Finite Group Databasebeta

← #20273807

Subgroups — #20273807

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#16811278C211yesmaximal, characteristic, complemented
2#16810970C211yesmaximal, characteristic, complemented
3#16810819C211yesmaximal, characteristic, complemented
4#16810818C211yesmaximal, characteristic, complemented
5#16810370C211yesmaximal, characteristic, complemented
6#16810352C211yesmaximal, characteristic, complemented
7C53×D4⋊C8C211yesmaximal, characteristic, complemented
8C2×#11719945—21no—
9C2×D424—21no—
10C2×#11719922—21no—
11#13133217C2211yescharacteristic
12#13133214C2211yescharacteristic
13#13133199—22no—
14#13133196—21no—
15#13133188—21no—
16C4×D212C2211yescharacteristic
17C4⋊C4×D53—21no—
18C4⋊#88322C2211yescharacteristic
19#13133139—21no—
20#13133139—22no—
21#13133137—21no—
22C4×D4×C53C2211yescharacteristic
23C4⋊C8×C53C2211yescharacteristic
24C4×C8×C53C2211yescharacteristic
25C2×D4×C53D411yescharacteristic
26C22×C4×C53—21no—
27C2×C22⋊D53—22no—
28#11719945—42no—
29#11719945—42no—
30C2×D212—21no—
31C2×D212—21no—
32C2×D212—21no—
33C2×D212D411yescharacteristic
34C2×C4×D53—22no—
35C2×C4×D53—21no—
36C2×Dic106—21no—
37C2×C8×C53—21no—
38C2×C8×C53—21no—
39C4⋊C4×C53D411yescharacteristic
40C22⋊C4×C53—21no—
41C42×C53C2311yescharacteristic
42C22⋊Dic53—22no—
43#11719922—42no—
44#11719921—21no—
45C4⋊Dic53—21no—
46C4⋊Dic53D411yescharacteristic
47C4⋊Dic53D412yes—
48#11719919—22no—
49C2×#88322—21no—
50D424—42no—
51C23×C53—21no—
52C22×D53—21no—
53C22×D53—41no—
54D4×C53—21no—
55D4×C53—21no—
56C2×C4×C53—21no—
57C2×C4×C53—41no—
58C2×C4×C53—41no—
59C2×C4×C53C2×D411yescharacteristic
60C2×C4×C53C2×D411yescharacteristic
61C2×C4×C53C2×D411yescharacteristic
62C22⋊D53—44no—
63C2×Dic53—22no—
64C2×Dic53—21no—
65D212—21no—
66D212—42no—
67D212—41no—
68D212—41no—
69D212—41no—
70D212—41no—
71D212D812yescomplemented
72C4×D53—44no—
73C4×D53—41no—
74C4×D53—41no—
75Dic106—41no—
76Dic106—41no—
77C8×C53—41no—
78C8×C53—41no—
79#88322—41no—
80C22×C53—41no—
81C22×C53—41no—
82C22×C53—41no—
83C22×C53#12011yescharacteristic
84C2×D53—22no—
85C2×D53—41no—
86C2×D53—41no—
87C2×D53—81no—
88C2×D53—81no—
89C2×D53—81no—
90C4×C53—21no—
91C4×C53—41no—
92C4×C53C2×D811yescharacteristic
93C4×C53#13511yescharacteristic
94C4×C53AGL(1,ℤ/8)11yescharacteristic
95C4×C53AGL(1,ℤ/8)11yescharacteristic
96Dic53—42no—
97Dic53—41no—
98#1625—531nomaximal, Sylow
99C2×C53—81no—
100C2×C53#44711yescharacteristic
101C2×C53#44911yescharacteristic
102C2×C53#45411yescharacteristic
103D53—42no—
104D53—161no—
105#547—531no—
106#486—531no—
107#459—531no—
108#459—531no—
109#332—531no—
110D4⋊C8—531no—
111D4⋊C8D5311yescharacteristic, complemented
112C53#162511yescharacteristic, complemented, Sylow
113C2×#41—1061no—
114C2×D8—1061no—
115C2×D8—1061no—
116C4⋊Q8—531no—
117#127—531no—
118#123—1062no—
119C22⋊Q8—1061no—
120#121—1061no—
121C4×D4—531no—
122C4×D4C2×D5311yescharacteristic
123C2×C4⋊C4—1061no—
124C4⋊C8—531no—
125C4⋊C8C2×D5311yescharacteristic
126#102—1061no—
127#102—1062no—
128#102—1061no—
129C4×C8C2×D5311yescharacteristic
130#41—2122no—
131#41—2122no—
132C2×Q8—1061no—
133C2×D4—531no—
134C2×D4—1062no—
135C2×D4—1061no—
136C2×D4—1061no—
137C2×D4—1061no—
138C2×D4D21211yescharacteristic
139C22×C4—21no—
140C22×C4—1062no—
141C22×C4—1061no—
142D8—2122no—
143D8—2122no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—1061no—
147C4⋊C4—531no—
148C4⋊C4—532no—
149C4⋊C4—1062no—
150C4⋊C4—1061no—
151C4⋊C4D21211yescharacteristic
152C22⋊C4—21no—
153C22⋊C4—1062no—
154C22⋊C4—1061no—
155C42C22×D5311yescharacteristic
156C23—21no—
157C23—1061no—
158C23—2121no—
159Q8—2121no—
160Q8—2121no—
161D4—21no—
162D4—21no—
163D4—532no—
164D4—1061no—
165D4—2122no—
166D4—2124no—
167D4—2121no—
168D4—2121no—
169D4—2121no—
170D4—2121no—
171C2×C4—21no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—1061no—
175C2×C4—1062no—
176C2×C4—2124no—
177C2×C4—2121no—
178C2×C4—2121no—
179C2×C4C2×D21211yescharacteristic
180C2×C4D4×D5311yescharacteristic
181C2×C4D4×D5311yescharacteristic
182C8—41no—
183C8—41no—
184C8—2121no—
185C22—41no—
186C22—41no—
187C22—41no—
188C22—1062no—
189C22—4241no—
190C22—4241no—
191C22—4241no—
192C22—2121no—
193C22—2121no—
194C22#1313318711yescharacteristic
195C4—21no—
196C4—41no—
197C4—2122no—
198C4—2121no—
199C4D8×D5311yescharacteristic
200C4#1313323911yescharacteristic
201C4#1313324411yescharacteristic
202C4#1313324511yescharacteristic
203C2—81no—
204C2—8481no—
205C2—2122no—
206C2#1681075811yescharacteristic
207C2#1681077011yescharacteristic
208C2#1681079711yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×D4⋊C8C2
socleC22×C53#120
hypercenterD4⋊C8D53
hypocenterC53#1625
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
1. #16811278C2semidirect
2. #16810970C2semidirect
3. #16810819C2semidirect
4. #16810818C2semidirect
5. #16810370C2semidirect
6. #16810352C2semidirect
7. C53×D4⋊C8C2semidirect
71. D212D8semidirect
111. D4⋊C8D53semidirect
112. C53#1625semidirect