Finite Group Databasebeta

← #20290627

Subgroups — #20290627

Order 6784.

296 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 62 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×#13133276C212yesmaximal
2#16811500C214yesmaximal, complemented
3#16811097C211yesmaximal, characteristic
4C2×C22⋊#6157C2214yes—
5#13133276—22no—
6#13133276—22no—
7C22×D212C2211yescharacteristic
8C4×D212—22no—
9#13133179—22no—
10#13133176C411yescharacteristic, complemented
11C22⋊C4×D53C412yescomplemented
12C22⋊C4×D53C2212yes—
13#117×C53C411yescharacteristic, complemented
14C23×D53C2×C411yescharacteristic
15C23×D53C2311yescharacteristic
16C22×#6157—24no—
17C22×C4×C53C2×C411yescharacteristic
18C2×D212—24no—
19C2×D212D412yes—
20C2×D212D412yes—
21C2×C4×D53—22no—
22C2×C4×D53—22no—
23C22⋊#6157—24no—
24C22⋊#6157—24no—
25C4⋊C4×C53—21no—
26C22⋊C4×C53C2×C412yes—
27C42×C53—21no—
28C22⋊Dic53C2×C412yes—
29#11719921—22no—
30#11719921—22no—
31C4⋊Dic53—21no—
32C4×Dic53—21no—
33C23×C53C22×C411yescharacteristic
34C22×D53—22no—
35C22×D53—81no—
36C22×D53—81no—
37C22×D53C22⋊C412yes—
38C22×D53C2×D412yes—
39C22×D53#4114yes—
40C2×#6157—44no—
41C2×#6157—84no—
42C2×C4×C53—22no—
43C2×C4×C53—22no—
44C2×C4×C53C22⋊C411yescharacteristic
45C2×C4×C53C22⋊C411yescharacteristic
46C2×Dic53—22no—
47D212—24no—
48D212—24no—
49C4×D53—82no—
50C4×D53—82no—
51C22×C53—41no—
52C22×C53C2×C22⋊C411yescharacteristic
53C22×C53#11712yes—
54C2×D53—22no—
55C2×D53—42no—
56C2×D53—44no—
57C2×D53—42no—
58C2×D53—82no—
59C2×D53#12314yes—
60#6157—84no—
61C4×C53—21no—
62C4×C53—21no—
63C4×C53—42no—
64Dic53—42no—
65#1828—531nomaximal, Sylow
66C2×C53—21no—
67C2×C53—22no—
68C2×C53#38111yescharacteristic
69D53—24no—
70D53—42no—
71#518—531no—
72C2×#99—532no—
73#342—534no—
74C53#182811yescharacteristic, complemented, Sylow
75C22×D4—531no—
76C4×D4—1062no—
77C4×D4—1062no—
78#117—531no—
79#117#615711yescharacteristic, complemented
80C2×C22⋊C4—532no—
81C2×C22⋊C4—532no—
82C2×C22⋊C4—534no—
83#99—1062no—
84#99—1062no—
85C24—531no—
86C24—531no—
87C2×D4—532no—
88C2×D4—532no—
89C2×D4—1064no—
90C22×C4—1062no—
91C22×C4—1062no—
92C22×C4—1064no—
93C22×C4C2×#615711yescharacteristic
94C4⋊C4—21no—
95C4⋊C4—1061no—
96C22⋊C4—532no—
97C22⋊C4—1062no—
98C22⋊C4—1062no—
99C22⋊C4—1064no—
100C22⋊C4—1064no—
101C22⋊C4C2×#615712yes—
102C42—21no—
103C42—1061no—
104C23—532no—
105C23—534no—
106C23—532no—
107C23—1062no—
108C23—4241no—
109C23—4241no—
110C23C22×#615711yescharacteristic
111D4—1064no—
112D4—1064no—
113C2×C4—22no—
114C2×C4—22no—
115C2×C4—1062no—
116C2×C4—4242no—
117C2×C4—4242no—
118C2×C4—4244no—
119C2×C4—2124no—
120C2×C4C22⋊#615711yescharacteristic
121C2×C4C22⋊#615711yescharacteristic
122C22—41no—
123C22—534no—
124C22—1062no—
125C22—4242no—
126C22—2122no—
127C22—2124no—
128C22—2122no—
129C22C2×C22⋊#615711yescharacteristic
130C22#1313329612yes—
131C4—21no—
132C4—21no—
133C4—42no—
134C4—4244no—
135C4—2122no—
136C2—21no—
137C2—22no—
138C2—1064no—
139C2—2122no—
140C2#1681156611yescharacteristic
subgroupG/H
centerC2#16811566
commutatorC23×C53C22×C4
FrattiniC23C22×#6157
Fitting#117×C53C4
socleC2×C53#381
hypercenter#117#6157
hypocenterC53#1828
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C23 ◁ #117

Complements

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

NH ≅ G/Ntype
2. #16811500C2semidirect
10. #13133176C4semidirect
11. C22⋊C4×D53C4semidirect
13. #117×C53C4semidirect
74. C53#1828semidirect
79. #117#6157semidirect