Finite Group Databasebeta

← #20276728

Subgroups — #20276728

Order 6784.

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

normal subgroups only

This group has 226 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#16810812C212yesmaximal
2#16810800C211yesmaximal, characteristic
3#16810418C211yesmaximal, characteristic, complemented
4#16810413C212yesmaximal, complemented
5C53×#563C211yesmaximal, characteristic, complemented
6#13133262C2211yescharacteristic
7#13133248—21no—
8#13133200—22no—
9#13133145—21no—
10#13133134C2211yescharacteristic
11#13133132C2212yes—
12#13133131—22no—
13C4⋊Q8×C53—21no—
14#126×C53—22no—
15#125×C53—21no—
16#124×C53C2211yescharacteristic
17#123×C53C2212yes—
18C22⋊Q8×C53—22no—
19C2×Q8×C53—21no—
20C2×Q8×C53D411yescharacteristic
21C2×D4×C53C2311yescharacteristic
22C22×C4×C53D412yes—
23C2×C22⋊D53—22no—
24#11719947—41no—
25D4×D53—41no—
26#11719945—21no—
27C4⋊C4×C53—21no—
28C4⋊C4×C53—22no—
29C4⋊C4×C53—22no—
30C4⋊C4×C53—21no—
31C4⋊C4×C53—22no—
32C22⋊C4×C53—22no—
33C22⋊C4×C53—22no—
34C22⋊C4×C53D412yes—
35C42×C53—21no—
36C42×C53D411yescharacteristic
37C22⋊Dic53—22no—
38#11719925—41no—
39#11719924—41no—
40#11719921—42no—
41#11719919—42no—
42#11719917—21no—
43C23×C53C2×D412yes—
44C22×D53—22no—
45Q8×C53—41no—
46Q8×C53—81no—
47D4×C53—41no—
48C2×C4×C53—22no—
49C2×C4×C53—21no—
50C2×C4×C53—22no—
51C2×C4×C53—21no—
52C2×C4×C53—42no—
53C2×C4×C53C2×D411yescharacteristic
54C22⋊D53—42no—
55C22⋊D53—41no—
56C2×Dic53—22no—
57C2×Dic53—42no—
58D212—21no—
59C4×D53—41no—
60Dic106—21no—
61#88322—41no—
62C22×C53—22no—
63C22×C53—42no—
64C22×C53#12011yescharacteristic
65C2×D53—41no—
66C2×D53—82no—
67C4×C53—21no—
68C4×C53—42no—
69C4×C53—41no—
70C4×C53—42no—
71C4×C53—41no—
72Dic53—41no—
73Dic53—82no—
74#2201—531nomaximal, Sylow
75C2×C53—21no—
76C2×C53—42no—
77C2×C53B(4,𝔽2)11yescharacteristic
78D53—81no—
79#563D5311yescharacteristic, complemented
80#458—532no—
81#455—531no—
82#354—531no—
83#352—532no—
84C53#220111yescharacteristic, complemented, Sylow
85H(5,𝔽2)—531no—
86#137—1061no—
87C4⋊Q8—21no—
88#126—22no—
89#125—21no—
90#124C2×D5311yescharacteristic
91#123—1062no—
92#123C2×D5312yes—
93C22⋊Q8—22no—
94#104—1061no—
95#100—531no—
96#99—532no—
97#99—1062no—
98#41—1061no—
99#41—2121no—
100C2×Q8—21no—
101C2×Q8C22⋊D5311yescharacteristic
102C2×D4—1062no—
103C2×D4—2121no—
104C2×D4C22×D5311yescharacteristic
105C22×C4C22⋊D5312yes—
106Q16—2121no—
107SD16—2121no—
108M16—1061no—
109C4⋊C4—21no—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—21no—
113C4⋊C4—22no—
114C4⋊C4—2122no—
115C22⋊C4—22no—
116C22⋊C4—22no—
117C22⋊C4—1062no—
118C22⋊C4—2122no—
119C22⋊C4C22⋊D5312yes—
120C42—21no—
121C42C22⋊D5311yescharacteristic
122C23—1062no—
123C23C2×C22⋊D5312yes—
124Q8—41no—
125Q8—81no—
126Q8—1061no—
127D4—41no—
128D4—1061no—
129D4—2122no—
130D4—2121no—
131C2×C4—21no—
132C2×C4—22no—
133C2×C4—21no—
134C2×C4—22no—
135C2×C4—42no—
136C2×C4—1062no—
137C2×C4—2121no—
138C2×C4—2122no—
139C2×C4C2×C22⋊D5311yescharacteristic
140C8—2121no—
141C22—22no—
142C22—42no—
143C22—4242no—
144C22—2121no—
145C22#1313318611yescharacteristic
146C4—21no—
147C4—42no—
148C4—41no—
149C4—42no—
150C4—41no—
151C4—4242no—
152C4—2121no—
153C2—21no—
154C2—42no—
155C2—4241no—
156C2#1681081011yescharacteristic
subgroupG/H
centerC2#16810810
commutatorC2×D4×C53C23
FrattiniC2×D4C22×D53
FittingC53×#563C2
socleC2×C53B(4,𝔽2)
hypercenter#563D53
hypocenterC53#2201
LCSG ▷ C2×D4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #563

Complements

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

NH ≅ G/Ntype
3. #16810418C2semidirect
4. #16810413C2semidirect
5. C53×#563C2semidirect
79. #563D53semidirect
84. C53#2201semidirect