Finite Group Databasebeta

← #20273674

Subgroups — #20273674

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#16810781C212yesmaximal, complemented
3#16810535C211yesmaximal, characteristic
4#16810361C212yesmaximal, complemented
5C53×#323C211yesmaximal, characteristic
6C22×Dic106C2211yescharacteristic
7C2×Dic212—22no—
8#13133196—21no—
9#13133196C2212yes—
10#13133194—22no—
11#13133194—22no—
12C2×C22⋊Dic53—21no—
13C2×C22⋊Dic53—21no—
14#13133142—22no—
15#13133124—21no—
16#13133124C2211yescharacteristic
17C2×C22⋊C4×C53C2211yescharacteristic
18C22⋊C8×C53C2212yes—
19C24×C53D411yescharacteristic
20C22×C4×C53D411yescharacteristic
21C22×C4×C53C2311yescharacteristic
22C22×Dic53—21no—
23C22×Dic53—21no—
24C22×Dic53—21no—
25C22×Dic53—21no—
26C2×Dic106—22no—
27C2×Dic106—21no—
28C2×Dic106—42no—
29C2×Dic106D412yes—
30C2×C8×C53—22no—
31C22⋊C4×C53—22no—
32C22⋊Dic53—22no—
33C22⋊Dic53—22no—
34C22⋊Dic53—42no—
35C4⋊Dic53—21no—
36C4⋊Dic53D412yes—
37#11719919—22no—
38#11719919—22no—
39Dic212—44no—
40C23×C53—21no—
41C23×C53—21no—
42C23×C53—41no—
43C23×C53—41no—
44C23×C53Q1612yescomplemented
45C23×C53C2×D411yescharacteristic
46C2×C4×C53—21no—
47C2×C4×C53—41no—
48C2×C4×C53—41no—
49C2×C4×C53C2×D412yes—
50C2×Dic53—22no—
51C2×Dic53—22no—
52C2×Dic53—22no—
53C2×Dic53—41no—
54C2×Dic53—41no—
55C2×Dic53—41no—
56C2×Dic53—41no—
57C2×Dic53—41no—
58C2×Dic53—81no—
59C2×Dic53—81no—
60Dic106—24no—
61Dic106—42no—
62Dic106—42no—
63Dic106—81no—
64C8×C53—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—
77Dic53—42no—
78Dic53—42no—
79Dic53—42no—
80Dic53—81no—
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#393—531no—
91#328—532no—
92#323D5311yescharacteristic
93C53#160211yescharacteristic, complemented, Sylow
94C22×Q8—531no—
95C2×Q16—1062no—
96C22⋊Q8—532no—
97C22⋊Q8—1062no—
98C22⋊Q8—1062no—
99C22⋊Q8—1061no—
100C2×C22⋊C4—1061no—
101C2×C22⋊C4—1061no—
102C2×C22⋊C4C2×D5311yescharacteristic
103#103—1062no—
104C22⋊C8C2×D5312yes—
105#95—531no—
106#95—1061no—
107C24D21211yescharacteristic
108C2×Q8—532no—
109C2×Q8—1061no—
110C2×Q8—1062no—
111C2×Q8—2122no—
112C22×C4—1061no—
113C22×C4—1061no—
114C22×C4—1061no—
115C22×C4—1061no—
116C22×C4D21211yescharacteristic
117C22×C4C22×D5311yescharacteristic
118Q16—2124no—
119C2×C8—22no—
120C4⋊C4—532no—
121C4⋊C4—1062no—
122C4⋊C4—1061no—
123C4⋊C4—1062no—
124C22⋊C4—22no—
125C22⋊C4—1062no—
126C22⋊C4—1062no—
127C22⋊C4—2122no—
128C23—21no—
129C23—21no—
130C23—41no—
131C23—41no—
132C23Dic21212yescomplemented
133C23C2×D21211yescharacteristic
134Q8—1064no—
135Q8—4241no—
136Q8—2122no—
137Q8—2122no—
138C2×C4—21no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—1062no—
142C2×C4—1062no—
143C2×C4—1062no—
144C2×C4—4241no—
145C2×C4—4241no—
146C2×C4—2121no—
147C2×C4—2121no—
148C2×C4—2121no—
149C2×C4—2121no—
150C2×C4—2121no—
151C2×C4D4×D5312yes—
152C8—42no—
153C22—21no—
154C22—21no—
155C22—22no—
156C22—42no—
157C22—41no—
158C22—41no—
159C22—81no—
160C22#1313318711yescharacteristic
161C22C2×Dic21211yescharacteristic
162C22#1313325011yescharacteristic
163C4—22no—
164C4—41no—
165C4—4241no—
166C4—2122no—
167C4—2122no—
168C4—2122no—
169C2—21no—
170C2—21no—
171C2—42no—
172C2#1681078111yescharacteristic
173C2#1681080311yescharacteristic
174C2#1681081111yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#323C2
socleC22×C53#120
hypercenter#323D53
hypocenterC53#1602
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #16810781C2semidirect
4. #16810361C2semidirect
44. C23×C53Q16semidirect
93. C53#1602semidirect
132. C23Dic212semidirect