Finite Group Databasebeta

← #20273723

Subgroups — #20273723

Order 6784.

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

normal subgroups only

This group has 210 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#16810951C212yesmaximal
2#16810553C211yesmaximal, characteristic
3#16810360C212yesmaximal
4#16810349C211yesmaximal, characteristic
5C53×#544C211yesmaximal, characteristic
6C2×C4⋊Dic53C2211yescharacteristic
7C2×#11719919—21no—
8#13133148—22no—
9#13133141—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—
26C4⋊C4×C53—21no—
27C4⋊C4×C53—21no—
28C4⋊C4×C53—21no—
29C4⋊C4×C53—22no—
30C4⋊C4×C53—22no—
31C4⋊C4×C53—22no—
32C4⋊C4×C53D412yes—
33C22⋊C4×C53—22no—
34C42×C53—22no—
35C4⋊Dic53—22no—
36#11719919—42no—
37C2×#88322—22no—
38C23×C53C2×D411yescharacteristic
39Q8×C53—42no—
40Q8×C53—81no—
41C2×C4×C53—21no—
42C2×C4×C53—21no—
43C2×C4×C53—21no—
44C2×C4×C53—22no—
45C2×C4×C53—22no—
46C2×C4×C53—41no—
47C2×C4×C53—41no—
48C2×C4×C53SD1612yes—
49C2×C4×C53C2×D412yes—
50C2×Dic53—22no—
51C2×Dic53—41no—
52C2×Dic53—41no—
53C2×Dic53—41no—
54C2×Dic53—41no—
55C2×Dic53—41no—
56#88322—42no—
57C22×C53—21no—
58C22×C53—21no—
59C22×C53#12011yescharacteristic
60C22×C53C2×SD1611yescharacteristic
61C22×C53#13711yescharacteristic
62C4×C53—22no—
63C4×C53—22no—
64C4×C53—42no—
65C4×C53—41no—
66C4×C53—42no—
67Dic53—42no—
68Dic53—81no—
69#1611—531nomaximal, Sylow
70C2×C53—21no—
71C2×C53—21no—
72C2×C53#44811yescharacteristic
73C2×C53#45611yescharacteristic
74C2×C53#45811yescharacteristic
75#544D5311yescharacteristic
76#483—532no—
77#398—531no—
78#328—532no—
79#324—531no—
80C53#161111yescharacteristic, complemented, Sylow
81C4⋊Q8—22no—
82#125—22no—
83C22⋊Q8—21no—
84C22⋊Q8C2×D5312yes—
85#117—21no—
86C2×C4⋊C4—531no—
87C2×C4⋊C4—1061no—
88C2×C4⋊C4C2×D5311yescharacteristic
89#106—1062no—
90#103—1062no—
91C22⋊C8—532no—
92#95—531no—
93#95—1061no—
94C2×Q8—21no—
95C2×Q8C22⋊D5312yes—
96C22×C4—1061no—
97C22×C4—1061no—
98C22×C4C22⋊D5311yescharacteristic
99C22×C4C22⋊D5311yescharacteristic
100C22×C4C22×D5311yescharacteristic
101C2×C8—1062no—
102C4⋊C4—21no—
103C4⋊C4—21no—
104C4⋊C4—21no—
105C4⋊C4—22no—
106C4⋊C4—22no—
107C4⋊C4—22no—
108C4⋊C4—1062no—
109C4⋊C4—2122no—
110C4⋊C4C22⋊D5312yes—
111C22⋊C4—22no—
112C42—22no—
113C23C2×C22⋊D5311yescharacteristic
114Q8—42no—
115Q8—81no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—22no—
119C2×C4—22no—
120C2×C4—21no—
121C2×C4—41no—
122C2×C4—41no—
123C2×C4—1062no—
124C2×C4—2121no—
125C2×C4—2121no—
126C2×C4—2121no—
127C2×C4—2121no—
128C2×C4—2121no—
129C2×C4#1171992412yes—
130C2×C4C2×C22⋊D5312yes—
131C8—2122no—
132C22—21no—
133C22—21no—
134C22#1313318611yescharacteristic
135C22C2×#1171992411yescharacteristic
136C22#1313324811yescharacteristic
137C4—22no—
138C4—22no—
139C4—42no—
140C4—41no—
141C4—42no—
142C4—4241no—
143C4—2122no—
144C2—21no—
145C2—21no—
146C2#1681076211yescharacteristic
147C2#1681080611yescharacteristic
148C2#1681081311yescharacteristic
subgroupG/H
centerC22#13133186
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#544C2
socleC22×C53#120
hypercenter#544D53
hypocenterC53#1611
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
80. C53#1611semidirect