Finite Group Databasebeta

← #20276377

Subgroups — #20276377

Order 6784.

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

normal subgroups only

This group has 258 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#16811171C211yesmaximal, characteristic
2#16810595C212yesmaximal, complemented
3#16810417C212yesmaximal, complemented
4#16810416C211yesmaximal, characteristic, complemented
5C53×#353C211yesmaximal, characteristic, complemented
6C2×#11719947C411yescharacteristic, complemented
7C2×#11719947C2211yescharacteristic
8#13133217C411yescharacteristic, complemented
9#13133204C411yescharacteristic, complemented
10#13133204C2211yescharacteristic
11#13133188—22no—
12D4×Dic53—22no—
13#13133173—22no—
14C42×D53C411yescharacteristic, complemented
15#13133132C2212yes—
16#13133131—22no—
17#127×C53C2211yescharacteristic
18#99×C53C2212yes—
19C2×D4×C53—22no—
20C2×D4×C53C2×C411yescharacteristic
21C2×D4×C53C2311yescharacteristic
22C2×C22⋊D53—21no—
23C2×C22⋊D53D412yes—
24C22×Dic53—21no—
25C22×Dic53D412yes—
26#11719947—41no—
27#11719947—41no—
28#11719947—41no—
29#11719947—41no—
30C2×C4×D53—21no—
31C2×C4×D53C2×C411yescharacteristic
32C2×Dic106C2×C411yescharacteristic
33C2×Dic106C2×C411yescharacteristic
34C22⋊C4×C53—22no—
35C42×C53C2×C411yescharacteristic
36C22⋊Dic53—22no—
37C22⋊Dic53—22no—
38C22⋊Dic53—22no—
39C4⋊Dic53—22no—
40#11719919—42no—
41C4×Dic53—21no—
42C4×Dic53—42no—
43C4×Dic53C2×C411yescharacteristic
44C23×C53—21no—
45C23×C53C2×D412yes—
46C22×D53C22⋊C411yescharacteristic
47D4×C53—41no—
48D4×C53—41no—
49D4×C53—44no—
50C2×C4×C53—21no—
51C2×C4×C53—42no—
52C2×C4×C53C22×C411yescharacteristic
53C22⋊D53—42no—
54C22⋊D53—42no—
55C2×Dic53—22no—
56C2×Dic53—22no—
57C2×Dic53—21no—
58C2×Dic53—21no—
59C2×Dic53—41no—
60C2×Dic53—41no—
61C2×Dic53—42no—
62C2×Dic53C22⋊C412yes—
63C2×Dic53C22⋊C411yescharacteristic
64C4×D53—21no—
65C4×D53—21no—
66C4×D53—42no—
67Dic106—41no—
68Dic106—41no—
69C22×C53—22no—
70C22×C53—42no—
71C22×C53—42no—
72C22×C53—41no—
73C22×C53C2×C22⋊C411yescharacteristic
74C2×D53—21no—
75C2×D53—41no—
76C4×C53—22no—
77C4×C53—21no—
78C4×C53—82no—
79Dic53—22no—
80Dic53—41no—
81Dic53—41no—
82Dic53—82no—
83Dic53#9911yescharacteristic
84Dic53#9911yescharacteristic
85#2126—531nomaximal, Sylow
86C2×C53—21no—
87C2×C53—42no—
88C2×C53—81no—
89C2×C53C2×#9911yescharacteristic
90D53—41no—
91#532—531no—
92#410—532no—
93#354—532no—
94#353—531no—
95#353D5311yescharacteristic, complemented
96C53#212611yescharacteristic, complemented, Sylow
97C2×#41—531no—
98C2×#41—531no—
99C4⋊Q8—531no—
100#127C2×D5311yescharacteristic
101#124—531no—
102#124—531no—
103#121—1062no—
104C4×D4—1062no—
105#117—1062no—
106C2×C42—531no—
107#99—532no—
108#99—1062no—
109#99C2×D5312yes—
110#41—2121no—
111#41—2121no—
112#41—2121no—
113#41—2121no—
114C2×Q8—531no—
115C2×Q8—531no—
116C2×D4—22no—
117C2×D4—532no—
118C2×D4—1061no—
119C2×D4C4×D5311yescharacteristic
120C2×D4C22×D5311yescharacteristic
121C22×C4—532no—
122C22×C4—531no—
123C22×C4—1061no—
124C22×C4—1061no—
125C4⋊C4—1062no—
126C4⋊C4—2122no—
127C22⋊C4—22no—
128C22⋊C4—1062no—
129C22⋊C4—1062no—
130C22⋊C4—1062no—
131C42—531no—
132C42—1061no—
133C42—2122no—
134C42C4×D5311yescharacteristic
135C23—21no—
136C23—531no—
137C23D4×D5312yes—
138Q8—2121no—
139Q8—2121no—
140D4—41no—
141D4—41no—
142D4—44no—
143D4—2122no—
144D4—2122no—
145C2×C4—21no—
146C2×C4—42no—
147C2×C4—531no—
148C2×C4—532no—
149C2×C4—1061no—
150C2×C4—1061no—
151C2×C4—1062no—
152C2×C4—1062no—
153C2×C4—1061no—
154C2×C4—1061no—
155C2×C4—2122no—
156C2×C4—2121no—
157C2×C4—2121no—
158C2×C4—2122no—
159C2×C4C2×C4×D5311yescharacteristic
160C22—22no—
161C22—42no—
162C22—42no—
163C22—41no—
164C22—1061no—
165C22—2121no—
166C22C22⋊C4×D5311yescharacteristic
167C4—22no—
168C4—21no—
169C4—82no—
170C4—531no—
171C4—531no—
172C4—1062no—
173C4—4242no—
174C4—2121no—
175C4—2121no—
176C2—21no—
177C2—42no—
178C2—81no—
179C2—2121no—
180C2#99×D5311yescharacteristic
subgroupG/H
centerC2#99×D53
commutatorC2×C4×C53C22×C4
FrattiniC2×D4C22×D53
FittingC53×#353C2
socleC2×C53C2×#99
hypercenter#353D53
hypocenterC53#2126
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. #16810417C2semidirect
4. #16810416C2semidirect
5. C53×#353C2semidirect
6. C2×#11719947C4semidirect
8. #13133217C4semidirect
9. #13133204C4semidirect
14. C42×D53C4semidirect
95. #353D53semidirect
96. C53#2126semidirect