Finite Group Databasebeta

← #20290197

Subgroups — #20290197

Order 6784.

196 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 58 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#16811643C211yesmaximal, characteristic
2#16811643C211yesmaximal, characteristic
3#16811480C214yesmaximal, complemented
4C2×#13133208C211yesmaximal, characteristic
5#13133284—21no—
6#13133284—21no—
7C22⋊#88324—21no—
8C22⋊#88324—21no—
9#13133272C2212yes—
10#13133272C2212yes—
11#13133208—21no—
12#13133208—21no—
13#13133208C412yes—
14#13133208C2212yes—
15C2×C4⋊Dic53—21no—
16C2×C4⋊Dic53C411yescharacteristic
17C2×#11719919—21no—
18C2×C4×Dic53C2211yescharacteristic
19C2×C4⋊C4×C53C411yescharacteristic
20C22×C4×C53—21no—
21C22×C4×C53C2×C411yescharacteristic
22C22×Dic53—21no—
23C22×Dic53D411yescharacteristic
24C22×Dic53D411yescharacteristic
25C2×#88324—22no—
26C2×#88324—22no—
27C4⋊C4×C53—21no—
28C4⋊C4×C53C2×C412yes—
29C4⋊Dic53—21no—
30C4⋊Dic53—22no—
31C4⋊Dic53—22no—
32C4⋊Dic53C2×C412yes—
33#11719919—22no—
34#11719919—22no—
35C4×Dic53C2×C411yescharacteristic
36C4×Dic53D411yescharacteristic
37C4×Dic53D411yescharacteristic
38C4×Dic53C2311yescharacteristic
39C23×C53C22⋊C411yescharacteristic
40C2×C4×C53—22no—
41C2×C4×C53—22no—
42C2×C4×C53—42no—
43C2×C4×C53C22⋊C411yescharacteristic
44C2×C4×C53C22×C411yescharacteristic
45C2×Dic53—22no—
46C2×Dic53—41no—
47C2×Dic53—41no—
48C2×Dic53—42no—
49C2×Dic53C22⋊C411yescharacteristic
50C2×Dic53C22⋊C411yescharacteristic
51C2×Dic53C2×D411yescharacteristic
52C2×Dic53C2×D411yescharacteristic
53#88324—42no—
54#88324—42no—
55C22×C53—21no—
56C22×C53—21no—
57C22×C53#10112yes—
58C22×C53C2×C22⋊C411yescharacteristic
59C4×C53—21no—
60C4×C53—21no—
61C4×C53—42no—
62Dic53—21no—
63Dic53—21no—
64Dic53—21no—
65Dic53—21no—
66Dic53—42no—
67#1534—531nomaximal, Sylow
68C2×C53—22no—
69C2×C53C2×#10111yescharacteristic
70C2×C53#42111yescharacteristic
71C2×C53#42111yescharacteristic
72C2×#125—531no—
73#432—531no—
74#432—531no—
75#330—534no—
76C53#153411yescharacteristic, complemented, Sylow
77#125—532no—
78#125—532no—
79#125—1061no—
80#125—1061no—
81C2×C4⋊C4—531no—
82C2×C4⋊C4—1061no—
83C2×C4⋊C4—1061no—
84C2×C4⋊C4#615711yescharacteristic
85C2×C42—531no—
86C4⋊C8—1061no—
87C4⋊C8—1061no—
88C22⋊C8—1061no—
89C22⋊C8—1061no—
90#97—532no—
91#97—532no—
92C22×C4—21no—
93C22×C4—531no—
94C22×C4—531no—
95C22×C4—1061no—
96C22×C4C2×#615711yescharacteristic
97C2×C8—1062no—
98C2×C8—1062no—
99C4⋊C4—21no—
100C4⋊C4—532no—
101C4⋊C4—1062no—
102C4⋊C4—1062no—
103C4⋊C4—1062no—
104C4⋊C4—1062no—
105C4⋊C4—1061no—
106C4⋊C4C2×#615712yes—
107C42—531no—
108C42—531no—
109C42—531no—
110C42—531no—
111C23C22⋊#615711yescharacteristic
112C2×C4—22no—
113C2×C4—22no—
114C2×C4—42no—
115C2×C4—531no—
116C2×C4—531no—
117C2×C4—531no—
118C2×C4—531no—
119C2×C4—1062no—
120C2×C4—2121no—
121C2×C4—2121no—
122C2×C4—2122no—
123C2×C4C22⋊#615711yescharacteristic
124C2×C4C22×#615711yescharacteristic
125C8—2122no—
126C8—2122no—
127C22—21no—
128C22—21no—
129C22#1313327812yes—
130C22C2×C22⋊#615711yescharacteristic
131C4—21no—
132C4—21no—
133C4—42no—
134C4—1061no—
135C4—1061no—
136C4—1061no—
137C4—1061no—
138C4—2122no—
139C2—22no—
140C2C2×#1313327811yescharacteristic
141C2#1681162111yescharacteristic
142C2#1681162211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
FittingC2×C4⋊C4×C53C4
socleC22×C53C2×C22⋊C4
hypercenterC2×C4⋊C4#6157
hypocenterC53#1534
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #16811480C2semidirect
76. C53#1534semidirect