22⋊C4—1062no—
128C22⋊C4—1062no—
129C42—531no—
130C42—2122no—
131C42C4×D5311yescharacteristic
132C23—21no—
133C23—531no—
134C23—1061no—
135C23—4242no—
136C23—2122no—
137C23D4×D5312yes—
138Q8—2121no—
139D4—41no—
140D4—41no—
141D4—82no—
142D4—4242no—
143D4—2121no—
144D4—2122no—
145D4—2122no—
146C2×C4—21no—
147C2×C4—42no—
148C2×C4—531no—
149C2×C4—532no—
150C2×C4—1062no—
151C2×C4—1062no—
152C2×C4—1062no—
153C2×C4—1061no—
154C2×C4—2122no—
155C2×C4C2×C4×D5311yescharacteristic
156C22—22no—
157C22—42no—
158C22—42no—
159C22—41no—
160C22—1061no—
161C22—4244no—
162C22—2122no—
163C22—2121no—
164C22—2121no—
165C22C22⋊C4×D5311yescharacteristic
166C4—21no—
167C4—41no—
168C4—82no—
169C4—532no—
170C4—1062no—
171C4—4242no—
172C4—2121no—
173C2—21no—
174C2—42no—
175C2—81no—
176C2—4241no—
177C2—2121no—
178C2#99×D5311yescharacteristic
subgroupG/H
centerC2#99×D53
commutatorC2×C4×C53C22×C4
FrattiniC2×D4C22×D53
FittingC53×#353C2
socleC2×C53C2×#99
hypercenter#353D53
hypocenterC53#2128
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #353

Complements

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

NH ≅ G/Ntype
2. #16810595C2semidirect
3. #16810416C2semidirect
4. #16810415C2semidirect
5. C53×#353C2semidirect
7. C2×D4×D53C4semidirect
9. #13133205C4semidirect
10. #13133204C4semidirect
13. #13133174C4semidirect
15. #13133132C22semidirect
18. #99×C53C22semidirect
94. #353D53semidirect
95. C53#2128semidirect
108. #99C2×D53semidirect