Finite Group Databasebeta

← #20273712

Subgroups — #20273712

Order 6784.

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

normal subgroups only

This group has 216 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#16810951C211yesmaximal, characteristic, complemented
2#16810934C211yesmaximal, characteristic
3#16810553C211yesmaximal, characteristic, complemented
4#16810360C211yesmaximal, characteristic, complemented
5#16810357C211yesmaximal, characteristic
6#16810349C211yesmaximal, characteristic, complemented
7C53×#539C211yesmaximal, characteristic
8C2×C4⋊Dic53C2211yescharacteristic
9C2×#11719919—21no—
10#13133150—21no—
11#13133148—21no—
12#13133141—21no—
13#13133138—21no—
14C22⋊#88322C2211yescharacteristic
15C22⋊#88322C2211yescharacteristic
16#13133124—21no—
17#13133124C2211yescharacteristic
18#126×C53—21no—
19#125×C53—21no—
20#123×C53—21no—
21#123×C53—21no—
22C22⋊Q8×C53—21no—
23C22⋊Q8×C53C2211yescharacteristic
24#121×C53C2211yescharacteristic
25C4×D4×C53—21no—
26C2×C4⋊C4×C53C2211yescharacteristic
27C2×Q8×C53D411yescharacteristic
28C2×D4×C53—21no—
29C2×D4×C53D411yescharacteristic
30C22×C4×C53—21no—
31C22×C4×C53D411yescharacteristic
32C22×C4×C53D411yescharacteristic
33C22×C4×C53C2311yescharacteristic
34C22×Dic53—21no—
35C22×Dic53—21no—
36C4⋊C4×C53—21no—
37C4⋊C4×C53—21no—
38C4⋊C4×C53—21no—
39C4⋊C4×C53—21no—
40C4⋊C4×C53—21no—
41C4⋊C4×C53—21no—
42C4⋊C4×C53D411yescharacteristic
43C4⋊C4×C53D411yescharacteristic
44C22⋊C4×C53—21no—
45C22⋊C4×C53—21no—
46C22⋊C4×C53—21no—
47C22⋊C4×C53—21no—
48C22⋊C4×C53—21no—
49C42×C53—21no—
50C4⋊Dic53—21no—
51C4⋊Dic53—21no—
52#11719919—42no—
53C2×#88322—21no—
54C2×#88322—21no—
55C23×C53—21no—
56C23×C53C2×D411yescharacteristic
57Q8×C53—41no—
58D4×C53—41no—
59D4×C53—41no—
60D4×C53—41no—
61C2×C4×C53—21no—
62C2×C4×C53—21no—
63C2×C4×C53—21no—
64C2×C4×C53—21no—
65C2×C4×C53—21no—
66C2×C4×C53—21no—
67C2×C4×C53—21no—
68C2×C4×C53—41no—
69C2×C4×C53—41no—
70C2×C4×C53—81no—
71C2×C4×C53C2×D411yescharacteristic
72C2×C4×C53C2×D411yescharacteristic
73C2×Dic53—22no—
74C2×Dic53—41no—
75C2×Dic53—41no—
76C2×Dic53—41no—
77C2×Dic53—41no—
78C2×Dic53—41no—
79#88322—41no—
80#88322—41no—
81C22×C53—21no—
82C22×C53—21no—
83C22×C53—41no—
84C22×C53—41no—
85C22×C53—41no—
86C22×C53#12011yescharacteristic
87C22×C53#13511yescharacteristic
88C22×C53AGL(1,ℤ/8)11yescharacteristic
89C4×C53—21no—
90C4×C53—21no—
91C4×C53—41no—
92C4×C53—41no—
93C4×C53—41no—
94C4×C53—41no—
95C4×C53—41no—
96Dic53—42no—
97Dic53—81no—
98#1609—531nomaximal, Sylow
99C2×C53—21no—
100C2×C53—21no—
101C2×C53—81no—
102C2×C53#44911yescharacteristic
103C2×C53#45611yescharacteristic
104C2×C53#45811yescharacteristic
105#539D5311yescharacteristic
106#483—531no—
107#480—531no—
108#398—531no—
109#328—531no—
110#327—531no—
111#324—531no—
112C53#160911yescharacteristic, complemented, Sylow
113#126—21no—
114#125—21no—
115#123—21no—
116#123—21no—
117C22⋊Q8—21no—
118C22⋊Q8C2×D5311yescharacteristic
119#121C2×D5311yescharacteristic
120C4×D4—21no—
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⋊C8—531no—
129C22⋊C8—531no—
130#95—531no—
131#95—1061no—
132C2×Q8C22⋊D5311yescharacteristic
133C2×D4—21no—
134C2×D4C22⋊D5311yescharacteristic
135C22×C4—21no—
136C22×C4—1061no—
137C22×C4—1061no—
138C22×C4C22⋊D5311yescharacteristic
139C22×C4C22⋊D5311yescharacteristic
140C22×C4C22×D5311yescharacteristic
141C2×C8—1061no—
142C2×C8—1061no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—21no—
146C4⋊C4—21no—
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4—1061no—
150C4⋊C4—1061no—
151C4⋊C4—2122no—
152C4⋊C4C22⋊D5311yescharacteristic
153C4⋊C4C22⋊D5311yescharacteristic
154C22⋊C4—21no—
155C22⋊C4—21no—
156C22⋊C4—21no—
157C22⋊C4—21no—
158C22⋊C4—21no—
159C42—21no—
160C23—21no—
161C23C2×C22⋊D5311yescharacteristic
162Q8—41no—
163D4—41no—
164D4—41no—
165D4—41no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—81no—
176C2×C4—1062no—
177C2×C4—2121no—
178C2×C4—2121no—
179C2×C4—2121no—
180C2×C4—2121no—
181C2×C4—2121no—
182C2×C4C2×C22⋊D5311yescharacteristic
183C2×C4C2×C22⋊D5311yescharacteristic
184C8—2121no—
185C8—2121no—
186C22—21no—
187C22—21no—
188C22—41no—
189C22—41no—
190C22—41no—
191C22#1313318611yescharacteristic
192C22#1313323711yescharacteristic
193C22#1313324211yescharacteristic
194C4—21no—
195C4—21no—
196C4—41no—
197C4—41no—
198C4—41no—
199C4—41no—
200C4—41no—
201C4—4241no—
202C4—2122no—
203C2—21no—
204C2—21no—
205C2—81no—
206C2#1681076811yescharacteristic
207C2#1681080611yescharacteristic
208C2#1681081311yescharacteristic
subgroupG/H
centerC22#13133186
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#539C2
socleC22×C53#120
hypercenter#539D53
hypocenterC53#1609
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #539

Complements

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

NH ≅ G/Ntype
1. #16810951C2semidirect
3. #16810553C2semidirect
4. #16810360C2semidirect
6. #16810349C2semidirect
112. C53#1609semidirect