Finite Group Databasebeta

← #20276381

Subgroups — #20276381

Order 6784.

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

normal subgroups only

This group has 286 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#99×D53C212yesmaximal, complemented
3#16810417C212yesmaximal, complemented
4#16810416C211yesmaximal, characteristic, complemented
5C53×#353C211yesmaximal, characteristic, complemented
6C2×#11719947C411yescharacteristic, complemented
7C2×D4×D53C2211yescharacteristic
8#13133213C411yescharacteristic, complemented
9#13133205C411yescharacteristic, complemented
10#13133204C2211yescharacteristic
11#13133189—22no—
12#13133185—22no—
13#13133174C411yescharacteristic, complemented
14C22⋊C4×D53—22no—
15#13133132C2212yes—
16#13133131—22no—
17#127×C53C2211yescharacteristic
18#99×C53C2212yes—
19C23×D53D412yes—
20C2×D4×C53—22no—
21C2×D4×C53C2×C411yescharacteristic, complemented
22C2×D4×C53C2311yescharacteristic
23C2×C22⋊D53—21no—
24C2×C22⋊D53—22no—
25C2×C22⋊D53D412yes—
26C22×Dic53—21no—
27#11719947—42no—
28D4×D53—42no—
29C2×D212C2×C411yescharacteristic
30C2×C4×D53—42no—
31C2×C4×D53C2×C411yescharacteristic
32C2×Dic106C2×C411yescharacteristic, complemented
33C22⋊C4×C53—22no—
34C42×C53C2×C411yescharacteristic, complemented
35C22⋊Dic53—22no—
36C22⋊Dic53—22no—
37#11719921—21no—
38#11719921—22no—
39#11719921—42no—
40#11719919—21no—
41C4×Dic53C2×C411yescharacteristic, complemented
42C23×C53—21no—
43C23×C53C2×D412yes—
44C22×D53—22no—
45C22×D53—22no—
46C22×D53—82no—
47C22×D53C22⋊C412yes—
48C22×D53C22⋊C411yescharacteristic
49D4×C53—41no—
50D4×C53—41no—
51D4×C53—82no—
52C2×C4×C53—21no—
53C2×C4×C53—42no—
54C2×C4×C53C22×C411yescharacteristic
55C22⋊D53—42no—
56C22⋊D53—42no—
57C22⋊D53—82no—
58C2×Dic53—21no—
59C2×Dic53—21no—
60C2×Dic53—42no—
61C2×Dic53—42no—
62C2×Dic53C22⋊C411yescharacteristic
63D212—41no—
64C4×D53—22no—
65C4×D53—84no—
66Dic106—41no—
67C22×C53—22no—
68C22×C53—42no—
69C22×C53—42no—
70C22×C53—41no—
71C22×C53C2×C22⋊C411yescharacteristic
72C2×D53—22no—
73C2×D53—41no—
74C2×D53—44no—
75C2×D53—42no—
76C2×D53—82no—
77C2×D53#9912yes—
78C4×C53—21no—
79C4×C53—41no—
80C4×C53—82no—
81Dic53—21no—
82Dic53—41no—
83Dic53—41no—
84Dic53—82no—
85#2127—531nomaximal, Sylow
86C2×C53—21no—
87C2×C53—42no—
88C2×C53—81no—
89C2×C53C2×#9911yescharacteristic
90D53—22no—
91D53—42no—
92#535—531no—
93C2×#99—532no—
94#354—532no—
95#353—531no—
96#353D5311yescharacteristic, complemented
97C53#212711yescharacteristic, complemented, Sylow
98C2×#41—531no—
99C22×D4—531no—
100#127—531no—
101#127C2×D5311yescharacteristic
102#124—531no—
103#124—531no—
104#121—1062no—
105#120—1062no—
106#117—531no—
107C2×C22⋊C4—1062no—
108#99—532no—
109#99—1062no—
110#99C2×D5312yes—
111C24—532no—
112#41—2122no—
113C2×Q8—531no—
114C2×D4—22no—
115C2×D4—532no—
116C2×D4—531no—
117C2×D4—1062no—
118C2×D4—1061no—
119C2×D4—2122no—
120C2×D4C4×D5311yescharacteristic, complemented
121C2×D4C22×D5311yescharacteristic
122C22×C4—531no—
123C22×C4—1061no—
124C22×C4—2122no—
125C4⋊C4—1061no—
126C22⋊C4—22no—
127C22⋊C4—1061no—
128C22⋊C4—1062no—
129C22⋊C4—1062no—
130C22⋊C4—1062no—
131C22⋊C4—2122no—
132C42—531no—
133C42C4×D5311yescharacteristic, complemented
134C23—21no—
135C23—531no—
136C23—532no—
137C23—1062no—
138C23—1062no—
139C23—4242no—
140C23D4×D5312yes—
141Q8—2121no—
142D4—41no—
143D4—41no—
144D4—82no—
145D4—4242no—
146D4—2121no—
147D4—2122no—
148D4—2122no—
149C2×C4—21no—
150C2×C4—42no—
151C2×C4—531no—
152C2×C4—1062no—
153C2×C4—1061no—
154C2×C4—1061no—
155C2×C4—4244no—
156C2×C4—2122no—
157C2×C4—2122no—
158C2×C4C2×C4×D5311yescharacteristic
159C22—22no—
160C22—42no—
161C22—42no—
162C22—41no—
163C22—532no—
164C22—1062no—
165C22—4242no—
166C22—2121no—
167C22—2122no—
168C22—2124no—
169C22C22⋊C4×D5311yescharacteristic
170C4—21no—
171C4—41no—
172C4—82no—
173C4—1061no—
174C4—4242no—
175C4—2121no—
176C4—2121no—
177C2—21no—
178C2—42no—
179C2—81no—
180C2—1062no—
181C2—2122no—
182C2#99×D5311yescharacteristic
subgroupG/H
centerC2#99×D53
commutatorC2×C4×C53C22×C4
FrattiniC2×D4C22×D53
FittingC53×#353C2
socleC2×C53C2×#99
hypercenter#353D53
hypocenterC53#2127
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. #99×D53C2semidirect
3. #16810417C2semidirect
4. #16810416C2semidirect
5. C53×#353C2semidirect
6. C2×#11719947C4semidirect
8. #13133213C4semidirect
9. #13133205C4semidirect
13. #13133174C4semidirect
21. C2×D4×C53C2×C4semidirect
32. C2×Dic106C2×C4semidirect
34. C42×C53C2×C4semidirect
41. C4×Dic53C2×C4semidirect
96. #353D53semidirect
97. C53#2127semidirect
120. C2×D4C4×D53semidirect
133. C42C4×D53semidirect