Finite Group Databasebeta

← #20273691

Subgroups — #20273691

Order 6784.

238 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 53 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#16811254C211yesmaximal, characteristic
2#16810781C212yesmaximal
3#16810543C211yesmaximal, characteristic
4#16810361C212yesmaximal
5C53×#324C211yesmaximal, characteristic
6C22×Dic106C2211yescharacteristic
7C2×Dic212—22no—
8#13133216—22no—
9#13133208—22no—
10#13133196—21no—
11#13133196C2212yes—
12#13133176—21no—
13C2×#11719919—21no—
14#13133142—22no—
15#13133124—21no—
16#13133124C2211yescharacteristic
17C2×C4⋊C4×C53C2211yescharacteristic
18C22⋊C8×C53C2212yes—
19C22×C4×C53D411yescharacteristic
20C22×C4×C53D411yescharacteristic
21C22×C4×C53C2311yescharacteristic
22C22×Dic53—21no—
23C22×Dic53—21no—
24C2×Dic106—22no—
25C2×Dic106—21no—
26C2×Dic106—42no—
27C2×Dic106D412yes—
28C2×C8×C53—22no—
29C4⋊C4×C53—22no—
30C22⋊Dic53—22no—
31C4⋊Dic53—21no—
32C4⋊Dic53—21no—
33C4⋊Dic53—21no—
34C4⋊Dic53D412yes—
35#11719919—22no—
36#11719919—22no—
37#11719919—42no—
38C4×Dic53—22no—
39Dic212—44no—
40C23×C53C2×D411yescharacteristic
41C2×C4×C53—21no—
42C2×C4×C53—21no—
43C2×C4×C53—21no—
44C2×C4×C53—41no—
45C2×C4×C53—41no—
46C2×C4×C53Q1612yescomplemented
47C2×C4×C53C2×D412yes—
48C2×Dic53—22no—
49C2×Dic53—22no—
50C2×Dic53—22no—
51C2×Dic53—22no—
52C2×Dic53—41no—
53C2×Dic53—41no—
54C2×Dic53—41no—
55C2×Dic53—41no—
56Dic106—24no—
57Dic106—42no—
58Dic106—42no—
59Dic106—81no—
60C8×C53—42no—
61C22×C53—21no—
62C22×C53—21no—
63C22×C53#12011yescharacteristic
64C22×C53C2×Q1611yescharacteristic
65C22×C53#13711yescharacteristic
66C4×C53—22no—
67C4×C53—22no—
68C4×C53—41no—
69Dic53—42no—
70Dic53—42no—
71Dic53—42no—
72Dic53—42no—
73#1605—531nomaximal, Sylow
74C2×C53—21no—
75C2×C53—21no—
76C2×C53#45111yescharacteristic
77C2×C53#45611yescharacteristic
78C2×C53B(4,𝔽2)11yescharacteristic
79#544—531no—
80#451—532no—
81#395—531no—
82#328—532no—
83#324D5311yescharacteristic
84C53#160511yescharacteristic, complemented, Sylow
85C22×Q8—531no—
86C2×Q16—1062no—
87C4⋊Q8—1062no—
88#125—1062no—
89C22⋊Q8—532no—
90C22⋊Q8—1061no—
91#117—1061no—
92C2×C4⋊C4—1061no—
93C2×C4⋊C4C2×D5311yescharacteristic
94#103—1062no—
95C22⋊C8C2×D5312yes—
96#95—531no—
97#95—1061no—
98C2×Q8—532no—
99C2×Q8—1061no—
100C2×Q8—1062no—
101C2×Q8—2122no—
102C22×C4—1061no—
103C22×C4—1061no—
104C22×C4D21211yescharacteristic
105C22×C4D21211yescharacteristic
106C22×C4C22×D5311yescharacteristic
107Q16—2124no—
108C2×C8—22no—
109C4⋊C4—22no—
110C4⋊C4—532no—
111C4⋊C4—1061no—
112C4⋊C4—1061no—
113C4⋊C4—1061no—
114C4⋊C4—1062no—
115C4⋊C4—1062no—
116C4⋊C4—2122no—
117C22⋊C4—1062no—
118C42—1062no—
119C23C2×D21211yescharacteristic
120Q8—1064no—
121Q8—4241no—
122Q8—2122no—
123Q8—2122no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—1062no—
130C2×C4—1062no—
131C2×C4—1062no—
132C2×C4—1062no—
133C2×C4—2121no—
134C2×C4—2121no—
135C2×C4—2121no—
136C2×C4—2121no—
137C2×C4Dic21212yescomplemented
138C2×C4D4×D5312yes—
139C8—42no—
140C22—21no—
141C22—21no—
142C22#1313318711yescharacteristic
143C22C2×Dic21211yescharacteristic
144C22#1313325011yescharacteristic
145C4—22no—
146C4—22no—
147C4—41no—
148C4—2122no—
149C4—2122no—
150C4—2122no—
151C4—2122no—
152C2—21no—
153C2—21no—
154C2#1681078111yescharacteristic
155C2#1681080811yescharacteristic
156C2#1681081111yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#324C2
socleC22×C53#120
hypercenter#324D53
hypocenterC53#1605
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
46. C2×C4×C53Q16semidirect
84. C53#1605semidirect
137. C2×C4Dic212semidirect