Finite Group Databasebeta

← #20273754

Subgroups — #20273754

Order 6784.

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

normal subgroups only

This group has 246 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#16811204C211yesmaximal, characteristic
2#16810942C212yesmaximal, complemented
3#16810562C211yesmaximal, characteristic, complemented
4#16810358C212yesmaximal, complemented
5C53×#324C211yesmaximal, characteristic, complemented
6C2×#11719945—21no—
7#13133197—22no—
8#13133192—22no—
9#13133191—21no—
10#13133191C2212yes—
11C2×C4⋊Dic53C2211yescharacteristic
12#13133149—22no—
13#13133139—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
22C2×C22⋊D53—22no—
23C22×Dic53—21no—
24C22×Dic53—21no—
25#11719945—42no—
26#11719945—42no—
27C2×D212—21no—
28C2×D212—21no—
29C2×D212D412yes—
30C2×C4×D53—22no—
31C2×Dic106—21no—
32C2×C8×C53—22no—
33C4⋊C4×C53—22no—
34#11719921—22no—
35#11719921—22no—
36C4⋊Dic53—21no—
37C4⋊Dic53—22no—
38C4⋊Dic53D412yes—
39C23×C53C2×D411yescharacteristic
40C22×D53—22no—
41C2×C4×C53—21no—
42C2×C4×C53—21no—
43C2×C4×C53—21no—
44C2×C4×C53—41no—
45C2×C4×C53—41no—
46C2×C4×C53SD1612yes—
47C2×C4×C53C2×D412yes—
48C22⋊D53—42no—
49C22⋊D53—42no—
50C2×Dic53—22no—
51C2×Dic53—41no—
52C2×Dic53—41no—
53C2×Dic53—42no—
54C2×Dic53—41no—
55C2×Dic53—41no—
56D212—42no—
57D212—42no—
58D212—81no—
59C4×D53—44no—
60Dic106—42no—
61C8×C53—42no—
62C22×C53—21no—
63C22×C53—21no—
64C22×C53#12011yescharacteristic
65C22×C53C2×SD1611yescharacteristic
66C22×C53AGL(1,ℤ/8)11yescharacteristic
67C2×D53—42no—
68C2×D53—42no—
69C2×D53—42no—
70C4×C53—22no—
71C4×C53—22no—
72C4×C53—41no—
73Dic53—42no—
74Dic53—81no—
75Dic53—81no—
76#1616—531nomaximal, Sylow
77C2×C53—21no—
78C2×C53—21no—
79C2×C53#45011yescharacteristic
80C2×C53#45411yescharacteristic
81C2×C53#45811yescharacteristic
82D53—82no—
83#537—531no—
84#481—532no—
85#400—531no—
86#327—532no—
87#324D5311yescharacteristic, complemented
88C53#161611yescharacteristic, complemented, Sylow
89C2×#41—1061no—
90C22⋊Q8—1062no—
91#121—532no—
92#121—1062no—
93#121—1061no—
94C2×C4⋊C4—531no—
95C2×C4⋊C4C2×D5311yescharacteristic
96#106—1062no—
97#102—1062no—
98C22⋊C8C2×D5312yes—
99#95—531no—
100#95—1061no—
101#95—1061no—
102#41—2122no—
103#41—2122no—
104C2×Q8—1061no—
105C2×D4—532no—
106C2×D4—1061no—
107C2×D4—1061no—
108C2×D4—1062no—
109C22×C4—1062no—
110C22×C4—1061no—
111C22×C4—1061no—
112C22×C4D21211yescharacteristic
113C22×C4D21211yescharacteristic
114C22×C4C22×D5311yescharacteristic
115C2×C8—22no—
116C4⋊C4—22no—
117C4⋊C4—532no—
118C4⋊C4—1061no—
119C4⋊C4—1062no—
120C22⋊C4—1062no—
121C22⋊C4—1062no—
122C23—1062no—
123C23C2×D21211yescharacteristic
124Q8—2122no—
125D4—4241no—
126D4—2122no—
127D4—2122no—
128D4—2122no—
129D4—2122no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—1062no—
136C2×C4—2124no—
137C2×C4—2121no—
138C2×C4—2121no—
139C2×C4—2122no—
140C2×C4—2121no—
141C2×C4—2121no—
142C2×C4#1171991312yes—
143C2×C4D4×D5312yes—
144C8—42no—
145C22—21no—
146C22—21no—
147C22—2122no—
148C22—2122no—
149C22—2122no—
150C22#1313318711yescharacteristic
151C22C2×#1171991311yescharacteristic
152C22#1313324511yescharacteristic
153C4—22no—
154C4—22no—
155C4—41no—
156C4—4241no—
157C4—4241no—
158C4—2122no—
159C2—21no—
160C2—21no—
161C2—4242no—
162C2#1681077611yescharacteristic
163C2#1681079811yescharacteristic
164C2#1681081411yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#324C2
socleC22×C53#120
hypercenter#324D53
hypocenterC53#1616
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
2. #16810942C2semidirect
3. #16810562C2semidirect
4. #16810358C2semidirect
5. C53×#324C2semidirect
87. #324D53semidirect
88. C53#1616semidirect