Finite Group Databasebeta

← #20276740

Subgroups — #20276740

Order 6784.

236 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 41 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#16811339C211yesmaximal, characteristic, complemented
2#16810796C212yesmaximal, complemented
3#16810422C211yesmaximal, characteristic, complemented
4#16810419C212yesmaximal, complemented
5C53×#456C211yesmaximal, characteristic, complemented
6#13133242—22no—
7#13133206—21no—
8#13133206C2212yes—
9#13133192—22no—
10C4×D212—21no—
11#13133175—22no—
12#13133144—22no—
13#13133136C2212yes—
142−1+4×C53C2211yescharacteristic
15#137×C53—21no—
16C4⋊Q8×C53C2211yescharacteristic
17#104×C53—21no—
18#101×C53C2211yescharacteristic
19#41×C53—21no—
20#41×C53—22no—
21#41×C53—41no—
22C2×Q8×C53—41no—
23C2×Q8×C53D411yescharacteristic
24C2×Q8×C53C2311yescharacteristic
25C2×D212—21no—
26C2×D212—21no—
27C2×D212D412yes—
28C2×C4×D53—22no—
29Q16×C53—41no—
30SD16×C53—41no—
31M16×C53—21no—
32C4⋊C4×C53—22no—
33C42×C53D411yescharacteristic
34#11719924—42no—
35#11719922—42no—
36#11719921—22no—
37#11719921—22no—
38#11719921—22no—
39C4⋊Dic53—21no—
40C4×Dic53—21no—
41C4×Dic53D412yes—
42#11719917—22no—
43C22×D53—22no—
44Q8×C53—22no—
45Q8×C53—21no—
46Q8×C53—41no—
47Q8×C53—41no—
48D4×C53—22no—
49D4×C53—21no—
50D4×C53—41no—
51C2×C4×C53—21no—
52C2×C4×C53—21no—
53C2×C4×C53—42no—
54C2×C4×C53—41no—
55C2×C4×C53C2×D412yes—
56C2×C4×C53C2×D411yescharacteristic
57C2×Dic53—22no—
58D212—42no—
59D212—42no—
60D212—81no—
61C4×D53—44no—
62C8×C53—41no—
63#88322—42no—
64C22×C53—41no—
65C22×C53#12011yescharacteristic
66C2×D53—42no—
67C2×D53—42no—
68C2×D53—42no—
69C4×C53—22no—
70C4×C53—22no—
71C4×C53—21no—
72C4×C53—41no—
73C4×C53—41no—
74Dic53—42no—
75#2204—531nomaximal, Sylow
76C2×C53—21no—
77C2×C53—81no—
78C2×C53B(4,𝔽2)11yescharacteristic
79D53—82no—
80#555—531no—
81#456D5311yescharacteristic, complemented
82#454—532no—
83#356—531no—
84#355—532no—
85C53#220411yescharacteristic, complemented, Sylow
862−1+4C2×D5311yescharacteristic
87#137—21no—
88AGL(1,ℤ/8)—1062no—
89C4⋊Q8C2×D5311yescharacteristic
90#124—532no—
91#124—1061no—
92#121—1062no—
93C4×D4—1061no—
94#117—1062no—
95#104—21no—
96#104—1062no—
97#101—532no—
98#101C2×D5311yescharacteristic
99#41—22no—
100#41—21no—
101#41—41no—
102C2×Q8—41no—
103C2×Q8C22⋊D5311yescharacteristic
104C2×Q8C22×D5311yescharacteristic
105C2×D4—532no—
106C2×D4—1061no—
107C2×D4—1061no—
108C22×C4—1062no—
109Q16—41no—
110SD16—41no—
111SD16—2122no—
112D8—2122no—
113M16—21no—
114M16—1062no—
115C4⋊C4—22no—
116C4⋊C4—1061no—
117C22⋊C4—1062no—
118C22⋊C4—1062no—
119C22⋊C4—1062no—
120C42—532no—
121C42—1061no—
122C42C22⋊D5311yescharacteristic
123C23—1062no—
124Q8—22no—
125Q8—21no—
126Q8—41no—
127Q8—41no—
128D4—22no—
129D4—21no—
130D4—41no—
131D4—4241no—
132D4—2122no—
133D4—2122no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—41no—
137C2×C4—42no—
138C2×C4—1062no—
139C2×C4—2124no—
140C2×C4D4×D5312yes—
141C2×C4C2×C22⋊D5311yescharacteristic
142C8—41no—
143C8—2122no—
144C22—41no—
145C22—2122no—
146C22—2122no—
147C22—2122no—
148C22#1313318511yescharacteristic
149C4—22no—
150C4—21no—
151C4—22no—
152C4—41no—
153C4—41no—
154C4—2122no—
155C2—21no—
156C2—81no—
157C2—4242no—
158C2#1681080911yescharacteristic
subgroupG/H
centerC2#16810809
commutatorC2×Q8×C53C23
FrattiniC2×Q8C22×D53
FittingC53×#456C2
socleC2×C53B(4,𝔽2)
hypercenter#456D53
hypocenterC53#2204
LCSG ▷ C2×Q8×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #16811339C2semidirect
2. #16810796C2semidirect
3. #16810422C2semidirect
4. #16810419C2semidirect
5. C53×#456C2semidirect
81. #456D53semidirect
85. C53#2204semidirect