Finite Group Databasebeta

← #20273752

Subgroups — #20273752

Order 6784.

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

normal subgroups only

This group has 246 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#16810939C212yesmaximal, complemented
2#16810562C211yesmaximal, characteristic, complemented
3#16810357C212yesmaximal, complemented
4#16810349C211yesmaximal, characteristic, complemented
5C53×#537C211yesmaximal, characteristic
6C2×C4⋊Dic53C2211yescharacteristic
7#13133148—22no—
8#13133138—22no—
9C22⋊#88322C2212yes—
10#13133124—21no—
11#13133124—21no—
12#13133124C2211yescharacteristic
13C2×#41×C53—21no—
14C22⋊Q8×C53—22no—
15#121×C53—22no—
16#121×C53—21no—
17#121×C53C2212yes—
18C2×C4⋊C4×C53C2211yescharacteristic
19#41×C53—42no—
20#41×C53—42no—
21C2×Q8×C53—21no—
22C2×D4×C53—21no—
23C2×D4×C53—22no—
24C2×D4×C53—21no—
25C2×D4×C53D412yes—
26C22×C4×C53—22no—
27C22×C4×C53D411yescharacteristic
28C22×C4×C53D411yescharacteristic
29C22×C4×C53C2311yescharacteristic
30C22×Dic53—21no—
31C22×Dic53—21no—
32C4⋊C4×C53—21no—
33C4⋊C4×C53—22no—
34C4⋊C4×C53D412yes—
35C22⋊C4×C53—22no—
36C22⋊C4×C53—22no—
37C4⋊Dic53—22no—
38C2×#88322—22no—
39C23×C53—22no—
40C23×C53C2×D411yescharacteristic
41Q8×C53—42no—
42D4×C53—42no—
43D4×C53—42no—
44D4×C53—42no—
45D4×C53—42no—
46D4×C53—81no—
47C2×C4×C53—21no—
48C2×C4×C53—21no—
49C2×C4×C53—22no—
50C2×C4×C53—21no—
51C2×C4×C53—41no—
52C2×C4×C53—44no—
53C2×C4×C53—41no—
54C2×C4×C53SD1612yes—
55C2×C4×C53C2×D412yes—
56C2×Dic53—41no—
57C2×Dic53—41no—
58C2×Dic53—42no—
59C2×Dic53—41no—
60C2×Dic53—41no—
61#88322—42no—
62C22×C53—21no—
63C22×C53—21no—
64C22×C53—42no—
65C22×C53—42no—
66C22×C53—42no—
67C22×C53#12011yescharacteristic
68C22×C53C2×SD1611yescharacteristic
69C22×C53AGL(1,ℤ/8)11yescharacteristic
70C4×C53—22no—
71C4×C53—22no—
72C4×C53—42no—
73C4×C53—41no—
74Dic53—81no—
75Dic53—81no—
76#1616—531nomaximal, Sylow
77C2×C53—21no—
78C2×C53—21no—
79C2×C53—82no—
80C2×C53#45011yescharacteristic
81C2×C53#45411yescharacteristic
82C2×C53#45811yescharacteristic
83#537D5311yescharacteristic
84#481—532no—
85#400—531no—
86#327—532no—
87#324—531no—
88C53#161611yescharacteristic, complemented, Sylow
89C2×#41—21no—
90C22⋊Q8—22no—
91#121—22no—
92#121—21no—
93#121C2×D5312yes—
94C2×C4⋊C4—531no—
95C2×C4⋊C4C2×D5311yescharacteristic
96#106—1062no—
97#102—1062no—
98C22⋊C8—532no—
99#95—531no—
100#95—1061no—
101#95—1061no—
102#41—42no—
103#41—42no—
104C2×Q8—21no—
105C2×D4—21no—
106C2×D4—22no—
107C2×D4—21no—
108C2×D4C22⋊D5312yes—
109C22×C4—22no—
110C22×C4—1061no—
111C22×C4—1061no—
112C22×C4C22⋊D5311yescharacteristic
113C22×C4C22⋊D5311yescharacteristic
114C22×C4C22×D5311yescharacteristic
115C2×C8—1062no—
116C4⋊C4—21no—
117C4⋊C4—22no—
118C4⋊C4—1062no—
119C4⋊C4C22⋊D5312yes—
120C22⋊C4—22no—
121C22⋊C4—22no—
122C23—22no—
123C23C2×C22⋊D5311yescharacteristic
124Q8—42no—
125D4—42no—
126D4—42no—
127D4—42no—
128D4—42no—
129D4—81no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—22no—
134C2×C4—41no—
135C2×C4—44no—
136C2×C4—41no—
137C2×C4—2121no—
138C2×C4—2121no—
139C2×C4—2122no—
140C2×C4—2121no—
141C2×C4—2121no—
142C2×C4#1171992312yes—
143C2×C4C2×C22⋊D5312yes—
144C8—2122no—
145C22—21no—
146C22—21no—
147C22—42no—
148C22—42no—
149C22—42no—
150C22#1313318611yescharacteristic
151C22C2×#1171992311yescharacteristic
152C22#1313324311yescharacteristic
153C4—22no—
154C4—22no—
155C4—42no—
156C4—41no—
157C4—4241no—
158C4—4241no—
159C2—21no—
160C2—21no—
161C2—82no—
162C2#1681077411yescharacteristic
163C2#1681079611yescharacteristic
164C2#1681081311yescharacteristic
subgroupG/H
centerC22#13133186
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#537C2
socleC22×C53#120
hypercenter#537D53
hypocenterC53#1616
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. #16810939C2semidirect
2. #16810562C2semidirect
3. #16810357C2semidirect
4. #16810349C2semidirect
88. C53#1616semidirect