Finite Group Databasebeta

← #20291012

Subgroups — #20291012

Order 6784.

230 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 54 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
1C2×#13133278C212yesmaximal, complemented
2#16811534C212yesmaximal, complemented
3#16811532C212yesmaximal, complemented
4#16811199C211yesmaximal, characteristic
5C2×#11719940—22no—
6#13133278—22no—
7#13133278C2214yes—
8C2×#11719945C411yescharacteristic
9C22×Dic106C2211yescharacteristic
10#13133217C411yescharacteristic
11#13133217C2211yescharacteristic
12#13133205C411yescharacteristic
13#13133205C2211yescharacteristic
14#13133201—22no—
15#13133197—22no—
16#117×C53C411yescharacteristic
17C22×C4×C53C2×C411yescharacteristic
18C2×C22⋊D53—21no—
19C22×Dic53D412yes—
20#11719945—42no—
21C2×D212C2×C411yescharacteristic
22C2×C4×D53—21no—
23C2×Dic106—42no—
24C2×Dic106C2×C411yescharacteristic
25C2×Dic106C2×C411yescharacteristic
26C2×Dic106D412yes—
27C2×Dic106C2311yescharacteristic
28#11719940—24no—
29#11719940—42no—
30C2×#88324—42no—
31C4⋊C4×C53—21no—
32C22⋊C4×C53—21no—
33C42×C53C2×C411yescharacteristic
34C42×C53C2×C411yescharacteristic
35#11719921—22no—
36#11719921—22no—
37C4⋊Dic53—22no—
38C4⋊Dic53—22no—
39C23×C53C22⋊C411yescharacteristic
40C22×D53—21no—
41C2×C4×C53—22no—
42C2×C4×C53—21no—
43C2×C4×C53—21no—
44C2×C4×C53C22⋊C411yescharacteristic
45C2×C4×C53C22×C411yescharacteristic
46C22⋊D53—42no—
47C2×Dic53—22no—
48C2×Dic53—22no—
49C2×Dic53—21no—
50C2×Dic53C22⋊C412yes—
51C2×Dic53C2×D412yes—
52D212—41no—
53C4×D53—42no—
54Dic106—41no—
55Dic106—42no—
56Dic106—41no—
57Dic106—41no—
58#88324—44no—
59C22×C53—41no—
60C22×C53#9912yes—
61C22×C53C2×C22⋊C411yescharacteristic
62C2×D53—42no—
63C2×D53—41no—
64C4×C53—21no—
65C4×C53—21no—
66C4×C53—41no—
67C4×C53—41no—
68Dic53—22no—
69Dic53—22no—
70Dic53—41no—
71#2133—531nomaximal, Sylow
72C2×C53—21no—
73C2×C53—22no—
74C2×C53C2×#9911yescharacteristic
75D53—81no—
76#536—531no—
77C2×#101—532no—
78#356—532no—
79#355—532no—
80C53#213311yescharacteristic, complemented, Sylow
81C2×#41—531no—
82C22×Q8—531no—
83C2×M16—1062no—
84C4⋊Q8—531no—
85C4⋊Q8—531no—
86#124—531no—
87#124—531no—
88#123—1062no—
89C22⋊Q8—1062no—
90#117#615711yescharacteristic
91#101—534no—
92#101—1062no—
93#41—2122no—
94C2×Q8—532no—
95C2×Q8—531no—
96C2×Q8—531no—
97C2×Q8—531no—
98C2×Q8—2122no—
99C2×D4—531no—
100C2×D4—1061no—
101C22×C4—532no—
102C22×C4—1061no—
103C22×C4C2×#615711yescharacteristic
104M16—1064no—
105M16—2122no—
106C2×C8—2122no—
107C4⋊C4—21no—
108C4⋊C4—1062no—
109C4⋊C4—1062no—
110C22⋊C4—21no—
111C22⋊C4—1062no—
112C22⋊C4—1062no—
113C42C2×#615711yescharacteristic
114C42C2×#615711yescharacteristic
115C23—1061no—
116C23C22⋊#615711yescharacteristic
117Q8—2122no—
118Q8—2121no—
119Q8—2121no—
120Q8—2121no—
121D4—2121no—
122D4—2122no—
123C2×C4—22no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—532no—
127C2×C4—532no—
128C2×C4—1062no—
129C2×C4—1062no—
130C2×C4—1061no—
131C2×C4—2122no—
132C2×C4C22⋊#615711yescharacteristic
133C2×C4C22×#615711yescharacteristic
134C8—2124no—
135C22—41no—
136C22—2122no—
137C22—2121no—
138C22#1313327612yes—
139C22C2×C22⋊#615711yescharacteristic
140C4—21no—
141C4—21no—
142C4—41no—
143C4—41no—
144C4—1062no—
145C4—1062no—
146C4—2121no—
147C2—21no—
148C2—22no—
149C2—4241no—
150C2C2×#1313327611yescharacteristic
subgroupG/H
centerC2C2×#13133276
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
Fitting#117×C53C4
socleC2×C53C2×#99
hypercenter#117#6157
hypocenterC53#2133
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. C2×#13133278C2semidirect
2. #16811534C2semidirect
3. #16811532C2semidirect
80. C53#2133semidirect