Finite Group Databasebeta

← #20273801

Subgroups — #20273801

Order 6784.

256 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 55 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#16811276C211yesmaximal, characteristic, complemented
2#16810815C211yesmaximal, characteristic, complemented
3#16810815C211yesmaximal, characteristic, complemented
4#16810369C211yesmaximal, characteristic, complemented
5#16810352C211yesmaximal, characteristic, complemented
6#16810352C211yesmaximal, characteristic, complemented
7C53×#486C211yesmaximal, characteristic, complemented
8C2×#11719949—21no—
9C2×#11719922—21no—
10C2×#11719922—21no—
11#13133202—22no—
12#13133193—21no—
13#13133192—21no—
14C4×D212C2211yescharacteristic
15C4×D212C2211yescharacteristic
16C4⋊C4×D53—21no—
17C4⋊#88322C2211yescharacteristic
18C4⋊#88322C2211yescharacteristic
19#13133138—21no—
20#13133138—21no—
21C4⋊Q8×C53C2211yescharacteristic
22#127×C53C2211yescharacteristic
23#102×C53—22no—
24C4×C8×C53C2211yescharacteristic
25C2×Q8×C53—21no—
26C2×D4×C53—21no—
27C2×D4×C53—21no—
28C2×D4×C53—21no—
29#11719949—42no—
30#11719949—42no—
31C2×D212—22no—
32C2×D212D411yescharacteristic
33C2×D212D411yescharacteristic
34C2×C4×D53—22no—
35C2×C4×D53—21no—
36C2×C4×D53—21no—
37C2×C8×C53—21no—
38C4⋊C4×C53—21no—
39C4⋊C4×C53D412yes—
40C42×C53C2311yescharacteristic
41#11719922—42no—
42#11719922—42no—
43#11719921—22no—
44#11719921—21no—
45#11719921—21no—
46C4⋊Dic53D411yescharacteristic
47C4⋊Dic53D411yescharacteristic
48#11719919—22no—
49C2×#88322—21no—
50C2×#88322—21no—
51C23×C53—41no—
52C22×D53—21no—
53C22×D53—21no—
54Q8×C53—41no—
55Q8×C53—41no—
56D4×C53—42no—
57D4×C53—41no—
58D4×C53—41no—
59D4×C53—41no—
60D4×C53—41no—
61C2×C4×C53—22no—
62C2×C4×C53C2×D411yescharacteristic
63C2×C4×C53C2×D411yescharacteristic
64C2×C4×C53C2×D411yescharacteristic
65C2×Dic53—21no—
66C2×Dic53—21no—
67D212—21no—
68D212—21no—
69D212—21no—
70D212—44no—
71D212D812yescomplemented
72C4×D53—44no—
73C4×D53—41no—
74C4×D53—41no—
75C4×D53—41no—
76C4×D53—41no—
77C8×C53—41no—
78#88322—41no—
79#88322—41no—
80C22×C53—81no—
81C22×C53—81no—
82C22×C53—81no—
83C22×C53#12011yescharacteristic
84C2×D53—22no—
85C2×D53—41no—
86C2×D53—41no—
87C2×D53—41no—
88C2×D53—41no—
89C2×D53—41no—
90C4×C53—21no—
91C4×C53—42no—
92C4×C53C2×D811yescharacteristic
93C4×C53#13511yescharacteristic
94C4×C53AGL(1,ℤ/8)11yescharacteristic
95C4×C53AGL(1,ℤ/8)11yescharacteristic
96Dic53—41no—
97Dic53—41no—
98#1625—531nomaximal, Sylow
99C2×C53—161no—
100C2×C53#44711yescharacteristic
101C2×C53#44911yescharacteristic
102C2×C53#45411yescharacteristic
103D53—42no—
104D53—81no—
105#547—531no—
106#486D5311yescharacteristic, complemented
107#459—531no—
108#459—531no—
109#332—531no—
110D4⋊C8—531no—
111D4⋊C8—531no—
112C53#162511yescharacteristic, complemented, Sylow
113C2×#41—1061no—
114C2×D8—1061no—
115C2×D8—1061no—
116C4⋊Q8C2×D5311yescharacteristic
117#127C2×D5311yescharacteristic
118#123—1062no—
119C22⋊Q8—1061no—
120#121—1061no—
121C4×D4—531no—
122C4×D4—531no—
123C2×C4⋊C4—1061no—
124C4⋊C8—531no—
125C4⋊C8—531no—
126#102—22no—
127#102—1061no—
128#102—1061no—
129C4×C8C2×D5311yescharacteristic
130#41—2122no—
131#41—2122no—
132C2×Q8—21no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—531no—
137C2×D4—531no—
138C2×D4—1062no—
139C22×C4—1062no—
140C22×C4—1061no—
141C22×C4—1061no—
142D8—2122no—
143D8—2122no—
144C2×C8—21no—
145C2×C8—1061no—
146C2×C8—1061no—
147C4⋊C4—21no—
148C4⋊C4—531no—
149C4⋊C4—531no—
150C4⋊C4—1062no—
151C4⋊C4C22⋊D5312yes—
152C22⋊C4—1062no—
153C22⋊C4—1061no—
154C22⋊C4—1061no—
155C42C22×D5311yescharacteristic
156C23—41no—
157C23—1061no—
158C23—1061no—
159Q8—41no—
160Q8—41no—
161D4—42no—
162D4—41no—
163D4—41no—
164D4—41no—
165D4—41no—
166D4—532no—
167D4—1061no—
168D4—1061no—
169D4—1061no—
170D4—2124no—
171C2×C4—22no—
172C2×C4—1061no—
173C2×C4—1061no—
174C2×C4—2124no—
175C2×C4—2121no—
176C2×C4—2121no—
177C2×C4—2121no—
178C2×C4—2121no—
179C2×C4D4×D5311yescharacteristic
180C2×C4D4×D5311yescharacteristic
181C2×C4C2×C22⋊D5311yescharacteristic
182C8—41no—
183C8—2121no—
184C8—2121no—
185C22—81no—
186C22—81no—
187C22—81no—
188C22—1062no—
189C22—2121no—
190C22—2121no—
191C22—2121no—
192C22—2121no—
193C22—2121no—
194C22#1313318511yescharacteristic
195C4—21no—
196C4—42no—
197C4—2121no—
198C4—2121no—
199C4D8×D5311yescharacteristic
200C4#1313323811yescharacteristic
201C4#1313324111yescharacteristic
202C4#1313324411yescharacteristic
203C2—161no—
204C2—4241no—
205C2—2122no—
206C2#1681075511yescharacteristic
207C2#1681076711yescharacteristic
208C2#1681079511yescharacteristic
subgroupG/H
centerC22#13133185
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#486C2
socleC22×C53#120
hypercenter#486D53
hypocenterC53#1625
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #16811276C2semidirect
2. #16810815C2semidirect
3. #16810815C2semidirect
4. #16810369C2semidirect
5. #16810352C2semidirect
6. #16810352C2semidirect
7. C53×#486C2semidirect
71. D212D8semidirect
106. #486D53semidirect
112. C53#1625semidirect