Finite Group Databasebeta

← #20273709

Subgroups — #20273709

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#16811218C211yesmaximal, characteristic
2#16810953C211yesmaximal, characteristic, complemented
3#16810554C211yesmaximal, characteristic, complemented
4#16810361C211yesmaximal, characteristic, complemented
5#16810357C211yesmaximal, characteristic
6#16810348C211yesmaximal, characteristic, complemented
7C53×#480C211yesmaximal, characteristic
8#13133210—21no—
9#13133207—21no—
10#13133198—21no—
11#13133198—21no—
12#13133196C2211yescharacteristic
13#13133194—21no—
14D4×Dic53—21no—
15C2×#11719919—21no—
16C2×#11719919C2211yescharacteristic
17#13133148—21no—
18#13133140—21no—
19C22⋊#88322C2211yescharacteristic
20#13133124—21no—
21#13133124C2211yescharacteristic
22#121×C53C2211yescharacteristic
23C2×C4⋊C4×C53C2211yescharacteristic
24#107×C53—21no—
25#102×C53—21no—
26C22⋊C8×C53C2211yescharacteristic
27C2×D4×C53—21no—
28C2×D4×C53D411yescharacteristic
29C22×C4×C53—21no—
30C22×C4×C53C2311yescharacteristic
31C22×Dic53—21no—
32C22×Dic53—21no—
33C22×Dic53D411yescharacteristic
34C22×Dic53D411yescharacteristic
35C2×Dic106D411yescharacteristic
36C2×C8×C53—21no—
37C4⋊C4×C53—21no—
38C4⋊C4×C53—21no—
39C4⋊C4×C53D411yescharacteristic
40C22⋊C4×C53—21no—
41C22⋊Dic53—21no—
42C22⋊Dic53—21no—
43C22⋊Dic53—21no—
44C22⋊Dic53—21no—
45C4⋊Dic53—21no—
46C4⋊Dic53D411yescharacteristic
47#11719919—21no—
48#11719919—21no—
49#11719919—21no—
50#11719919—21no—
51#11719919—21no—
52#11719919—42no—
53C4×Dic53—21no—
54C2×#88322—21no—
55C23×C53—21no—
56C23×C53C2×D411yescharacteristic
57D4×C53—41no—
58D4×C53—41no—
59D4×C53—41no—
60C2×C4×C53—21no—
61C2×C4×C53—41no—
62C2×C4×C53—41no—
63C2×C4×C53—41no—
64C2×C4×C53—41no—
65C2×C4×C53C2×D411yescharacteristic
66C2×C4×C53C2×D411yescharacteristic
67C2×Dic53—22no—
68C2×Dic53—21no—
69C2×Dic53—21no—
70C2×Dic53—21no—
71C2×Dic53—21no—
72C2×Dic53—21no—
73C2×Dic53—21no—
74C2×Dic53—41no—
75C2×Dic53—41no—
76C2×Dic53—41no—
77C2×Dic53—81no—
78Dic106—41no—
79C8×C53—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—81no—
93Dic53—42no—
94Dic53—41no—
95Dic53—41no—
96Dic53—41no—
97Dic53—41no—
98#1609—531nomaximal, Sylow
99C2×C53—21no—
100C2×C53—21no—
101C2×C53—81no—
102C2×C53#44911yescharacteristic
103C2×C53#45611yescharacteristic
104C2×C53#45811yescharacteristic
105#539—531no—
106#483—531no—
107#480D5311yescharacteristic
108#398—531no—
109#328—531no—
110#327—531no—
111#324—531no—
112C53#160911yescharacteristic, complemented, Sylow
113#126—1061no—
114#125—1061no—
115#123—1061no—
116#123—1061no—
117C22⋊Q8—531no—
118C22⋊Q8—1061no—
119#121C2×D5311yescharacteristic
120C4×D4—1061no—
121C2×C4⋊C4—531no—
122C2×C4⋊C4—1061no—
123C2×C4⋊C4C2×D5311yescharacteristic
124#107—21no—
125#106—1061no—
126#103—1061no—
127#102—21no—
128C22⋊C8—531no—
129C22⋊C8C2×D5311yescharacteristic
130#95—531no—
131#95—1061no—
132C2×Q8—531no—
133C2×D4—21no—
134C2×D4C22⋊D5311yescharacteristic
135C22×C4—21no—
136C22×C4—531no—
137C22×C4—531no—
138C22×C4—1061no—
139C22×C4—1061no—
140C22×C4C22×D5311yescharacteristic
141C2×C8—21no—
142C2×C8—1061no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—531no—
146C4⋊C4—1061no—
147C4⋊C4—1061no—
148C4⋊C4—1061no—
149C4⋊C4—1061no—
150C4⋊C4—1061no—
151C4⋊C4—1061no—
152C4⋊C4—2122no—
153C4⋊C4C22⋊D5311yescharacteristic
154C22⋊C4—21no—
155C22⋊C4—1061no—
156C22⋊C4—1061no—
157C22⋊C4—1061no—
158C22⋊C4—1061no—
159C42—1061no—
160C23—21no—
161C23D4×D5311yescharacteristic
162Q8—2121no—
163D4—41no—
164D4—41no—
165D4—41no—
166C2×C4—21no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—1062no—
172C2×C4—1061no—
173C2×C4—1061no—
174C2×C4—1061no—
175C2×C4—1061no—
176C2×C4—1061no—
177C2×C4—1061no—
178C2×C4—4241no—
179C2×C4—2121no—
180C2×C4—2121no—
181C2×C4—2121no—
182C2×C4D4×D5311yescharacteristic
183C2×C4C2×C22⋊D5311yescharacteristic
184C8—41no—
185C8—2121no—
186C22—21no—
187C22—21no—
188C22—41no—
189C22—41no—
190C22—41no—
191C22#1313318511yescharacteristic
192C22#1313323611yescharacteristic
193C22#1313324111yescharacteristic
194C4—21no—
195C4—21no—
196C4—41no—
197C4—81no—
198C4—2122no—
199C4—2121no—
200C4—2121no—
201C4—2121no—
202C4—2121no—
203C2—21no—
204C2—21no—
205C2—81no—
206C2#1681076511yescharacteristic
207C2#1681080411yescharacteristic
208C2#1681081211yescharacteristic
subgroupG/H
centerC22#13133185
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#480C2
socleC22×C53#120
hypercenter#480D53
hypocenterC53#1609
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. #16810953C2semidirect
3. #16810554C2semidirect
4. #16810361C2semidirect
6. #16810348C2semidirect
112. C53#1609semidirect