Finite Group Databasebeta

← #20273337

Subgroups — #20273337

Order 6784.

248 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 59 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#124×D53C211yesmaximal, characteristic
2#16810678C212yesmaximal, complemented
3#16810363C211yesmaximal, characteristic, complemented
4#16810362C212yesmaximal, complemented
5C53×#329C211yesmaximal, characteristic, complemented
6C2×Q8×D53C411yescharacteristic
7C2×D4×D53C411yescharacteristic
8#13133206C411yescharacteristic
9#13133205C2211yescharacteristic
10#13133204C411yescharacteristic
11#13133203—22no—
12C22⋊C4×D53—22no—
13C42×D53C2211yescharacteristic
14#13133146—22no—
15#13133129—22no—
16#13133128C2212yescomplemented
17#124×C53C2211yescharacteristic
18#97×C53C2212yescomplemented
19C23×D53—21no—
20C2×Q8×C53C2×C411yescharacteristic
21C2×D4×C53C2×C411yescharacteristic
22C2×C22⋊D53—21no—
23Q8×D53—42no—
24D4×D53—42no—
25C2×D212C2×C411yescharacteristic
26C2×C4×D53—21no—
27C2×C4×D53C2×C411yescharacteristic
28C2×C4×D53D412yes—
29C2×Dic106—21no—
30C2×Dic106C2×C411yescharacteristic
31C2×C8×C53—22no—
32C22⋊C4×C53—22no—
33C42×C53C2311yescharacteristic
34C22⋊Dic53—22no—
35#11719921—22no—
36#11719921—22no—
37C4×Dic53C2×C411yescharacteristic
38C4×Dic53D412yes—
39C2×#88322—22no—
40C23×C53—21no—
41C22×D53—21no—
42C22×D53—44no—
43C22×D53—42no—
44C22×D53C22⋊C411yescharacteristic
45Q8×C53—41no—
46D4×C53—41no—
47C2×C4×C53—21no—
48C2×C4×C53C22×C411yescharacteristic
49C2×C4×C53C2×D412yes—
50C22⋊D53—42no—
51C2×Dic53—21no—
52C2×Dic53C22⋊C411yescharacteristic
53C2×Dic53C22⋊C412yes—
54D212—41no—
55C4×D53—22no—
56C4×D53—42no—
57C4×D53—42no—
58Dic106—42no—
59Dic106—41no—
60C8×C53—42no—
61#88322—42no—
62C22×C53—41no—
63C22×C53—42no—
64C22×C53C2×C22⋊C411yescharacteristic
65C2×D53—22no—
66C2×D53—41no—
67C2×D53—42no—
68C2×D53—84no—
69C2×D53#10012yes—
70C4×C53—21no—
71C4×C53—22no—
72C4×C53—41no—
73Dic53—21no—
74Dic53—22no—
75Dic53—41no—
76#1533—531nomaximal, Sylow
77C2×C53—81no—
78C2×C53C2×#10011yescharacteristic
79C2×C53#42112yes—
80D53—22no—
81D53—81no—
82C2×#124—531no—
83#432—532no—
84#329—531no—
85#329—532no—
86#329D5311yescharacteristic, complemented
87C53#153311yescharacteristic, complemented, Sylow
88C22×Q8—531no—
89C22×D4—531no—
90#124—531no—
91#124—531no—
92#124—531no—
93#124—1062no—
94#124C2×D5311yescharacteristic
95C2×C22⋊C4—1062no—
96C2×C42—531no—
97C4⋊C8—1062no—
98C22⋊C8—1062no—
99#97—532no—
100#97C2×D5312yescomplemented
101C24—1061no—
102C2×Q8—531no—
103C2×Q8—1061no—
104C2×Q8—2122no—
105C2×Q8C4×D5311yescharacteristic
106C2×D4—531no—
107C2×D4—1061no—
108C2×D4—2122no—
109C2×D4C4×D5311yescharacteristic
110C22×C4—531no—
111C22×C4—532no—
112C22×C4—1061no—
113C2×C8—22no—
114C2×C8—1062no—
115C22⋊C4—22no—
116C22⋊C4—1062no—
117C22⋊C4—1062no—
118C22⋊C4—1062no—
119C42—531no—
120C42—532no—
121C42C22×D5311yescharacteristic
122C23—21no—
123C23—531no—
124C23—1061no—
125C23—2122no—
126C23—2124no—
127Q8—41no—
128Q8—2121no—
129Q8—2122no—
130D4—41no—
131D4—2121no—
132D4—2122no—
133C2×C4—21no—
134C2×C4—532no—
135C2×C4—531no—
136C2×C4—1062no—
137C2×C4—1061no—
138C2×C4—2122no—
139C2×C4—2122no—
140C2×C4C2×C4×D5311yescharacteristic
141C2×C4D4×D5312yes—
142C8—42no—
143C8—2122no—
144C22—41no—
145C22—42no—
146C22—532no—
147C22—1062no—
148C22—4244no—
149C22—2121no—
150C22—2122no—
151C22C22⋊C4×D5311yescharacteristic
152C4—21no—
153C4—22no—
154C4—41no—
155C4—1061no—
156C4—1062no—
157C4—2121no—
158C2—81no—
159C2—1062no—
160C2—4241no—
161C2#100×D5311yescharacteristic
162C2#1681063812yes—
subgroupG/H
centerC22C22⋊C4×D53
commutatorC2×C4×C53C22×C4
FrattiniC42C22×D53
FittingC53×#329C2
socleC22×C53C2×C22⋊C4
hypercenter#329D53
hypocenterC53#1533
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #16810678C2semidirect
3. #16810363C2semidirect
4. #16810362C2semidirect
5. C53×#329C2semidirect
16. #13133128C22semidirect
18. #97×C53C22semidirect
86. #329D53semidirect
87. C53#1533semidirect
100. #97C2×D53semidirect