Finite Group Databasebeta

← #20276741

Subgroups — #20276741

Order 6784.

236 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 45 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#16810805C211yesmaximal, characteristic, complemented
2#16810795C212yesmaximal, complemented
3#16810422C211yesmaximal, characteristic, complemented
4#16810420C212yesmaximal, complemented
5C53×#555C211yesmaximal, characteristic, complemented
6#13133265C2211yescharacteristic
7#13133248—21no—
8#13133243—22no—
9#13133145—22no—
10#13133145—21no—
11#13133136C2212yes—
12#13133136C2211yescharacteristic
13C4⋊Q8×C53C2211yescharacteristic
14#124×C53—21no—
15#124×C53C2212yes—
16#121×C53—22no—
17C4×D4×C53—21no—
18#117×C53—22no—
19C2×Q8×C53D411yescharacteristic
20C2×Q8×C53C2311yescharacteristic
21C2×D4×C53—21no—
22C2×D4×C53—21no—
23C2×D4×C53D412yes—
24C22×C4×C53—22no—
25#11719949—41no—
26Q8×D53—41no—
27#11719945—22no—
28#11719945—21no—
29C4⋊C4×C53—21no—
30C4⋊C4×C53—22no—
31C22⋊C4×C53—22no—
32C22⋊C4×C53—22no—
33C22⋊C4×C53—22no—
34C42×C53—21no—
35C42×C53D412yes—
36C42×C53D411yescharacteristic
37#11719925—41no—
38#11719924—41no—
39#11719923—42no—
40#11719922—42no—
41#11719917—22no—
42#11719917—21no—
43C23×C53—22no—
44Q8×C53—41no—
45Q8×C53—41no—
46D4×C53—42no—
47D4×C53—42no—
48D4×C53—81no—
49C2×C4×C53—21no—
50C2×C4×C53—22no—
51C2×C4×C53—21no—
52C2×C4×C53—44no—
53C2×C4×C53C2×D412yes—
54C2×C4×C53C2×D411yescharacteristic
55C22⋊D53—41no—
56D212—22no—
57D212—21no—
58C4×D53—42no—
59C4×D53—41no—
60Dic106—22no—
61Dic106—21no—
62#88322—42no—
63#88322—41no—
64C22×C53—42no—
65C22×C53—42no—
66C22×C53—42no—
67C22×C53#12011yescharacteristic
68C2×D53—41no—
69C4×C53—22no—
70C4×C53—22no—
71C4×C53—21no—
72C4×C53—42no—
73C4×C53—41no—
74Dic53—41no—
75#2204—531nomaximal, Sylow
76C2×C53—21no—
77C2×C53—82no—
78C2×C53B(4,𝔽2)11yescharacteristic
79D53—81no—
80#555D5311yescharacteristic, complemented
81#456—531no—
82#454—532no—
83#356—531no—
84#355—532no—
85C53#220411yescharacteristic, complemented, Sylow
862−1+4—531no—
87#137—1061no—
88AGL(1,ℤ/8)—1062no—
89C4⋊Q8C2×D5311yescharacteristic
90#124—21no—
91#124C2×D5312yes—
92#121—22no—
93C4×D4—21no—
94#117—22no—
95#104—1062no—
96#104—1061no—
97#101—531no—
98#101—532no—
99#41—1062no—
100#41—1061no—
101#41—2121no—
102C2×Q8—2121no—
103C2×Q8C22⋊D5311yescharacteristic
104C2×Q8C22×D5311yescharacteristic
105C2×D4—21no—
106C2×D4—21no—
107C2×D4C22⋊D5312yes—
108C22×C4—22no—
109Q16—2121no—
110SD16—2122no—
111SD16—2121no—
112D8—2122no—
113M16—1062no—
114M16—1061no—
115C4⋊C4—21no—
116C4⋊C4—22no—
117C22⋊C4—22no—
118C22⋊C4—22no—
119C22⋊C4—22no—
120C42—21no—
121C42C22⋊D5312yes—
122C42C22⋊D5311yescharacteristic
123C23—22no—
124Q8—41no—
125Q8—41no—
126Q8—1062no—
127Q8—1061no—
128D4—42no—
129D4—42no—
130D4—81no—
131D4—1062no—
132D4—1061no—
133D4—2121no—
134C2×C4—21no—
135C2×C4—22no—
136C2×C4—21no—
137C2×C4—44no—
138C2×C4—2122no—
139C2×C4—2121no—
140C2×C4C2×C22⋊D5311yescharacteristic
141C2×C4C2×C22⋊D5312yes—
142C8—2122no—
143C8—2121no—
144C22—42no—
145C22—42no—
146C22—42no—
147C22—2121no—
148C22#1313318611yescharacteristic
149C4—22no—
150C4—22no—
151C4—21no—
152C4—42no—
153C4—41no—
154C4—2121no—
155C2—21no—
156C2—82no—
157C2—4241no—
158C2#1681081011yescharacteristic
subgroupG/H
centerC2#16810810
commutatorC2×Q8×C53C23
FrattiniC2×Q8C22×D53
FittingC53×#555C2
socleC2×C53B(4,𝔽2)
hypercenter#555D53
hypocenterC53#2204
LCSG ▷ C2×Q8×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #555

Complements

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

NH ≅ G/Ntype
1. #16810805C2semidirect
2. #16810795C2semidirect
3. #16810422C2semidirect
4. #16810420C2semidirect
5. C53×#555C2semidirect
80. #555D53semidirect
85. C53#2204semidirect