Finite Group Databasebeta

← #20290195

Subgroups — #20290195

Order 6784.

248 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 56 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#16811645C212yesmaximal, complemented
2#16811479C212yesmaximal, complemented
3#16811478C212yesmaximal, complemented
4#124×D53C211yesmaximal, characteristic
5C4⋊#88324—22no—
6#13133274—22no—
7#13133272C2214yescomplemented
8C2×Q8×D53C411yescharacteristic
9C2×D4×D53C411yescharacteristic
10#13133206C2211yescharacteristic
11#13133205C411yescharacteristic
12#13133204C2211yescharacteristic
13#13133203—22no—
14C22⋊C4×D53—22no—
15C42×D53C2211yescharacteristic
16#124×C53C411yescharacteristic
17C23×D53—21no—
18C2×Q8×C53C2×C411yescharacteristic
19C2×D4×C53C2×C411yescharacteristic
20C2×C22⋊D53—21no—
21Q8×D53—42no—
22D4×D53—42no—
23C2×D212C2×C411yescharacteristic
24C2×C4×D53—21no—
25C2×C4×D53C2×C411yescharacteristic
26C2×C4×D53D412yes—
27C2×Dic106—21no—
28C2×Dic106C2×C411yescharacteristic
29C2×#88324—24no—
30C22⋊C4×C53—22no—
31C42×C53C2×C411yescharacteristic
32C22⋊Dic53—22no—
33#11719921—22no—
34#11719921—22no—
35C4×Dic53D412yes—
36C4×Dic53C2311yescharacteristic
37C23×C53—21no—
38C22×D53—21no—
39C22×D53—44no—
40C22×D53—42no—
41C22×D53C22⋊C411yescharacteristic
42Q8×C53—41no—
43D4×C53—41no—
44C2×C4×C53—21no—
45C2×C4×C53C22⋊C412yes—
46C2×C4×C53C22×C411yescharacteristic
47C22⋊D53—42no—
48C2×Dic53—21no—
49C2×Dic53C22⋊C411yescharacteristic
50C2×Dic53C2×D412yes—
51D212—41no—
52C4×D53—22no—
53C4×D53—42no—
54C4×D53—42no—
55Dic106—42no—
56Dic106—41no—
57#88324—44no—
58C22×C53—41no—
59C22×C53—42no—
60C22×C53C2×C22⋊C411yescharacteristic
61C2×D53—22no—
62C2×D53—41no—
63C2×D53—42no—
64C2×D53—84no—
65C2×D53#10012yes—
66C4×C53—21no—
67C4×C53—22no—
68C4×C53—41no—
69Dic53—21no—
70Dic53—22no—
71Dic53—41no—
72#1533—531nomaximal, Sylow
73C2×C53—81no—
74C2×C53C2×#10011yescharacteristic
75C2×C53#42112yes—
76D53—22no—
77D53—81no—
78C2×#124—531no—
79#432—532no—
80#329—532no—
81#329—532no—
82C53#153311yescharacteristic, complemented, Sylow
83C22×Q8—531no—
84C22×D4—531no—
85#124—531no—
86#124—531no—
87#124—531no—
88#124—1062no—
89#124#615711yescharacteristic
90C2×C22⋊C4—1062no—
91C2×C42—531no—
92C4⋊C8—1062no—
93C22⋊C8—1062no—
94#97—534no—
95C24—1061no—
96C2×Q8—531no—
97C2×Q8—1061no—
98C2×Q8—2122no—
99C2×Q8C2×#615711yescharacteristic
100C2×D4—531no—
101C2×D4—1061no—
102C2×D4—2122no—
103C2×D4C2×#615711yescharacteristic
104C22×C4—531no—
105C22×C4—532no—
106C22×C4—1061no—
107C2×C8—1064no—
108C22⋊C4—22no—
109C22⋊C4—1062no—
110C22⋊C4—1062no—
111C22⋊C4—1062no—
112C42—532no—
113C42—531no—
114C42C2×#615711yescharacteristic
115C23—21no—
116C23—531no—
117C23—1061no—
118C23—2122no—
119C23—2124no—
120Q8—41no—
121Q8—2121no—
122Q8—2122no—
123D4—41no—
124D4—2121no—
125D4—2122no—
126C2×C4—21no—
127C2×C4—531no—
128C2×C4—532no—
129C2×C4—1062no—
130C2×C4—1061no—
131C2×C4—2122no—
132C2×C4—2122no—
133C2×C4C22⋊#615712yes—
134C2×C4C22×#615711yescharacteristic
135C8—2124no—
136C22—41no—
137C22—42no—
138C22—532no—
139C22—1062no—
140C22—4244no—
141C22—2121no—
142C22—2122no—
143C22C2×C22⋊#615711yescharacteristic
144C4—21no—
145C4—22no—
146C4—41no—
147C4—1061no—
148C4—1062no—
149C4—2121no—
150C2—81no—
151C2—1062no—
152C2—4241no—
153C2D53⋊#10011yescharacteristic
154C2#1681162012yes—
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
Fitting#124×C53C4
socleC22×C53C2×C22⋊C4
hypercenter#124#6157
hypocenterC53#1533
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #124

Complements

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

NH ≅ G/Ntype
1. #16811645C2semidirect
2. #16811479C2semidirect
3. #16811478C2semidirect
7. #13133272C22semidirect
82. C53#1533semidirect