Finite Group Databasebeta

← #20273743

Subgroups — #20273743

Order 6784.

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

normal subgroups only

This group has 208 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#16810958C211yesmaximal, characteristic, complemented
3#16810561C211yesmaximal, characteristic, complemented
4#16810361C211yesmaximal, characteristic, complemented
5#16810357C211yesmaximal, characteristic
6#16810348C211yesmaximal, characteristic, complemented
7C53×#481C211yesmaximal, characteristic
8#13133210—21no—
9#13133207—21no—
10#13133198—21no—
11#13133198—21no—
12#13133196C2211yescharacteristic
13#13133194—21no—
14D4×Dic53—21no—
15C2×#11719919C2211yescharacteristic
16#13133150—21no—
17#13133140—21no—
18C22⋊#88322C2211yescharacteristic
19#13133124—21no—
20#13133124—21no—
21#13133124C2211yescharacteristic
22#121×C53C2211yescharacteristic
23C2×C4⋊C4×C53C2211yescharacteristic
24#106×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—
52C4×Dic53—21no—
53C2×#88322—21no—
54C23×C53—21no—
55C23×C53C2×D411yescharacteristic
56D4×C53—41no—
57D4×C53—41no—
58D4×C53—41no—
59C2×C4×C53—21no—
60C2×C4×C53—41no—
61C2×C4×C53—42no—
62C2×C4×C53—41no—
63C2×C4×C53C2×D411yescharacteristic
64C2×C4×C53C2×D411yescharacteristic
65C2×Dic53—21no—
66C2×Dic53—21no—
67C2×Dic53—21no—
68C2×Dic53—21no—
69C2×Dic53—21no—
70C2×Dic53—21no—
71C2×Dic53—41no—
72C2×Dic53—41no—
73C2×Dic53—41no—
74C2×Dic53—41no—
75C2×Dic53—81no—
76Dic106—41no—
77C8×C53—41no—
78#88322—41no—
79C22×C53—21no—
80C22×C53—21no—
81C22×C53—41no—
82C22×C53—41no—
83C22×C53—41no—
84C22×C53#12011yescharacteristic
85C22×C53#13511yescharacteristic
86C22×C53#13711yescharacteristic
87C4×C53—21no—
88C4×C53—21no—
89C4×C53—41no—
90C4×C53—81no—
91Dic53—41no—
92Dic53—41no—
93Dic53—41no—
94Dic53—41no—
95Dic53—81no—
96#1615—531nomaximal, Sylow
97C2×C53—21no—
98C2×C53—21no—
99C2×C53—81no—
100C2×C53#45211yescharacteristic
101C2×C53#45411yescharacteristic
102C2×C53#45811yescharacteristic
103#539—531no—
104#484—531no—
105#481D5311yescharacteristic
106#400—531no—
107#328—531no—
108#327—531no—
109#324—531no—
110C53#161511yescharacteristic, complemented, Sylow
111#126—1061no—
112#125—1061no—
113#123—1061no—
114#123—1061no—
115C22⋊Q8—531no—
116C22⋊Q8—1061no—
117#121C2×D5311yescharacteristic
118C4×D4—1061no—
119C2×C4⋊C4—531no—
120C2×C4⋊C4C2×D5311yescharacteristic
121#107—1061no—
122#106—21no—
123#103—1061no—
124#102—21no—
125C22⋊C8—531no—
126C22⋊C8C2×D5311yescharacteristic
127#95—531no—
128#95—1061no—
129#95—1061no—
130C2×Q8—531no—
131C2×D4—21no—
132C2×D4C22⋊D5311yescharacteristic
133C22×C4—21no—
134C22×C4—531no—
135C22×C4—531no—
136C22×C4—1061no—
137C22×C4—1061no—
138C22×C4C22×D5311yescharacteristic
139C2×C8—21no—
140C2×C8—1061no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—531no—
144C4⋊C4—1061no—
145C4⋊C4—1061no—
146C4⋊C4—1061no—
147C4⋊C4—1061no—
148C4⋊C4—1061no—
149C4⋊C4—1061no—
150C4⋊C4C22⋊D5311yescharacteristic
151C22⋊C4—21no—
152C22⋊C4—1061no—
153C22⋊C4—1061no—
154C22⋊C4—1061no—
155C22⋊C4—1061no—
156C42—1061no—
157C23—21no—
158C23D4×D5311yescharacteristic
159Q8—2121no—
160D4—41no—
161D4—41no—
162D4—41no—
163C2×C4—21no—
164C2×C4—41no—
165C2×C4—42no—
166C2×C4—41no—
167C2×C4—1061no—
168C2×C4—1061no—
169C2×C4—1061no—
170C2×C4—1061no—
171C2×C4—1061no—
172C2×C4—1061no—
173C2×C4—4241no—
174C2×C4—2121no—
175C2×C4—2121no—
176C2×C4—2121no—
177C2×C4—2121no—
178C2×C4D4×D5311yescharacteristic
179C2×C4C2×C22⋊D5311yescharacteristic
180C8—41no—
181C8—2121no—
182C22—21no—
183C22—21no—
184C22—41no—
185C22—41no—
186C22—41no—
187C22#1313318511yescharacteristic
188C22#1313323811yescharacteristic
189C22#1313324611yescharacteristic
190C4—21no—
191C4—21no—
192C4—41no—
193C4—81no—
194C4—4241no—
195C4—2121no—
196C4—2121no—
197C4—2121no—
198C4—2121no—
199C2—21no—
200C2—21no—
201C2—81no—
202C2#1681078211yescharacteristic
203C2#1681079411yescharacteristic
204C2#1681081211yescharacteristic
subgroupG/H
centerC22#13133185
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#481C2
socleC22×C53#120
hypercenter#481D53
hypocenterC53#1615
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
2. #16810958C2semidirect
3. #16810561C2semidirect
4. #16810361C2semidirect
6. #16810348C2semidirect
110. C53#1615semidirect