Finite Group Databasebeta

← #20273797

Subgroups — #20273797

Order 6784.

240 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 53 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#16811338C211yesmaximal, characteristic, complemented
2#16810995C211yesmaximal, characteristic, complemented
3#16810815C211yesmaximal, characteristic, complemented
4#16810371C211yesmaximal, characteristic, complemented
5D4⋊#88322C211yesmaximal, characteristic, complemented
6#16810350C211yesmaximal, characteristic, complemented
7C53×#459C211yesmaximal, characteristic, complemented
8C2×#11719923—22no—
9C2×#11719922—21no—
10#13133217C2211yescharacteristic
11#13133203—22no—
12#13133191—21no—
13#13133188—21no—
14C4×D212C2211yescharacteristic
15#13133176—21no—
16#13133175—21no—
17C4⋊#88322C2211yescharacteristic
18#13133138—21no—
19C4×#88322C2211yescharacteristic
20C2×D8×C53—21no—
21#127×C53C2211yescharacteristic
22C4×D4×C53C2211yescharacteristic
23C4⋊C8×C53C2211yescharacteristic
24#102×C53—21no—
25C2×D4×C53—21no—
26C2×D4×C53—21no—
27C2×D4×C53—21no—
28C2×D4×C53D411yescharacteristic
29C22×C4×C53—21no—
30C2×C22⋊D53—22no—
31C2×D212D411yescharacteristic
32C2×C4×D53—21no—
33C2×Dic106D412yes—
34D8×C53—42no—
35C2×C8×C53—21no—
36C4⋊C4×C53D411yescharacteristic
37C22⋊C4×C53—21no—
38C42×C53C2311yescharacteristic
39C22⋊Dic53—22no—
40#11719923—44no—
41#11719922—42no—
42#11719921—22no—
43#11719921—21no—
44C4⋊Dic53—21no—
45C4⋊Dic53—21no—
46C4⋊Dic53D411yescharacteristic
47C4×Dic53—22no—
48C2×#88322—21no—
49C2×#88322—21no—
50C23×C53—21no—
51C23×C53—41no—
52C22×D53—21no—
53D4×C53—21no—
54D4×C53—21no—
55D4×C53—42no—
56D4×C53—41no—
57D4×C53—41no—
58D4×C53—41no—
59D4×C53—41no—
60C2×C4×C53—21no—
61C2×C4×C53—41no—
62C2×C4×C53—41no—
63C2×C4×C53C2×D411yescharacteristic
64C2×C4×C53C2×D411yescharacteristic
65C2×C4×C53C2×D411yescharacteristic
66C22⋊D53—82no—
67C2×Dic53—22no—
68C2×Dic53—21no—
69D212—21no—
70D212—21no—
71C4×D53—41no—
72C4×D53—41no—
73Dic106—42no—
74C8×C53—41no—
75#88322—22no—
76#88322—41no—
77C22×C53—41no—
78C22×C53—41no—
79C22×C53—41no—
80C22×C53—81no—
81C22×C53—81no—
82C22×C53—81no—
83C22×C53#12011yescharacteristic
84C2×D53—41no—
85C2×D53—41no—
86C2×D53—41no—
87C4×C53—21no—
88C4×C53—41no—
89C4×C53#13511yescharacteristic
90C4×C53#13511yescharacteristic
91C4×C53AGL(1,ℤ/8)11yescharacteristic
92C4×C53AGL(1,ℤ/8)11yescharacteristic
93Dic53—41no—
94Dic53—42no—
95#1624—531nomaximal, Sylow
96C2×C53—81no—
97C2×C53—161no—
98C2×C53#44911yescharacteristic
99C2×C53#44911yescharacteristic
100C2×C53#45311yescharacteristic
101D53—81no—
102#555—531no—
103#492—531no—
104#459—531no—
105#459D5311yescharacteristic, complemented
106#333—531no—
107D4⋊C8—531no—
108D4⋊C8—531no—
109C53#162411yescharacteristic, complemented, Sylow
110C2×SD16—1062no—
111C2×D8—21no—
112C2×D8—1061no—
113C4⋊Q8—531no—
114#127C2×D5311yescharacteristic
115#124—1062no—
116#121—1061no—
117#121—1061no—
118C4×D4—531no—
119C4×D4C2×D5311yescharacteristic
120#117—1061no—
121#117—1061no—
122C4⋊C8—531no—
123C4⋊C8C2×D5311yescharacteristic
124#102—21no—
125#102—1061no—
126C4×C8—531no—
127C2×Q8—532no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—531no—
132C2×D4—1062no—
133C2×D4C22⋊D5311yescharacteristic
134C22×C4—21no—
135C22×C4—1061no—
136SD16—2124no—
137D8—42no—
138D8—2122no—
139C2×C8—21no—
140C2×C8—1061no—
141C2×C8—1061no—
142C4⋊C4—531no—
143C4⋊C4—1061no—
144C4⋊C4—1061no—
145C4⋊C4C22⋊D5311yescharacteristic
146C22⋊C4—21no—
147C22⋊C4—1062no—
148C22⋊C4—1062no—
149C22⋊C4—1061no—
150C42—1062no—
151C42C22×D5311yescharacteristic
152C23—21no—
153C23—41no—
154C23—1061no—
155Q8—2122no—
156D4—21no—
157D4—21no—
158D4—42no—
159D4—41no—
160D4—41no—
161D4—41no—
162D4—41no—
163D4—1061no—
164D4—1061no—
165D4—4242no—
166C2×C4—21no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—1061no—
170C2×C4—1062no—
171C2×C4—2121no—
172C2×C4—2121no—
173C2×C4D4×D5311yescharacteristic
174C2×C4D4×D5311yescharacteristic
175C2×C4C2×C22⋊D5311yescharacteristic
176C8—41no—
177C8—1062no—
178C8—2121no—
179C22—41no—
180C22—41no—
181C22—41no—
182C22—81no—
183C22—81no—
184C22—81no—
185C22—2121no—
186C22—2121no—
187C22—2121no—
188C22#1313318511yescharacteristic
189C4—21no—
190C4—41no—
191C4—2121no—
192C4—2122no—
193C4#1313323611yescharacteristic
194C4#1313323711yescharacteristic
195C4#1313324111yescharacteristic
196C4#1313324311yescharacteristic
197C2—81no—
198C2—161no—
199C2—4241no—
200C2#1681076511yescharacteristic
201C2#1681076611yescharacteristic
202C2#1681078911yescharacteristic
subgroupG/H
centerC22#13133185
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#459C2
socleC22×C53#120
hypercenter#459D53
hypocenterC53#1624
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #16811338C2semidirect
2. #16810995C2semidirect
3. #16810815C2semidirect
4. #16810371C2semidirect
5. D4⋊#88322C2semidirect
6. #16810350C2semidirect
7. C53×#459C2semidirect
105. #459D53semidirect
109. C53#1624semidirect