Finite Group Databasebeta

← #20290997

Subgroups — #20290997

Order 6784.

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

normal subgroups only

This group has 238 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
1#16811602C212yesmaximal
2#16811529C214yesmaximal, complemented
3#16811197C211yesmaximal, characteristic
4#13133296—22no—
5#13133276C2214yes—
6#13133275—22no—
7C2×#11719945C2211yescharacteristic
8#13133215C412yescomplemented
9#13133206C2212yes—
10#13133200—22no—
11#13133193—22no—
12#13133176C411yescharacteristic, complemented
13C22×Q8×C53C411yescharacteristic, complemented
14C2×Q8×C53—21no—
15C2×Q8×C53—42no—
16C2×Q8×C53C2×C412yes—
17C22×C4×C53—21no—
18C22×C4×C53C2×C411yescharacteristic
19C2×C22⋊D53D412yes—
20#11719945—42no—
21C2×D212C2311yescharacteristic
22C2×C4×D53D412yes—
23C2×Dic106C2×C411yescharacteristic
24C22⋊#6157—24no—
25C4⋊#6157—42no—
26C4×#6157—42no—
27C22⋊Dic53—21no—
28#11719921—24no—
29C4⋊Dic53—21no—
30#11719919—24no—
31C4×Dic53C2×C412yes—
32C23×C53C22⋊C411yescharacteristic
33C22×D53C2×D412yes—
34C2×#6157—44no—
35Q8×C53—42no—
36Q8×C53—42no—
37C2×C4×C53—22no—
38C2×C4×C53—22no—
39C2×C4×C53—42no—
40C2×C4×C53C22⋊C411yescharacteristic
41C2×C4×C53C22×C411yescharacteristic
42C22⋊D53—42no—
43C2×Dic53—22no—
44C2×Dic53C22⋊C412yes—
45D212—41no—
46C4×D53—22no—
47C4×D53—22no—
48Dic106—41no—
49C22×C53—21no—
50C22×C53—41no—
51C22×C53C2×C22⋊C411yescharacteristic
52C2×D53—22no—
53C2×D53—42no—
54#6157—84no—
55C4×C53—21no—
56C4×C53—42no—
57C4×C53#9912yes—
58Dic53—22no—
59Dic53—42no—
60#2129—531nomaximal, Sylow
61C2×C53—21no—
62C2×C53—41no—
63C2×C53C2×#9911yescharacteristic
64D53—42no—
65#536—531no—
66#410—532no—
67#354—534no—
68C53#212911yescharacteristic, complemented, Sylow
69C2×#41—531no—
70C22×Q8#615711yescharacteristic, complemented
71C4⋊Q8—532no—
72#124—532no—
73#123—1062no—
74C22⋊Q8—1062no—
75#117—531no—
76#117—1062no—
77#99—534no—
78#99—1062no—
79#41—2122no—
80C2×Q8—21no—
81C2×Q8—42no—
82C2×Q8—531no—
83C2×Q8C2×#615712yes—
84C2×D4—532no—
85C2×D4—531no—
86C22×C4—21no—
87C22×C4—532no—
88C22×C4C2×#615711yescharacteristic
89C4⋊C4—1064no—
90C4⋊C4—1061no—
91C4⋊C4—2122no—
92C22⋊C4—1064no—
93C22⋊C4—1061no—
94C22⋊C4—1064no—
95C42—532no—
96C42—2122no—
97C23—532no—
98C23C22⋊#615711yescharacteristic
99Q8—42no—
100Q8—42no—
101Q8—2121no—
102D4—2121no—
103D4—2122no—
104C2×C4—22no—
105C2×C4—22no—
106C2×C4—42no—
107C2×C4—532no—
108C2×C4—1062no—
109C2×C4—1062no—
110C2×C4—1062no—
111C2×C4—2124no—
112C2×C4C22⋊#615711yescharacteristic
113C2×C4C22×#615711yescharacteristic
114C22—21no—
115C22—41no—
116C22—1062no—
117C22—2122no—
118C22C2×C22⋊#615711yescharacteristic
119C4—21no—
120C4—42no—
121C4—1062no—
122C4—4244no—
123C4—2122no—
124C4#1313327512yes—
125C2—21no—
126C2—41no—
127C2—2122no—
128C2C2×#1313327511yescharacteristic
subgroupG/H
centerC2C2×#13133275
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
FittingC22×Q8×C53C4
socleC2×C53C2×#99
hypercenterC22×Q8#6157
hypocenterC53#2129
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
2. #16811529C2semidirect
8. #13133215C4semidirect
12. #13133176C4semidirect
13. C22×Q8×C53C4semidirect
68. C53#2129semidirect
70. C22×Q8#6157semidirect