Finite Group Databasebeta

← #20273690

Subgroups — #20273690

Order 6784.

238 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 51 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#16811252C211yesmaximal, characteristic
2#16810781C211yesmaximal, characteristic
3#16810778C211yesmaximal, characteristic
4#16810544C211yesmaximal, characteristic
5#16810359C211yesmaximal, characteristic
6#16810348C211yesmaximal, characteristic
7C53×#328C211yesmaximal, characteristic
8C22×Dic106C2211yescharacteristic
9C2×Dic212—21no—
10C2×#11719925—21no—
11#13133216—21no—
12#13133215—21no—
13#13133207—21no—
14#13133207—21no—
15#13133196C2211yescharacteristic
16#13133194—21no—
17#13133173—21no—
18C2×C4⋊Dic53—21no—
19C2×#11719919C2211yescharacteristic
20#13133142—21no—
21#13133140—21no—
22C22⋊#88322C2211yescharacteristic
23#13133124—21no—
24C22⋊Q8×C53C2211yescharacteristic
25C22⋊C8×C53C2211yescharacteristic
26#95×C53C2211yescharacteristic
27C2×Q8×C53D411yescharacteristic
28C22×C4×C53—21no—
29C22×C4×C53C2311yescharacteristic
30C22×Dic53—21no—
31C22×Dic53D411yescharacteristic
32C22×Dic53D411yescharacteristic
33C2×Dic106—21no—
34C2×Dic106—21no—
35C2×Dic106—21no—
36C2×Dic106—42no—
37C2×Dic106D411yescharacteristic
38C2×C8×C53—21no—
39C4⋊C4×C53—21no—
40C4⋊C4×C53D411yescharacteristic
41C22⋊C4×C53—21no—
42C22⋊Dic53—21no—
43#11719925—42no—
44C4⋊Dic53—21no—
45C4⋊Dic53—42no—
46C4⋊Dic53D411yescharacteristic
47#11719919—21no—
48#11719919—21no—
49#11719919—21no—
50#11719919—21no—
51#11719919—21no—
52#11719919—21no—
53#11719919—21no—
54C4×Dic53—21no—
55C4×Dic53—21no—
56C2×#88322—21no—
57Dic212—42no—
58C23×C53C2×D411yescharacteristic
59Q8×C53—41no—
60C2×C4×C53—22no—
61C2×C4×C53—21no—
62C2×C4×C53—21no—
63C2×C4×C53—41no—
64C2×C4×C53—41no—
65C2×C4×C53—41no—
66C2×C4×C53C2×D411yescharacteristic
67C2×C4×C53C2×D411yescharacteristic
68C2×Dic53—21no—
69C2×Dic53—21no—
70C2×Dic53—22no—
71C2×Dic53—21no—
72C2×Dic53—21no—
73C2×Dic53—21no—
74C2×Dic53—41no—
75C2×Dic53—41no—
76C2×Dic53—41no—
77C2×Dic53Q1612yescomplemented
78Dic106—22no—
79Dic106—22no—
80Dic106—41no—
81Dic106—41no—
82Dic106—41no—
83Dic106—81no—
84C8×C53—41no—
85#88322—41no—
86C22×C53—21no—
87C22×C53—21no—
88C22×C53#12011yescharacteristic
89C22×C53C2×Q1611yescharacteristic
90C22×C53#13711yescharacteristic
91C4×C53—21no—
92C4×C53—21no—
93C4×C53—42no—
94C4×C53—41no—
95C4×C53—41no—
96Dic53—22no—
97Dic53—41no—
98Dic53—42no—
99Dic53—41no—
100Dic53—41no—
101#1605—531nomaximal, Sylow
102C2×C53—21no—
103C2×C53—21no—
104C2×C53#45111yescharacteristic
105C2×C53#45611yescharacteristic
106C2×C53B(4,𝔽2)11yescharacteristic
107#544—531no—
108#451—531no—
109#451—531no—
110#395—531no—
111#328—531no—
112#328D5311yescharacteristic
113#324—531no—
114C53#160511yescharacteristic, complemented, Sylow
115C22×Q8—531no—
116C2×Q16—1061no—
117C2×Q16—1061no—
118C4⋊Q8—1061no—
119C4⋊Q8—1061no—
120#125—1061no—
121#125—1061no—
122C22⋊Q8—531no—
123C22⋊Q8—1061no—
124C22⋊Q8C2×D5311yescharacteristic
125#117—1061no—
126C2×C4⋊C4—531no—
127C2×C4⋊C4—1061no—
128#103—1061no—
129#103—1061no—
130C22⋊C8—531no—
131C22⋊C8C2×D5311yescharacteristic
132#95—1061no—
133#95C2×D5311yescharacteristic
134C2×Q8—531no—
135C2×Q8—1061no—
136C2×Q8—1061no—
137C2×Q8—1061no—
138C2×Q8—2122no—
139C2×Q8D21211yescharacteristic
140C22×C4—21no—
141C22×C4—531no—
142C22×C4—531no—
143C22×C4—1061no—
144C22×C4C22×D5311yescharacteristic
145Q16—2122no—
146Q16—2122no—
147C2×C8—21no—
148C2×C8—1061no—
149C4⋊C4—21no—
150C4⋊C4—531no—
151C4⋊C4—1061no—
152C4⋊C4—1061no—
153C4⋊C4—1061no—
154C4⋊C4—1061no—
155C4⋊C4—1061no—
156C4⋊C4—1061no—
157C4⋊C4—1061no—
158C4⋊C4—1061no—
159C4⋊C4—2122no—
160C4⋊C4D21211yescharacteristic
161C22⋊C4—21no—
162C22⋊C4—1061no—
163C42—1061no—
164C42—1061no—
165C23D4×D5311yescharacteristic
166Q8—41no—
167Q8—1062no—
168Q8—1062no—
169Q8—4241no—
170Q8—2121no—
171Q8—2121no—
172Q8—2121no—
173C2×C4—22no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—532no—
180C2×C4—1061no—
181C2×C4—1061no—
182C2×C4—1062no—
183C2×C4—1061no—
184C2×C4—1061no—
185C2×C4—1061no—
186C2×C4—2121no—
187C2×C4—2121no—
188C2×C4—2121no—
189C2×C4C2×D21211yescharacteristic
190C2×C4D4×D5311yescharacteristic
191C8—41no—
192C8—2121no—
193C22—21no—
194C22—21no—
195C22#1313318711yescharacteristic
196C22Q16×D5311yescharacteristic
197C22#1313324611yescharacteristic
198C4—21no—
199C4—21no—
200C4—42no—
201C4—41no—
202C4—41no—
203C4—1062no—
204C4—2121no—
205C4—2122no—
206C4—2121no—
207C4—2121no—
208C2—21no—
209C2—21no—
210C2#1681078011yescharacteristic
211C2#1681080711yescharacteristic
212C2#1681081111yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#328C2
socleC22×C53#120
hypercenter#328D53
hypocenterC53#1605
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
77. C2×Dic53Q16semidirect
114. C53#1605semidirect