Finite Group Databasebeta

← #20273714

Subgroups — #20273714

Order 6784.

216 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 49 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#16811220C211yesmaximal, characteristic
2#16810954C211yesmaximal, characteristic, complemented
3#16810937C211yesmaximal, characteristic
4#16810553C211yesmaximal, characteristic, complemented
5#16810361C211yesmaximal, characteristic, complemented
6#16810358C211yesmaximal, characteristic
7C53×#324C211yesmaximal, characteristic, complemented
8#13133212—21no—
9#13133207—21no—
10#13133202—21no—
11#13133200—21no—
12#13133196C2211yescharacteristic
13#13133195—21no—
14#13133191C2211yescharacteristic
15C4×C22⋊D53—21no—
16C2×C4⋊Dic53C2211yescharacteristic
17C2×#11719919—21no—
18#13133151—21no—
19#13133149—21no—
20#13133142—21no—
21#13133139—21no—
22#13133124—21no—
23#13133124C2211yescharacteristic
24C2×C4⋊C4×C53C2211yescharacteristic
25C22⋊C8×C53C2211yescharacteristic
26C22⋊C8×C53C2211yescharacteristic
27C22×C4×C53D411yescharacteristic
28C22×C4×C53D411yescharacteristic
29C22×C4×C53C2311yescharacteristic
30C2×C22⋊D53—21no—
31C22×Dic53—21no—
32C22×Dic53—21no—
33C2×D212D411yescharacteristic
34C2×C4×D53—21no—
35C2×Dic106D411yescharacteristic
36C2×C8×C53—21no—
37C2×C8×C53—21no—
38C4⋊C4×C53—21no—
39C4⋊C4×C53—21no—
40C22⋊Dic53—21no—
41#11719921—21no—
42#11719921—21no—
43#11719921—21no—
44#11719921—21no—
45C4⋊Dic53—21no—
46C4⋊Dic53—21no—
47C4⋊Dic53D411yescharacteristic
48C4⋊Dic53D411yescharacteristic
49#11719919—21no—
50#11719919—21no—
51#11719919—21no—
52#11719919—21no—
53#11719919—42no—
54C4×Dic53—21no—
55C23×C53C2×D411yescharacteristic
56C22×D53—21no—
57C2×C4×C53—21no—
58C2×C4×C53—21no—
59C2×C4×C53—21no—
60C2×C4×C53—21no—
61C2×C4×C53—41no—
62C2×C4×C53—41no—
63C2×C4×C53C2×D411yescharacteristic
64C2×C4×C53C2×D411yescharacteristic
65C22⋊D53—41no—
66C22⋊D53—41no—
67C2×Dic53—22no—
68C2×Dic53—21no—
69C2×Dic53—21no—
70C2×Dic53—21no—
71C2×Dic53—41no—
72C2×Dic53—41no—
73C2×Dic53—41no—
74C2×Dic53—41no—
75C2×Dic53—41no—
76D212—41no—
77C4×D53—81no—
78Dic106—41no—
79C8×C53—41no—
80C8×C53—41no—
81C22×C53—21no—
82C22×C53—21no—
83C22×C53#12011yescharacteristic
84C22×C53#13511yescharacteristic
85C22×C53AGL(1,ℤ/8)11yescharacteristic
86C2×D53—41no—
87C2×D53—41no—
88C2×D53—41no—
89C4×C53—21no—
90C4×C53—21no—
91C4×C53—41no—
92C4×C53—41no—
93Dic53—42no—
94Dic53—41no—
95Dic53—41no—
96Dic53—41no—
97Dic53—81no—
98#1609—531nomaximal, Sylow
99C2×C53—21no—
100C2×C53—21no—
101C2×C53#44911yescharacteristic
102C2×C53#45611yescharacteristic
103C2×C53#45811yescharacteristic
104D53—81no—
105#539—531no—
106#483—531no—
107#480—531no—
108#398—531no—
109#328—531no—
110#327—531no—
111#324D5311yescharacteristic, complemented
112C53#160911yescharacteristic, complemented, Sylow
113#126—1061no—
114#125—1061no—
115#123—1061no—
116#123—1061no—
117C22⋊Q8—531no—
118C22⋊Q8—1061no—
119#121—531no—
120C4×D4—1061no—
121C2×C4⋊C4—531no—
122C2×C4⋊C4—1061no—
123C2×C4⋊C4C2×D5311yescharacteristic
124#107—1061no—
125#106—1061no—
126#103—1061no—
127#102—1061no—
128C22⋊C8C2×D5311yescharacteristic
129C22⋊C8C2×D5311yescharacteristic
130#95—531no—
131#95—1061no—
132C2×Q8—531no—
133C2×D4—531no—
134C2×D4—1061no—
135C22×C4—1061no—
136C22×C4—1061no—
137C22×C4—1061no—
138C22×C4D21211yescharacteristic
139C22×C4D21211yescharacteristic
140C22×C4C22×D5311yescharacteristic
141C2×C8—21no—
142C2×C8—21no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—531no—
146C4⋊C4—531no—
147C4⋊C4—1061no—
148C4⋊C4—1061no—
149C4⋊C4—1061no—
150C4⋊C4—1061no—
151C4⋊C4—1061no—
152C4⋊C4—1061no—
153C4⋊C4—2122no—
154C22⋊C4—1061no—
155C22⋊C4—1061no—
156C22⋊C4—1061no—
157C22⋊C4—1061no—
158C22⋊C4—1061no—
159C42—1061no—
160C23—1061no—
161C23C2×D21211yescharacteristic
162Q8—2121no—
163D4—2121no—
164D4—2121no—
165D4—2121no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—1062no—
173C2×C4—1061no—
174C2×C4—1061no—
175C2×C4—1061no—
176C2×C4—4241no—
177C2×C4—2121no—
178C2×C4—2121no—
179C2×C4—2121no—
180C2×C4—2121no—
181C2×C4—2121no—
182C2×C4D4×D5311yescharacteristic
183C2×C4D4×D5311yescharacteristic
184C8—41no—
185C8—41no—
186C22—21no—
187C22—21no—
188C22—2121no—
189C22—2121no—
190C22—2121no—
191C22#1313318711yescharacteristic
192C22#1313323911yescharacteristic
193C22#1313324511yescharacteristic
194C4—21no—
195C4—21no—
196C4—41no—
197C4—41no—
198C4—4241no—
199C4—2122no—
200C4—2121no—
201C4—2121no—
202C4—2121no—
203C2—21no—
204C2—21no—
205C2—4241no—
206C2#1681077011yescharacteristic
207C2#1681080811yescharacteristic
208C2#1681081411yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#324C2
socleC22×C53#120
hypercenter#324D53
hypocenterC53#1609
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #16810954C2semidirect
4. #16810553C2semidirect
5. #16810361C2semidirect
7. C53×#324C2semidirect
111. #324D53semidirect
112. C53#1609semidirect