Finite Group Databasebeta

← #20273689

Subgroups — #20273689

Order 6784.

238 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 57 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#16810778C212yesmaximal
2#16810543C211yesmaximal, characteristic
3#16810360C212yesmaximal
4#16810349C211yesmaximal, characteristic
5C53×#544C211yesmaximal, characteristic
6C22×Dic106C2211yescharacteristic
7C2×#11719925—22no—
8C2×#11719919—21no—
9#13133140—22no—
10C22⋊#88322C2212yes—
11#13133124—21no—
12#13133124C2211yescharacteristic
13C4⋊Q8×C53—22no—
14#125×C53—22no—
15C22⋊Q8×C53—21no—
16C22⋊Q8×C53C2212yes—
17#117×C53—21no—
18C2×C4⋊C4×C53C2211yescharacteristic
19C2×Q8×C53—21no—
20C2×Q8×C53D412yes—
21C22×C4×C53D411yescharacteristic
22C22×C4×C53D411yescharacteristic
23C22×C4×C53C2311yescharacteristic
24C22×Dic53—21no—
25C22×Dic53—21no—
26C2×Dic106—22no—
27C2×Dic106—42no—
28C4⋊C4×C53—21no—
29C4⋊C4×C53—21no—
30C4⋊C4×C53—21no—
31C4⋊C4×C53—22no—
32C4⋊C4×C53—22no—
33C4⋊C4×C53—22no—
34C4⋊C4×C53D412yes—
35C22⋊C4×C53—22no—
36C42×C53—22no—
37#11719925—44no—
38#11719919—42no—
39C2×#88322—22no—
40C23×C53C2×D411yescharacteristic
41Q8×C53—42no—
42Q8×C53—81no—
43C2×C4×C53—21no—
44C2×C4×C53—21no—
45C2×C4×C53—21no—
46C2×C4×C53—22no—
47C2×C4×C53—22no—
48C2×C4×C53—41no—
49C2×C4×C53—41no—
50C2×C4×C53Q1612yescomplemented
51C2×C4×C53C2×D412yes—
52C2×Dic53—22no—
53C2×Dic53—22no—
54C2×Dic53—41no—
55C2×Dic53—41no—
56C2×Dic53—41no—
57C2×Dic53—41no—
58Dic106—24no—
59Dic106—42no—
60#88322—42no—
61C22×C53—21no—
62C22×C53—21no—
63C22×C53#12011yescharacteristic
64C22×C53C2×Q1611yescharacteristic
65C22×C53#13711yescharacteristic
66C4×C53—22no—
67C4×C53—22no—
68C4×C53—42no—
69C4×C53—41no—
70C4×C53—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#544D5311yescharacteristic
80#451—532no—
81#395—531no—
82#328—532no—
83#324—531no—
84C53#160511yescharacteristic, complemented, Sylow
85C22×Q8—531no—
86C2×Q16—1062no—
87C4⋊Q8—22no—
88#125—22no—
89C22⋊Q8—21no—
90C22⋊Q8C2×D5312yes—
91#117—21no—
92C2×C4⋊C4—1061no—
93C2×C4⋊C4C2×D5311yescharacteristic
94#103—1062no—
95C22⋊C8—532no—
96#95—531no—
97#95—1061no—
98C2×Q8—21no—
99C2×Q8—1062no—
100C2×Q8—2122no—
101C2×Q8C22⋊D5312yes—
102C22×C4—1061no—
103C22×C4—1061no—
104C22×C4C22⋊D5311yescharacteristic
105C22×C4C22⋊D5311yescharacteristic
106C22×C4C22×D5311yescharacteristic
107Q16—2124no—
108C2×C8—1062no—
109C4⋊C4—21no—
110C4⋊C4—21no—
111C4⋊C4—21no—
112C4⋊C4—22no—
113C4⋊C4—22no—
114C4⋊C4—22no—
115C4⋊C4—2122no—
116C4⋊C4C22⋊D5312yes—
117C22⋊C4—22no—
118C42—22no—
119C23C2×C22⋊D5311yescharacteristic
120Q8—42no—
121Q8—81no—
122Q8—1064no—
123Q8—2122no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—22no—
127C2×C4—22no—
128C2×C4—21no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—1062no—
132C2×C4—1062no—
133C2×C4—2121no—
134C2×C4—2121no—
135C2×C4—2121no—
136C2×C4—2121no—
137C2×C4#1171992512yescomplemented
138C2×C4C2×C22⋊D5312yes—
139C8—2122no—
140C22—21no—
141C22—21no—
142C22#1313318611yescharacteristic
143C22C2×#1171992511yescharacteristic
144C22#1313324811yescharacteristic
145C4—22no—
146C4—22no—
147C4—42no—
148C4—41no—
149C4—42no—
150C4—2122no—
151C4—2122no—
152C2—21no—
153C2—21no—
154C2#1681077911yescharacteristic
155C2#1681080611yescharacteristic
156C2#1681081011yescharacteristic
subgroupG/H
centerC22#13133186
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#544C2
socleC22×C53#120
hypercenter#544D53
hypocenterC53#1605
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #544

Complements

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

NH ≅ G/Ntype
50. C2×C4×C53Q16semidirect
84. C53#1605semidirect
137. C2×C4#11719925semidirect