Finite Group Databasebeta

← #20291018

Subgroups — #20291018

Order 6784.

230 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 52 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#16811610C212yesmaximal
2#16811534C212yesmaximal, complemented
3#16811533C212yesmaximal, complemented
4#16811193C211yesmaximal, characteristic
5C2×#11719940—22no—
6#13133278C2214yes—
7#13133277—22no—
8C2×#11719947C2211yescharacteristic
9C2×Q8×D53C411yescharacteristic
10#13133217C2211yescharacteristic
11#13133215C2211yescharacteristic
12#13133204C411yescharacteristic
13#13133199—22no—
14#13133194—22no—
15#13133174C411yescharacteristic
16#124×C53C411yescharacteristic
17C2×Q8×C53C2×C411yescharacteristic
18C2×D4×C53C2×C411yescharacteristic
19C2×C22⋊D53D412yes—
20C22×Dic53D412yes—
21Q8×D53—42no—
22#11719947—42no—
23C2×C4×D53—21no—
24C2×C4×D53C2×C411yescharacteristic
25C2×Dic106—21no—
26C2×Dic106C2×C411yescharacteristic
27C2×Dic106C2311yescharacteristic
28#11719940—24no—
29#11719940—42no—
30C2×#88324—42no—
31C22⋊C4×C53—22no—
32C42×C53C2×C411yescharacteristic
33C22⋊Dic53—22no—
34#11719921—21no—
35C4⋊Dic53—22no—
36#11719919—21no—
37#11719919—22no—
38C4×Dic53C2×C411yescharacteristic
39C23×C53C22⋊C412yes—
40C22×D53C22⋊C411yescharacteristic
41Q8×C53—41no—
42D4×C53—41no—
43C2×C4×C53—21no—
44C2×C4×C53—21no—
45C2×C4×C53C22×C411yescharacteristic
46C22⋊D53—42no—
47C2×Dic53—22no—
48C2×Dic53—22no—
49C2×Dic53—21no—
50C2×Dic53—21no—
51C2×Dic53C22⋊C411yescharacteristic
52C2×Dic53C2×D412yes—
53C4×D53—22no—
54C4×D53—42no—
55Dic106—41no—
56Dic106—42no—
57Dic106—41no—
58#88324—44no—
59C22×C53—22no—
60C22×C53—42no—
61C22×C53C2×C22⋊C411yescharacteristic
62C2×D53—21no—
63C2×D53—41no—
64C4×C53—21no—
65C4×C53—41no—
66C4×C53—41no—
67Dic53—22no—
68Dic53—41no—
69Dic53—41no—
70Dic53#9912yes—
71#2135—531nomaximal, Sylow
72C2×C53—21no—
73C2×C53—42no—
74C2×C53C2×#9911yescharacteristic
75D53—41no—
76#536—531no—
77#413—532no—
78#356—532no—
79#356—532no—
80C53#213511yescharacteristic, complemented, Sylow
81C2×#41—531no—
82C22×Q8—531no—
83C2×M16—1062no—
84C4⋊Q8—531no—
85C4⋊Q8—531no—
86#124—531no—
87#124#615711yescharacteristic
88#123—1062no—
89C22⋊Q8—1062no—
90#117—531no—
91#101—534no—
92#100—1062no—
93#41—2122no—
94C2×Q8—531no—
95C2×Q8—531no—
96C2×Q8—1061no—
97C2×Q8—2122no—
98C2×Q8C2×#615711yescharacteristic
99C2×D4—532no—
100C2×D4C2×#615711yescharacteristic
101C22×C4—532no—
102C22×C4—531no—
103C22×C4—1061no—
104M16—1064no—
105M16—2122no—
106C2×C8—2122no—
107C4⋊C4—1061no—
108C4⋊C4—1062no—
109C4⋊C4—1062no—
110C22⋊C4—22no—
111C22⋊C4—1061no—
112C22⋊C4—1062no—
113C42—531no—
114C42C2×#615711yescharacteristic
115C23—531no—
116C23C22⋊#615712yes—
117Q8—41no—
118Q8—2121no—
119Q8—2121no—
120Q8—2122no—
121D4—41no—
122D4—2122no—
123C2×C4—21no—
124C2×C4—21no—
125C2×C4—531no—
126C2×C4—532no—
127C2×C4—1062no—
128C2×C4—1062no—
129C2×C4—1062no—
130C2×C4—1061no—
131C2×C4—1061no—
132C2×C4—2122no—
133C2×C4C22×#615711yescharacteristic
134C8—2124no—
135C22—22no—
136C22—42no—
137C22—1061no—
138C22—2121no—
139C22C2×C22⋊#615711yescharacteristic
140C4—21no—
141C4—41no—
142C4—41no—
143C4—532no—
144C4—1062no—
145C4—2121no—
146C4—2121no—
147C2—21no—
148C2—42no—
149C2—2121no—
150C2D53⋊#9911yescharacteristic
subgroupG/H
centerC2D53⋊#99
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
Fitting#124×C53C4
socleC2×C53C2×#99
hypercenter#124#6157
hypocenterC53#2135
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #124

Complements

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

NH ≅ G/Ntype
2. #16811534C2semidirect
3. #16811533C2semidirect
80. C53#2135semidirect