Finite Group Databasebeta

← #20273655

Subgroups — #20273655

Order 6784.

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

normal subgroups only

This group has 266 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#16810773C212yesmaximal
2#16810540C211yesmaximal, characteristic
3#16810360C212yesmaximal, complemented
4#16810349C211yesmaximal, characteristic, complemented
5C53×#544C211yesmaximal, characteristic, complemented
6C22×D212C2211yescharacteristic
7C2×#11719924—22no—
8C2×#11719921—21no—
9C2×#11719921—21no—
10#13133137—22no—
11C22⋊#88322C2212yes—
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
19C23×D53—21no—
20C2×Q8×C53—21no—
21C2×Q8×C53D412yes—
22C22×C4×C53D411yescharacteristic
23C22×C4×C53D411yescharacteristic
24C22×C4×C53C2311yescharacteristic
25C22×Dic53—21no—
26C2×D212—22no—
27C2×D212—42no—
28C4⋊C4×C53—21no—
29C4⋊C4×C53—21no—
30C4⋊C4×C53—21no—
31C4⋊C4×C53—22no—
32C4⋊C4×C53—22no—
33C4⋊C4×C53—22no—
34C4⋊C4×C53D412yes—
35C22⋊C4×C53—22no—
36C42×C53—22no—
37#11719924—44no—
38#11719921—42no—
39#11719921—42no—
40C2×#88322—22no—
41C23×C53C2×D411yescharacteristic
42C22×D53—22no—
43C22×D53—41no—
44C22×D53—41no—
45C22×D53—41no—
46C22×D53—41no—
47C22×D53—41no—
48C22×D53—41no—
49Q8×C53—42no—
50Q8×C53—81no—
51C2×C4×C53—21no—
52C2×C4×C53—21no—
53C2×C4×C53—21no—
54C2×C4×C53—22no—
55C2×C4×C53—22no—
56C2×C4×C53—41no—
57C2×C4×C53—41no—
58C2×C4×C53SD1612yescomplemented
59C2×C4×C53C2×D412yes—
60C2×Dic53—41no—
61C2×Dic53—41no—
62C2×Dic53—41no—
63D212—24no—
64D212—42no—
65#88322—42no—
66C22×C53—21no—
67C22×C53—21no—
68C22×C53#12011yescharacteristic
69C22×C53C2×SD1611yescharacteristic
70C22×C53#13711yescharacteristic
71C2×D53—42no—
72C2×D53—42no—
73C2×D53—42no—
74C2×D53—81no—
75C2×D53—81no—
76C2×D53—81no—
77C2×D53—81no—
78C4×C53—22no—
79C4×C53—22no—
80C4×C53—42no—
81C4×C53—41no—
82C4×C53—42no—
83Dic53—81no—
84#1599—531nomaximal, Sylow
85C2×C53—21no—
86C2×C53—21no—
87C2×C53#44811yescharacteristic
88C2×C53#45411yescharacteristic
89C2×C53B(4,𝔽2)11yescharacteristic
90D53—82no—
91#544D5311yescharacteristic, complemented
92#450—532no—
93#394—531no—
94#328—532no—
95#324—531no—
96C53#159911yescharacteristic, complemented, Sylow
97C22×D4—531no—
98C2×SD16—1062no—
99C4⋊Q8—22no—
100#125—22no—
101C22⋊Q8—21no—
102C22⋊Q8C2×D5312yes—
103#117—21no—
104C2×C4⋊C4C2×D5311yescharacteristic
105C2×C22⋊C4—1061no—
106C2×C22⋊C4—1061no—
107#102—1062no—
108C22⋊C8—532no—
109#95—531no—
110C24—1061no—
111C2×Q8—21no—
112C2×Q8C22⋊D5312yes—
113C2×D4—1062no—
114C2×D4—2122no—
115C22×C4—1061no—
116C22×C4C22⋊D5311yescharacteristic
117C22×C4C22⋊D5311yescharacteristic
118C22×C4C22×D5311yescharacteristic
119SD16—2124no—
120C2×C8—1062no—
121C4⋊C4—21no—
122C4⋊C4—21no—
123C4⋊C4—21no—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4C22⋊D5312yes—
128C22⋊C4—22no—
129C22⋊C4—2122no—
130C22⋊C4—2122no—
131C42—22no—
132C23—1062no—
133C23—2121no—
134C23—2121no—
135C23—2121no—
136C23—2121no—
137C23—2121no—
138C23—2121no—
139C23C2×C22⋊D5311yescharacteristic
140Q8—42no—
141Q8—81no—
142D4—1064no—
143D4—2122no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—21no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—2121no—
152C2×C4—2121no—
153C2×C4—2121no—
154C2×C4#1171992412yescomplemented
155C2×C4C2×C22⋊D5312yes—
156C8—2122no—
157C22—21no—
158C22—21no—
159C22—4241no—
160C22—4241no—
161C22—4241no—
162C22—4241no—
163C22—2122no—
164C22—2122no—
165C22—2122no—
166C22#1313318611yescharacteristic
167C22C2×#1171992411yescharacteristic
168C22#1313324811yescharacteristic
169C4—22no—
170C4—22no—
171C4—42no—
172C4—41no—
173C4—42no—
174C4—4241no—
175C2—21no—
176C2—21no—
177C2—4242no—
178C2#1681076211yescharacteristic
179C2#1681079611yescharacteristic
180C2#1681081011yescharacteristic
subgroupG/H
centerC22#13133186
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#544C2
socleC22×C53#120
hypercenter#544D53
hypocenterC53#1599
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
3. #16810360C2semidirect
4. #16810349C2semidirect
5. C53×#544C2semidirect
58. C2×C4×C53SD16semidirect
91. #544D53semidirect
96. C53#1599semidirect
154. C2×C4#11719924semidirect