Finite Group Databasebeta

← #20273748

Subgroups — #20273748

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#16811220C211yesmaximal, characteristic
2#16810959C211yesmaximal, characteristic, complemented
3#16810942C211yesmaximal, characteristic
4#16810562C211yesmaximal, 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
17#13133151—21no—
18#13133149—21no—
19#13133142—21no—
20#13133139—21no—
21#13133124—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—
53C4×Dic53—21no—
54C23×C53C2×D411yescharacteristic
55C22×D53—21no—
56C2×C4×C53—21no—
57C2×C4×C53—21no—
58C2×C4×C53—21no—
59C2×C4×C53—21no—
60C2×C4×C53—41no—
61C2×C4×C53—41no—
62C2×C4×C53C2×D411yescharacteristic
63C2×C4×C53C2×D411yescharacteristic
64C22⋊D53—41no—
65C22⋊D53—41no—
66C2×Dic53—21no—
67C2×Dic53—21no—
68C2×Dic53—21no—
69C2×Dic53—41no—
70C2×Dic53—41no—
71C2×Dic53—42no—
72C2×Dic53—41no—
73C2×Dic53—41no—
74D212—41no—
75C4×D53—81no—
76Dic106—41no—
77C8×C53—41no—
78C8×C53—41no—
79C22×C53—21no—
80C22×C53—21no—
81C22×C53#12011yescharacteristic
82C22×C53#13511yescharacteristic
83C22×C53#13711yescharacteristic
84C2×D53—41no—
85C2×D53—41no—
86C2×D53—41no—
87C4×C53—21no—
88C4×C53—21no—
89C4×C53—41no—
90C4×C53—41no—
91Dic53—41no—
92Dic53—41no—
93Dic53—41no—
94Dic53—81no—
95Dic53—81no—
96#1615—531nomaximal, Sylow
97C2×C53—21no—
98C2×C53—21no—
99C2×C53#45211yescharacteristic
100C2×C53#45411yescharacteristic
101C2×C53#45811yescharacteristic
102D53—81no—
103#539—531no—
104#484—531no—
105#481—531no—
106#400—531no—
107#328—531no—
108#327—531no—
109#324D5311yescharacteristic, complemented
110C53#161511yescharacteristic, complemented, Sylow
111#126—1061no—
112#125—1061no—
113#123—1061no—
114#123—1061no—
115C22⋊Q8—531no—
116C22⋊Q8—1061no—
117#121—531no—
118C4×D4—1061no—
119C2×C4⋊C4—531no—
120C2×C4⋊C4C2×D5311yescharacteristic
121#107—1061no—
122#106—1061no—
123#103—1061no—
124#102—1061no—
125C22⋊C8C2×D5311yescharacteristic
126C22⋊C8C2×D5311yescharacteristic
127#95—531no—
128#95—1061no—
129#95—1061no—
130C2×Q8—531no—
131C2×D4—531no—
132C2×D4—1061no—
133C22×C4—1061no—
134C22×C4—1061no—
135C22×C4—1061no—
136C22×C4D21211yescharacteristic
137C22×C4D21211yescharacteristic
138C22×C4C22×D5311yescharacteristic
139C2×C8—21no—
140C2×C8—21no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—531no—
144C4⋊C4—531no—
145C4⋊C4—1061no—
146C4⋊C4—1061no—
147C4⋊C4—1061no—
148C4⋊C4—1061no—
149C4⋊C4—1061no—
150C4⋊C4—1061no—
151C22⋊C4—1061no—
152C22⋊C4—1061no—
153C22⋊C4—1061no—
154C22⋊C4—1061no—
155C22⋊C4—1061no—
156C42—1061no—
157C23—1061no—
158C23C2×D21211yescharacteristic
159Q8—2121no—
160D4—2121no—
161D4—2121no—
162D4—2121no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—1061no—
170C2×C4—1061no—
171C2×C4—1061no—
172C2×C4—4241no—
173C2×C4—2121no—
174C2×C4—2121no—
175C2×C4—2122no—
176C2×C4—2121no—
177C2×C4—2121no—
178C2×C4D4×D5311yescharacteristic
179C2×C4D4×D5311yescharacteristic
180C8—41no—
181C8—41no—
182C22—21no—
183C22—21no—
184C22—2121no—
185C22—2121no—
186C22—2121no—
187C22#1313318711yescharacteristic
188C22#1313323911yescharacteristic
189C22#1313325011yescharacteristic
190C4—21no—
191C4—21no—
192C4—41no—
193C4—41no—
194C4—4241no—
195C4—4241no—
196C4—2121no—
197C4—2121no—
198C4—2121no—
199C2—21no—
200C2—21no—
201C2—4241no—
202C2#1681078711yescharacteristic
203C2#1681079811yescharacteristic
204C2#1681081411yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#324C2
socleC22×C53#120
hypercenter#324D53
hypocenterC53#1615
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. #16810959C2semidirect
4. #16810562C2semidirect
5. #16810361C2semidirect
7. C53×#324C2semidirect
109. #324D53semidirect
110. C53#1615semidirect