Finite Group Databasebeta

← #20273671

Subgroups — #20273671

Order 6784.

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

normal subgroups only

This group has 256 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#16811249C211yesmaximal, characteristic
2#16810779C212yesmaximal, complemented
3#16810359C212yesmaximal, complemented
4#16810347C211yesmaximal, characteristic
5C53×#393C211yesmaximal, characteristic
6C2×#11719925—22no—
7#13133196—21no—
8#13133196C2212yes—
9#13133194—22no—
10#13133194—22no—
11C2×C22⋊Dic53—21no—
12#13133141—22no—
13C22⋊#88322C2212yes—
14C22×Q8×C53C2211yescharacteristic
15C2×C22⋊C4×C53—21no—
16C2×C22⋊C4×C53C2211yescharacteristic
17#95×C53—21no—
18#95×C53C2211yescharacteristic
19C24×C53D411yescharacteristic
20C2×Q8×C53—22no—
21C2×Q8×C53—42no—
22C22×C4×C53—21no—
23C22×C4×C53—21no—
24C22×C4×C53D411yescharacteristic
25C22×C4×C53C2311yescharacteristic
26C22×Dic53—21no—
27C22×Dic53—21no—
28C2×Dic106—21no—
29C2×Dic106D412yes—
30C22⋊C4×C53—22no—
31C22⋊C4×C53—42no—
32C22⋊Dic53—22no—
33C22⋊Dic53—22no—
34#11719925—44no—
35C4⋊Dic53—21no—
36C4⋊Dic53D412yes—
37#11719919—22no—
38#11719919—22no—
39C2×#88322—22no—
40C23×C53—21no—
41C23×C53—21no—
42C23×C53—41no—
43C23×C53—41no—
44C23×C53Q1612yescomplemented
45C23×C53C2×D411yescharacteristic
46Q8×C53—24no—
47Q8×C53—42no—
48C2×C4×C53—22no—
49C2×C4×C53—21no—
50C2×C4×C53—41no—
51C2×C4×C53—41no—
52C2×C4×C53—41no—
53C2×C4×C53—41no—
54C2×C4×C53—41no—
55C2×C4×C53—41no—
56C2×C4×C53—41no—
57C2×C4×C53C2×D412yes—
58C2×Dic53—22no—
59C2×Dic53—22no—
60C2×Dic53—81no—
61C2×Dic53—81no—
62Dic106—42no—
63Dic106—81no—
64#88322—42no—
65C22×C53—21no—
66C22×C53—21no—
67C22×C53—22no—
68C22×C53—42no—
69C22×C53—41no—
70C22×C53—41no—
71C22×C53—81no—
72C22×C53#12011yescharacteristic
73C22×C53C2×Q1611yescharacteristic
74C22×C53#13711yescharacteristic
75C4×C53—22no—
76C4×C53—41no—
77C4×C53—42no—
78C4×C53—81no—
79Dic53—42no—
80Dic53—42no—
81#1602—531nomaximal, Sylow
82C2×C53—21no—
83C2×C53—21no—
84C2×C53—42no—
85C2×C53#45111yescharacteristic
86C2×C53#45511yescharacteristic
87C2×C53B(4,𝔽2)11yescharacteristic
88#543—531no—
89#451—532no—
90#393D5311yescharacteristic
91#328—532no—
92#323—531no—
93C53#160211yescharacteristic, complemented, Sylow
94C22×Q8C2×D5311yescharacteristic
95C2×Q16—1062no—
96C22⋊Q8—532no—
97C22⋊Q8—1062no—
98C22⋊Q8—1062no—
99C22⋊Q8—1061no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—1061no—
102C2×C22⋊C4C2×D5311yescharacteristic
103#103—1062no—
104C22⋊C8—532no—
105#95—21no—
106#95C2×D5311yescharacteristic
107C24C22⋊D5311yescharacteristic
108C2×Q8—22no—
109C2×Q8—42no—
110C2×Q8—532no—
111C2×Q8—1061no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—1061no—
115C22×C4—1061no—
116C22×C4C22⋊D5311yescharacteristic
117C22×C4C22×D5311yescharacteristic
118Q16—2124no—
119C2×C8—1062no—
120C4⋊C4—532no—
121C4⋊C4—1062no—
122C4⋊C4—1061no—
123C4⋊C4—1062no—
124C22⋊C4—22no—
125C22⋊C4—42no—
126C22⋊C4—1062no—
127C22⋊C4—1062no—
128C23—21no—
129C23—21no—
130C23—41no—
131C23—41no—
132C23#1171992512yescomplemented
133C23C2×C22⋊D5311yescharacteristic
134Q8—24no—
135Q8—42no—
136Q8—4241no—
137Q8—2122no—
138C2×C4—22no—
139C2×C4—21no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—1062no—
148C2×C4—1062no—
149C2×C4—4241no—
150C2×C4—4241no—
151C2×C4D4×D5312yes—
152C8—2122no—
153C22—21no—
154C22—21no—
155C22—22no—
156C22—42no—
157C22—41no—
158C22—41no—
159C22—81no—
160C22#1313318511yescharacteristic
161C22C2×#1171992511yescharacteristic
162C22#1313324711yescharacteristic
163C4—22no—
164C4—41no—
165C4—42no—
166C4—81no—
167C4—2122no—
168C4—2122no—
169C2—21no—
170C2—21no—
171C2—42no—
172C2#1681077811yescharacteristic
173C2#1681080011yescharacteristic
174C2#1681080911yescharacteristic
subgroupG/H
centerC22#13133185
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#393C2
socleC22×C53#120
hypercenter#393D53
hypocenterC53#1602
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #393

Complements

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

NH ≅ G/Ntype
2. #16810779C2semidirect
3. #16810359C2semidirect
44. C23×C53Q16semidirect
93. C53#1602semidirect
132. C23#11719925semidirect