Finite Group Databasebeta

← #20273759

Subgroups — #20273759

Order 6784.

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

normal subgroups only

This group has 202 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#16810959C212yesmaximal
3#16810562C211yesmaximal, characteristic
4#16810361C212yesmaximal
5C53×#324C211yesmaximal, characteristic
6#13133216—22no—
7#13133208—22no—
8#13133196—21no—
9#13133196C2212yes—
10#13133176—21no—
11C2×C4⋊Dic53C2211yescharacteristic
12#13133151—22no—
13#13133142—22no—
14#13133124—21no—
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—
36C4×Dic53—22no—
37C23×C53C2×D411yescharacteristic
38C2×C4×C53—21no—
39C2×C4×C53—21no—
40C2×C4×C53—21no—
41C2×C4×C53—41no—
42C2×C4×C53—41no—
43C2×C4×C53Q1612yes—
44C2×C4×C53C2×D412yes—
45C2×Dic53—22no—
46C2×Dic53—22no—
47C2×Dic53—41no—
48C2×Dic53—41no—
49C2×Dic53—42no—
50C2×Dic53—41no—
51C2×Dic53—41no—
52Dic106—42no—
53Dic106—81no—
54C8×C53—42no—
55C22×C53—21no—
56C22×C53—21no—
57C22×C53#12011yescharacteristic
58C22×C53C2×Q1611yescharacteristic
59C22×C53#13711yescharacteristic
60C4×C53—22no—
61C4×C53—22no—
62C4×C53—41no—
63Dic53—42no—
64Dic53—42no—
65Dic53—81no—
66Dic53—81no—
67#1617—531nomaximal, Sylow
68C2×C53—21no—
69C2×C53—21no—
70C2×C53#45111yescharacteristic
71C2×C53#45411yescharacteristic
72C2×C53#45811yescharacteristic
73#544—531no—
74#484—532no—
75#400—531no—
76#328—532no—
77#324D5311yescharacteristic
78C53#161711yescharacteristic, complemented, Sylow
79C4⋊Q8—1062no—
80#125—1062no—
81C22⋊Q8—532no—
82C22⋊Q8—1061no—
83#117—1061no—
84C2×C4⋊C4—531no—
85C2×C4⋊C4C2×D5311yescharacteristic
86#107—1062no—
87#103—1062no—
88C22⋊C8C2×D5312yes—
89#95—531no—
90#95—1061no—
91#95—1061no—
92C2×Q8—532no—
93C2×Q8—1061no—
94C22×C4—1061no—
95C22×C4—1061no—
96C22×C4D21211yescharacteristic
97C22×C4D21211yescharacteristic
98C22×C4C22×D5311yescharacteristic
99C2×C8—22no—
100C4⋊C4—22no—
101C4⋊C4—532no—
102C4⋊C4—1061no—
103C4⋊C4—1061no—
104C4⋊C4—1061no—
105C4⋊C4—1062no—
106C4⋊C4—1062no—
107C4⋊C4—1062no—
108C22⋊C4—1062no—
109C42—1062no—
110C23C2×D21211yescharacteristic
111Q8—4241no—
112Q8—2122no—
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4—1062no—
119C2×C4—1062no—
120C2×C4—2121no—
121C2×C4—2121no—
122C2×C4—2122no—
123C2×C4—2121no—
124C2×C4—2121no—
125C2×C4Dic21212yes—
126C2×C4D4×D5312yes—
127C8—42no—
128C22—21no—
129C22—21no—
130C22#1313318711yescharacteristic
131C22C2×Dic21211yescharacteristic
132C22#1313325011yescharacteristic
133C4—22no—
134C4—22no—
135C4—41no—
136C4—4241no—
137C4—4241no—
138C4—2122no—
139C4—2122no—
140C2—21no—
141C2—21no—
142C2#1681078111yescharacteristic
143C2#1681079811yescharacteristic
144C2#1681081411yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#324C2
socleC22×C53#120
hypercenter#324D53
hypocenterC53#1617
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
78. C53#1617semidirect