Finite Group Databasebeta

← #20273725

Subgroups — #20273725

Order 6784.

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

normal subgroups only

This group has 210 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#16810954C212yesmaximal
3#16810553C211yesmaximal, characteristic
4#16810361C212yesmaximal
5C53×#324C211yesmaximal, characteristic
6#13133216—22no—
7#13133208—22no—
8#13133196—21no—
9#13133196C2212yes—
10#13133176—21no—
11C2×C4⋊Dic53C2211yescharacteristic
12C2×#11719919—21no—
13#13133149—22no—
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—21no—
25C2×Dic106D412yes—
26C2×C8×C53—22no—
27C4⋊C4×C53—22no—
28C22⋊Dic53—22no—
29C4⋊Dic53—21no—
30C4⋊Dic53—21no—
31C4⋊Dic53—21no—
32C4⋊Dic53—22no—
33C4⋊Dic53D412yes—
34#11719919—22no—
35#11719919—22no—
36#11719919—42no—
37C4×Dic53—22no—
38C23×C53C2×D411yescharacteristic
39C2×C4×C53—21no—
40C2×C4×C53—21no—
41C2×C4×C53—21no—
42C2×C4×C53—41no—
43C2×C4×C53—41no—
44C2×C4×C53SD1612yes—
45C2×C4×C53C2×D412yes—
46C2×Dic53—22no—
47C2×Dic53—22no—
48C2×Dic53—22no—
49C2×Dic53—41no—
50C2×Dic53—41no—
51C2×Dic53—41no—
52C2×Dic53—41no—
53C2×Dic53—41no—
54Dic106—42no—
55Dic106—81no—
56C8×C53—42no—
57C22×C53—21no—
58C22×C53—21no—
59C22×C53#12011yescharacteristic
60C22×C53C2×SD1611yescharacteristic
61C22×C53#13711yescharacteristic
62C4×C53—22no—
63C4×C53—22no—
64C4×C53—41no—
65Dic53—42no—
66Dic53—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#544—531no—
76#483—532no—
77#398—531no—
78#328—532no—
79#324D5311yescharacteristic
80C53#161111yescharacteristic, complemented, Sylow
81C4⋊Q8—1062no—
82#125—1062no—
83C22⋊Q8—532no—
84C22⋊Q8—1061no—
85#117—1061no—
86C2×C4⋊C4—531no—
87C2×C4⋊C4—1061no—
88C2×C4⋊C4C2×D5311yescharacteristic
89#106—1062no—
90#103—1062no—
91C22⋊C8C2×D5312yes—
92#95—531no—
93#95—1061no—
94C2×Q8—532no—
95C2×Q8—1061no—
96C22×C4—1061no—
97C22×C4—1061no—
98C22×C4D21211yescharacteristic
99C22×C4D21211yescharacteristic
100C22×C4C22×D5311yescharacteristic
101C2×C8—22no—
102C4⋊C4—22no—
103C4⋊C4—532no—
104C4⋊C4—1061no—
105C4⋊C4—1061no—
106C4⋊C4—1061no—
107C4⋊C4—1062no—
108C4⋊C4—1062no—
109C4⋊C4—1062no—
110C4⋊C4—2122no—
111C22⋊C4—1062no—
112C42—1062no—
113C23C2×D21211yescharacteristic
114Q8—4241no—
115Q8—2122no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—21no—
119C2×C4—41no—
120C2×C4—41no—
121C2×C4—1062no—
122C2×C4—1062no—
123C2×C4—1062no—
124C2×C4—2121no—
125C2×C4—2121no—
126C2×C4—2121no—
127C2×C4—2121no—
128C2×C4—2121no—
129C2×C4#1171991312yes—
130C2×C4D4×D5312yes—
131C8—42no—
132C22—21no—
133C22—21no—
134C22#1313318711yescharacteristic
135C22C2×#1171991311yescharacteristic
136C22#1313325011yescharacteristic
137C4—22no—
138C4—22no—
139C4—41no—
140C4—4241no—
141C4—2122no—
142C4—2122no—
143C4—2122no—
144C2—21no—
145C2—21no—
146C2#1681076411yescharacteristic
147C2#1681080811yescharacteristic
148C2#1681081411yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#324C2
socleC22×C53#120
hypercenter#324D53
hypocenterC53#1611
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
80. C53#1611semidirect