Finite Group Databasebeta

← #20276379

Subgroups — #20276379

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#16811172C211yesmaximal, characteristic
2#16810595C212yesmaximal, complemented
3#16810418C211yesmaximal, characteristic, complemented
4#16810415C212yesmaximal, complemented
5C53×#354C211yesmaximal, characteristic, complemented
6C2×#11719949C411yescharacteristic, complemented
7C2×#11719947C2211yescharacteristic
8#13133215C411yescharacteristic, complemented
9#13133213C2211yescharacteristic
10#13133205C411yescharacteristic, complemented
11#13133190—22no—
12#13133177—22no—
13#13133173—22no—
14C42×D53C411yescharacteristic, complemented
15#13133132C2212yes—
16#13133131—22no—
17#124×C53C2211yescharacteristic
18#99×C53C2212yes—
19C2×Q8×C53C2×C411yescharacteristic
20C2×D4×C53C2311yescharacteristic
21C2×C22⋊D53—22no—
22C2×C22⋊D53D412yes—
23C22×Dic53D412yes—
24#11719949—41no—
25#11719949—41no—
26#11719947—41no—
27#11719947—41no—
28C2×D212—21no—
29C2×D212C2×C411yescharacteristic
30C2×C4×D53—21no—
31C2×C4×D53—21no—
32C2×C4×D53C2×C411yescharacteristic
33C2×Dic106C2×C411yescharacteristic
34C22⋊C4×C53—22no—
35C22⋊C4×C53—22no—
36C42×C53C2×C411yescharacteristic
37C22⋊Dic53—22no—
38#11719921—22no—
39#11719919—22no—
40#11719919—42no—
41C4×Dic53—21no—
42C4×Dic53—42no—
43C4×Dic53C2×C411yescharacteristic
44C23×C53C2×D412yes—
45C22×D53—21no—
46C22×D53C22⋊C411yescharacteristic
47Q8×C53—41no—
48D4×C53—41no—
49C2×C4×C53—21no—
50C2×C4×C53—21no—
51C2×C4×C53—42no—
52C2×C4×C53C22×C411yescharacteristic
53C22⋊D53—42no—
54C22⋊D53—44no—
55C2×Dic53—21no—
56C2×Dic53—22no—
57C2×Dic53—22no—
58C2×Dic53—42no—
59C2×Dic53C22⋊C412yes—
60C2×Dic53C22⋊C411yescharacteristic
61D212—41no—
62D212—42no—
63C4×D53—21no—
64C4×D53—21no—
65C4×D53—42no—
66C4×D53—41no—
67C4×D53—41no—
68Dic106—41no—
69C22×C53—22no—
70C22×C53—42no—
71C22×C53C2×C22⋊C411yescharacteristic
72C2×D53—21no—
73C2×D53—41no—
74C2×D53—42no—
75C2×D53—41no—
76C4×C53—21no—
77C4×C53—41no—
78C4×C53—41no—
79C4×C53—82no—
80Dic53—22no—
81Dic53—22no—
82Dic53—82no—
83Dic53#9911yescharacteristic
84Dic53#9911yescharacteristic
85#2126—531nomaximal, Sylow
86C2×C53—21no—
87C2×C53—42no—
88C2×C53C2×#9911yescharacteristic
89D53—41no—
90D53—81no—
91#532—531no—
92#410—532no—
93#354—531no—
94#354D5311yescharacteristic, complemented
95#353—532no—
96C53#212611yescharacteristic, complemented, Sylow
97C2×#41—531no—
98C2×#41—531no—
99C4⋊Q8—531no—
100#127—531no—
101#124—531no—
102#124C2×D5311yescharacteristic
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×Q8C4×D5311yescharacteristic
116C2×D4—532no—
117C2×D4—531no—
118C2×D4—1061no—
119C2×D4—1062no—
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—22no—
129C22⋊C4—1062no—
130C22⋊C4—1062no—
131C42—531no—
132C42—1061no—
133C42—2122no—
134C42C4×D5311yescharacteristic
135C23—531no—
136C23—1061no—
137C23D4×D5312yes—
138Q8—41no—
139Q8—2121no—
140D4—41no—
141D4—2121no—
142D4—2122no—
143D4—2122no—
144D4—2124no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—42no—
148C2×C4—531no—
149C2×C4—532no—
150C2×C4—1061no—
151C2×C4—1061no—
152C2×C4—1062no—
153C2×C4—1062no—
154C2×C4—1061no—
155C2×C4—2121no—
156C2×C4—2121no—
157C2×C4—2122no—
158C2×C4—2122no—
159C2×C4C2×C4×D5311yescharacteristic
160C22—22no—
161C22—42no—
162C22—1061no—
163C22—2122no—
164C22—2121no—
165C22—2121no—
166C22C22⋊C4×D5311yescharacteristic
167C4—21no—
168C4—41no—
169C4—41no—
170C4—82no—
171C4—531no—
172C4—531no—
173C4—1062no—
174C4—1062no—
175C4—4242no—
176C2—21no—
177C2—42no—
178C2—4241no—
179C2—2121no—
180C2#99×D5311yescharacteristic
subgroupG/H
centerC2#99×D53
commutatorC2×C4×C53C22×C4
FrattiniC2×D4C22×D53
FittingC53×#354C2
socleC2×C53C2×#99
hypercenter#354D53
hypocenterC53#2126
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #354

Complements

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

NH ≅ G/Ntype
2. #16810595C2semidirect
3. #16810418C2semidirect
4. #16810415C2semidirect
5. C53×#354C2semidirect
6. C2×#11719949C4semidirect
8. #13133215C4semidirect
10. #13133205C4semidirect
14. C42×D53C4semidirect
94. #354D53semidirect
96. C53#2126semidirect