Finite Group Databasebeta

← #20276716

Subgroups — #20276716

Order 6784.

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

normal subgroups only

This group has 270 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#16810809C212yesmaximal, complemented
2#16810800C211yesmaximal, characteristic, complemented
3#16810418C211yesmaximal, characteristic, complemented
4#16810410C212yesmaximal, complemented
5C53×H(3,𝔽4)C211yesmaximal, characteristic, complemented
6#13133262C2211yescharacteristic
7#13133248—21no—
8#13133186—22no—
9#13133145—21no—
10#13133134C2211yescharacteristic
11#13133132C2212yes—
12#13133131—22no—
13#124×C53—22no—
14#124×C53—21no—
15#124×C53—21no—
16#124×C53C2211yescharacteristic
17#120×C53—22no—
18#120×C53C2212yes—
19C24×C53D412yescomplemented
20C2×Q8×C53—21no—
21C2×Q8×C53D411yescharacteristic, complemented
22C2×D4×C53—21no—
23C2×D4×C53—21no—
24C2×D4×C53—22no—
25C2×D4×C53C2311yescharacteristic
26C2×C22⋊D53—22no—
27C2×C22⋊D53—42no—
28#11719947—41no—
29D4×D53—41no—
30#11719945—21no—
31C22⋊C4×C53—22no—
32C22⋊C4×C53—22no—
33C22⋊C4×C53—22no—
34C22⋊C4×C53—22no—
35C22⋊C4×C53D412yescomplemented
36C42×C53—21no—
37C42×C53D411yescharacteristic, complemented
38C22⋊Dic53—22no—
39C22⋊Dic53—42no—
40#11719925—41no—
41#11719924—41no—
42#11719917—21no—
43C23×C53—22no—
44C23×C53—42no—
45C23×C53—82no—
46C23×C53C2×D412yes—
47C22×D53—22no—
48Q8×C53—41no—
49Q8×C53—81no—
50D4×C53—41no—
51D4×C53—82no—
52D4×C53—81no—
53D4×C53—81no—
54C2×C4×C53—22no—
55C2×C4×C53—21no—
56C2×C4×C53—21no—
57C2×C4×C53C2×D411yescharacteristic
58C22⋊D53—42no—
59C22⋊D53—41no—
60C22⋊D53—84no—
61C2×Dic53—22no—
62C2×Dic53—42no—
63D212—21no—
64C4×D53—41no—
65Dic106—21no—
66#88322—41no—
67C22×C53—22no—
68C22×C53—42no—
69C22×C53—42no—
70C22×C53—42no—
71C22×C53—44no—
72C22×C53—42no—
73C22×C53—82no—
74C22×C53#12011yescharacteristic
75C2×D53—41no—
76C2×D53—82no—
77C4×C53—21no—
78C4×C53—42no—
79C4×C53—41no—
80C4×C53—41no—
81Dic53—41no—
82Dic53—82no—
83#2199—531nomaximal, Sylow
84C2×C53—21no—
85C2×C53—42no—
86C2×C53—82no—
87C2×C53B(4,𝔽2)11yescharacteristic
88D53—81no—
89H(3,𝔽4)D5311yescharacteristic, complemented
90B(4,𝔽2)—532no—
91#455—531no—
92#354—531no—
93AΣL(1,𝔽16)—532no—
94C53#219911yescharacteristic, complemented, Sylow
95H(5,𝔽2)—531no—
96#137—1061no—
97#124—21no—
98#124—22no—
99#124—21no—
100#124C2×D5311yescharacteristic
101#120—22no—
102#120—1062no—
103#120C2×D5312yes—
104#104—1061no—
105#100—531no—
106#99—532no—
107#99—1062no—
108C24C22⋊D5312yescomplemented
109#41—1061no—
110#41—2121no—
111C2×Q8—21no—
112C2×Q8C22⋊D5311yescharacteristic, complemented
113C2×D4—21no—
114C2×D4—21no—
115C2×D4—22no—
116C2×D4—1062no—
117C2×D4—2121no—
118C2×D4—2122no—
119C2×D4C22×D5311yescharacteristic
120Q16—2121no—
121SD16—2121no—
122M16—1061no—
123C22⋊C4—22no—
124C22⋊C4—22no—
125C22⋊C4—22no—
126C22⋊C4—22no—
127C22⋊C4—1062no—
128C22⋊C4—2122no—
129C22⋊C4C22⋊D5312yescomplemented
130C42—21no—
131C42C22⋊D5311yescharacteristic, complemented
132C23—22no—
133C23—42no—
134C23—82no—
135C23—1062no—
136C23C2×C22⋊D5312yes—
137Q8—41no—
138Q8—81no—
139Q8—1061no—
140D4—41no—
141D4—82no—
142D4—81no—
143D4—81no—
144D4—1061no—
145D4—4244no—
146D4—2122no—
147D4—2121no—
148C2×C4—21no—
149C2×C4—22no—
150C2×C4—21no—
151C2×C4—1062no—
152C2×C4—2121no—
153C2×C4—2122no—
154C2×C4C2×C22⋊D5311yescharacteristic
155C8—2121no—
156C22—22no—
157C22—42no—
158C22—42no—
159C22—42no—
160C22—44no—
161C22—42no—
162C22—82no—
163C22—4242no—
164C22—2121no—
165C22#1313318611yescharacteristic
166C4—21no—
167C4—42no—
168C4—41no—
169C4—41no—
170C4—4242no—
171C4—2121no—
172C2—21no—
173C2—42no—
174C2—82no—
175C2—4241no—
176C2#1681081011yescharacteristic
subgroupG/H
centerC2#16810810
commutatorC2×D4×C53C23
FrattiniC2×D4C22×D53
FittingC53×H(3,𝔽4)C2
socleC2×C53B(4,𝔽2)
hypercenterH(3,𝔽4)D53
hypocenterC53#2199
LCSG ▷ C2×D4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. #16810809C2semidirect
2. #16810800C2semidirect
3. #16810418C2semidirect
4. #16810410C2semidirect
5. C53×H(3,𝔽4)C2semidirect
19. C24×C53D4semidirect
21. C2×Q8×C53D4semidirect
35. C22⋊C4×C53D4semidirect
37. C42×C53D4semidirect
89. H(3,𝔽4)D53semidirect
94. C53#2199semidirect
108. C24C22⋊D53semidirect
112. C2×Q8C22⋊D53semidirect
129. C22⋊C4C22⋊D53semidirect
131. C42C22⋊D53semidirect