Finite Group Databasebeta

← #20273912

Subgroups — #20273912

Order 6784.

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

normal subgroups only

This group has 250 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#16811293C211yesmaximal, characteristic
2#16810905C211yesmaximal, characteristic, complemented
3#16810825C211yesmaximal, characteristic, complemented
4#16810367C211yesmaximal, characteristic, complemented
5#16810355C211yesmaximal, characteristic, complemented
6#16810352C211yesmaximal, characteristic, complemented
7C53×#486C211yesmaximal, characteristic, complemented
8C2×#11719947—21no—
9C2×#11719924—21no—
10#13133213—21no—
11#13133189—22no—
12#13133188—21no—
13C4×Dic106C2211yescharacteristic
14C4×D212C2211yescharacteristic
15D4×Dic53—21no—
16#13133148—21no—
17C4⋊#88322C2211yescharacteristic
18C4⋊#88322C2211yescharacteristic
19#13133141—21no—
20#13133140—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×C53D412yes—
29C2×C22⋊D53—22no—
30C2×C22⋊D53—22no—
31C22×Dic53—22no—
32#11719947—42no—
33#11719947—42no—
34C2×D212D411yescharacteristic
35C2×C4×D53—21no—
36C2×Dic106D411yescharacteristic
37C2×C8×C53—21no—
38C4⋊C4×C53—21no—
39C4⋊C4×C53—21no—
40C42×C53C2311yescharacteristic
41C22⋊Dic53—22no—
42#11719924—42no—
43#11719921—21no—
44C4⋊Dic53D411yescharacteristic
45C4⋊Dic53D411yescharacteristic
46#11719919—21no—
47C4×Dic53—21no—
48C2×#88322—21no—
49C2×#88322—21no—
50C23×C53—22no—
51C22×D53—21no—
52Q8×C53—41no—
53Q8×C53—41no—
54D4×C53—42no—
55D4×C53—41no—
56D4×C53—41no—
57D4×C53—41no—
58D4×C53—41no—
59C2×C4×C53—41no—
60C2×C4×C53C2×D411yescharacteristic
61C2×C4×C53C2×D411yescharacteristic
62C2×C4×C53C2×D411yescharacteristic
63C22⋊D53—44no—
64C22⋊D53—82no—
65C2×Dic53—21no—
66C2×Dic53—21no—
67C2×Dic53—21no—
68C2×Dic53—44no—
69D212—21no—
70D212—21no—
71C4×D53—41no—
72C4×D53—41no—
73Dic106—21no—
74Dic106SD1612yescomplemented
75C8×C53—41no—
76#88322—41no—
77#88322—41no—
78C22×C53—42no—
79C22×C53—42no—
80C22×C53—42no—
81C22×C53#12011yescharacteristic
82C2×D53—41no—
83C2×D53—41no—
84C2×D53—41no—
85C4×C53—21no—
86C4×C53—81no—
87C4×C53C2×SD1611yescharacteristic
88C4×C53#13511yescharacteristic
89C4×C53AGL(1,ℤ/8)11yescharacteristic
90C4×C53AGL(1,ℤ/8)11yescharacteristic
91Dic53—22no—
92Dic53—41no—
93Dic53—41no—
94#1642—531nomaximal, Sylow
95C2×C53—82no—
96C2×C53#44911yescharacteristic
97C2×C53#45011yescharacteristic
98C2×C53#45511yescharacteristic
99D53—81no—
100#550—531no—
101#486D5311yescharacteristic, complemented
102#475—531no—
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—1061no—
112#127C2×D5311yescharacteristic
113#121—1061no—
114#121—1062no—
115C4×Q8—531no—
116C4×D4—531no—
117C4×D4—1061no—
118#106—1061no—
119C4⋊C8—531no—
120C4⋊C8—531no—
121#103—1061no—
122#103—1061no—
123#102—22no—
124C4×C8C2×D5311yescharacteristic
125#41—2122no—
126#41—2122no—
127C2×Q8—21no—
128C2×Q8—531no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—531no—
132C2×D4—1062no—
133C2×D4—1062no—
134C2×D4C22⋊D5312yes—
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—531no—
144C4⋊C4—531no—
145C4⋊C4—1061no—
146C22⋊C4—1062no—
147C22⋊C4—1061no—
148C42—1061no—
149C42C22×D5311yescharacteristic
150C23—22no—
151C23—1061no—
152Q8—41no—
153Q8—41no—
154Q8—532no—
155Q8—1061no—
156D4—42no—
157D4—41no—
158D4—41no—
159D4—41no—
160D4—41no—
161D4—1061no—
162D4—1061no—
163D4—4242no—
164D4—2124no—
165C2×C4—41no—
166C2×C4—1061no—
167C2×C4—1061no—
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—42no—
179C22—42no—
180C22—42no—
181C22—2121no—
182C22—2121no—
183C22—2121no—
184C22#1313318511yescharacteristic
185C4—21no—
186C4—81no—
187C4—1062no—
188C4—2121no—
189C4—2121no—
190C4SD16×D5311yescharacteristic
191C4#1313323611yescharacteristic
192C4#1313324111yescharacteristic
193C4#1313324411yescharacteristic
194C2—82no—
195C2—4241no—
196C2#1681076511yescharacteristic
197C2#1681077211yescharacteristic
198C2#1681080011yescharacteristic
subgroupG/H
centerC22#13133185
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#486C2
socleC22×C53#120
hypercenter#486D53
hypocenterC53#1642
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
2. #16810905C2semidirect
3. #16810825C2semidirect
4. #16810367C2semidirect
5. #16810355C2semidirect
6. #16810352C2semidirect
7. C53×#486C2semidirect
74. Dic106SD16semidirect
101. #486D53semidirect
107. C53#1642semidirect