Finite Group Databasebeta

← #20273914

Subgroups — #20273914

Order 6784.

250 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 57 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#16811295C211yesmaximal, characteristic
2#16810971C211yesmaximal, characteristic, complemented
3#16810825C211yesmaximal, characteristic, complemented
4#16810366C211yesmaximal, characteristic, complemented
5Q8⋊#88322C211yesmaximal, characteristic, complemented
6#16810350C211yesmaximal, characteristic, complemented
7C53×#475C211yesmaximal, characteristic, complemented
8C2×#11719949—21no—
9C2×#11719924—21no—
10#13133214—21no—
11#13133214C2211yescharacteristic
12#13133192—21no—
13#13133192—22no—
14C4×D212—21no—
15C4×D212C2211yescharacteristic
16C4⋊#88322C2211yescharacteristic
17#13133141—21no—
18#13133137—22no—
19C4×#88322C2211yescharacteristic
20C4⋊Q8×C53C2211yescharacteristic
21C4×Q8×C53C2211yescharacteristic
22#106×C53—21no—
23C4⋊C8×C53C2211yescharacteristic
24#103×C53—21no—
25C2×Q8×C53—21no—
26C2×Q8×C53D411yescharacteristic
27#11719949—42no—
28#11719949—42no—
29C2×D212—22no—
30C2×D212—21no—
31C2×D212—21no—
32C2×D212—22no—
33C2×D212D411yescharacteristic
34C2×D212D412yes—
35C2×C4×D53—22no—
36C2×C4×D53—21no—
37C2×C8×C53—21no—
38C4⋊C4×C53—21no—
39C4⋊C4×C53—21no—
40C4⋊C4×C53—21no—
41C4⋊C4×C53D411yescharacteristic
42C42×C53—21no—
43C42×C53C2311yescharacteristic
44#11719924—42no—
45#11719921—22no—
46#11719921—21no—
47C4⋊Dic53D411yescharacteristic
48C2×#88322—21no—
49C2×#88322—21no—
50C22×D53—21no—
51C22×D53—22no—
52Q8×C53—21no—
53Q8×C53—41no—
54Q8×C53—41no—
55Q8×C53SD1612yescomplemented
56C2×C4×C53—21no—
57C2×C4×C53—21no—
58C2×C4×C53—41no—
59C2×C4×C53C2×D411yescharacteristic
60C2×C4×C53C2×D411yescharacteristic
61C2×C4×C53C2×D411yescharacteristic
62C2×Dic53—21no—
63D212—21no—
64D212—21no—
65D212—42no—
66D212—44no—
67D212—41no—
68D212—41no—
69D212—41no—
70D212—41no—
71D212—82no—
72C4×D53—44no—
73C4×D53—41no—
74C4×D53—41no—
75C8×C53—41no—
76#88322—41no—
77#88322—41no—
78C22×C53#12011yescharacteristic
79C2×D53—41no—
80C2×D53—42no—
81C2×D53—42no—
82C2×D53—41no—
83C2×D53—42no—
84C2×D53—41no—
85C4×C53—22no—
86C4×C53—21no—
87C4×C53—41no—
88C4×C53—81no—
89C4×C53C2×SD1611yescharacteristic
90C4×C53#13511yescharacteristic
91C4×C53AGL(1,ℤ/8)11yescharacteristic
92C4×C53AGL(1,ℤ/8)11yescharacteristic
93Dic53—41no—
94#1642—531nomaximal, Sylow
95C2×C53#44911yescharacteristic
96C2×C53#45011yescharacteristic
97C2×C53#45511yescharacteristic
98D53—81no—
99D53—82no—
100#550—531no—
101#486—531no—
102#475D5311yescharacteristic, complemented
103#461—531no—
104#331—531no—
105Q8⋊C8—531no—
106D4⋊C8—531no—
107C53#164211yescharacteristic, complemented, Sylow
108C2×#41—1061no—
109C2×SD16—1061no—
110C4⋊Q8C2×D5311yescharacteristic
111#127—531no—
112#127—1061no—
113#121—1062no—
114#121—1061no—
115C4×Q8C2×D5311yescharacteristic
116C4×D4—531no—
117C4×D4—1061no—
118#106—21no—
119C4⋊C8—531no—
120C4⋊C8C2×D5311yescharacteristic
121#103—21no—
122#103—1061no—
123#102—1062no—
124C4×C8—531no—
125#41—2122no—
126#41—2122no—
127C2×Q8—21no—
128C2×Q8C22⋊D5311yescharacteristic
129C2×D4—531no—
130C2×D4—532no—
131C2×D4—1062no—
132C2×D4—1062no—
133C2×D4—1061no—
134C2×D4—1061no—
135C22×C4—1062no—
136C22×C4—1061no—
137SD16—2122no—
138C2×C8—21no—
139C2×C8—1061no—
140C2×C8—1061no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—531no—
145C4⋊C4C22⋊D5311yescharacteristic
146C22⋊C4—1062no—
147C22⋊C4—1061no—
148C42—21no—
149C42C22×D5311yescharacteristic
150C23—1061no—
151C23—1062no—
152Q8—21no—
153Q8—41no—
154Q8—41no—
155Q8#1171992412yescomplemented
156D4—1061no—
157D4—1061no—
158D4—4242no—
159D4—2122no—
160D4—2124no—
161D4—2121no—
162D4—2121no—
163D4—2121no—
164D4—2121no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—41no—
168C2×C4—1061no—
169C2×C4—2124no—
170C2×C4—2121no—
171C2×C4—2121no—
172C2×C4D4×D5311yescharacteristic
173C2×C4D4×D5311yescharacteristic
174C2×C4C2×C22⋊D5311yescharacteristic
175C8—41no—
176C8—2121no—
177C8—2121no—
178C22—2121no—
179C22—2122no—
180C22—2122no—
181C22—2121no—
182C22—2121no—
183C22—2122no—
184C22#1313318511yescharacteristic
185C4—22no—
186C4—21no—
187C4—41no—
188C4—81no—
189C4—2121no—
190C4C2×#1171992411yescharacteristic
191C4#1313323811yescharacteristic
192C4#1313324211yescharacteristic
193C4#1313324411yescharacteristic
194C2—4241no—
195C2—4242no—
196C2#1681076711yescharacteristic
197C2#1681077311yescharacteristic
198C2#1681079911yescharacteristic
subgroupG/H
centerC22#13133185
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#475C2
socleC22×C53#120
hypercenter#475D53
hypocenterC53#1642
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
2. #16810971C2semidirect
3. #16810825C2semidirect
4. #16810366C2semidirect
5. Q8⋊#88322C2semidirect
6. #16810350C2semidirect
7. C53×#475C2semidirect
55. Q8×C53SD16semidirect
102. #475D53semidirect
107. C53#1642semidirect
155. Q8#11719924semidirect