Finite Group Databasebeta

← #20273677

Subgroups — #20273677

Order 6784.

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

normal subgroups only

This group has 244 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#16811222C211yesmaximal, characteristic
2#16810762C211yesmaximal, characteristic
3#16810542C211yesmaximal, characteristic, complemented
4#16810360C211yesmaximal, characteristic, complemented
5#16810358C211yesmaximal, characteristic
6#16810348C211yesmaximal, characteristic, complemented
7C53×#451C211yesmaximal, characteristic, complemented
8C2×#11719924—21no—
9#13133212—21no—
10#13133208—21no—
11#13133202—21no—
12#13133201—21no—
13#13133193—21no—
14#13133191C2211yescharacteristic
15#13133177—21no—
16C2×#11719919—21no—
17C2×#11719919C2211yescharacteristic
18#13133141—21no—
19C22⋊#88322C2211yescharacteristic
20#13133124—21no—
21#13133124C2211yescharacteristic
22C22×Q8×C53C2211yescharacteristic
23C2×Q16×C53—21no—
24C22⋊Q8×C53C2211yescharacteristic
25#103×C53—21no—
26C22⋊C8×C53C2211yescharacteristic
27C2×Q8×C53—21no—
28C2×Q8×C53—21no—
29C2×Q8×C53—42no—
30C2×Q8×C53D411yescharacteristic
31C22×C4×C53—21no—
32C22×C4×C53C2311yescharacteristic
33C2×C22⋊D53—21no—
34C22×Dic53—21no—
35C22×Dic53D411yescharacteristic
36C22×Dic53D411yescharacteristic
37C2×D212D411yescharacteristic
38C2×C4×D53—21no—
39Q16×C53—42no—
40C2×C8×C53—21no—
41C4⋊C4×C53—21no—
42C4⋊C4×C53D411yescharacteristic
43C22⋊C4×C53—21no—
44#11719924—42no—
45#11719921—21no—
46#11719921—21no—
47#11719921—21no—
48#11719921—21no—
49C4⋊Dic53—21no—
50C4⋊Dic53—21no—
51C4⋊Dic53D411yescharacteristic
52#11719919—21no—
53#11719919—21no—
54#11719919—21no—
55#11719919—42no—
56C4×Dic53—21no—
57C2×#88322—21no—
58C23×C53C2×D411yescharacteristic
59C22×D53—21no—
60Q8×C53—22no—
61Q8×C53—22no—
62Q8×C53—41no—
63Q8×C53—41no—
64Q8×C53—41no—
65C2×C4×C53—22no—
66C2×C4×C53—21no—
67C2×C4×C53—21no—
68C2×C4×C53—41no—
69C2×C4×C53—41no—
70C2×C4×C53—41no—
71C2×C4×C53C2×D411yescharacteristic
72C2×C4×C53C2×D411yescharacteristic
73C22⋊D53—41no—
74C22⋊D53—41no—
75C2×Dic53—22no—
76C2×Dic53—21no—
77C2×Dic53—21no—
78C2×Dic53—21no—
79C2×Dic53—21no—
80C2×Dic53—21no—
81C2×Dic53—41no—
82C2×Dic53—41no—
83C2×Dic53—41no—
84D212—41no—
85C4×D53—81no—
86C8×C53—41no—
87#88322—41no—
88C22×C53—21no—
89C22×C53—21no—
90C22×C53#12011yescharacteristic
91C22×C53#13511yescharacteristic
92C22×C53#13711yescharacteristic
93C2×D53—41no—
94C2×D53—41no—
95C2×D53—41no—
96C4×C53—21no—
97C4×C53—21no—
98C4×C53—41no—
99C4×C53—41no—
100C4×C53—42no—
101Dic53—42no—
102Dic53—41no—
103Dic53—41no—
104Dic53—41no—
105#1603—531nomaximal, Sylow
106C2×C53—21no—
107C2×C53—21no—
108C2×C53#45211yescharacteristic
109C2×C53#45611yescharacteristic
110C2×C53B(4,𝔽2)11yescharacteristic
111D53—81no—
112#539—531no—
113#451D5311yescharacteristic, complemented
114#448—531no—
115#395—531no—
116#328—531no—
117#327—531no—
118#324—531no—
119C53#160311yescharacteristic, complemented, Sylow
120C22×Q8C2×D5311yescharacteristic
121C2×Q16—21no—
122C2×SD16—1061no—
123#126—1061no—
124#125—1061no—
125#123—1061no—
126#123—1061no—
127C22⋊Q8—1061no—
128C22⋊Q8C2×D5311yescharacteristic
129#121—531no—
130C4×D4—1061no—
131C2×C4⋊C4—531no—
132C2×C4⋊C4—1061no—
133#103—21no—
134#103—1061no—
135C22⋊C8—531no—
136C22⋊C8C2×D5311yescharacteristic
137#95—531no—
138#95—1061no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—42no—
142C2×Q8C22⋊D5311yescharacteristic
143C2×D4—531no—
144C2×D4—1061no—
145C22×C4—21no—
146C22×C4—531no—
147C22×C4—531no—
148C22×C4—1061no—
149C22×C4—1061no—
150C22×C4C22×D5311yescharacteristic
151Q16—42no—
152SD16—2122no—
153C2×C8—21no—
154C2×C8—1061no—
155C4⋊C4—21no—
156C4⋊C4—531no—
157C4⋊C4—1061no—
158C4⋊C4—1061no—
159C4⋊C4—1061no—
160C4⋊C4—1061no—
161C4⋊C4—1061no—
162C4⋊C4—2122no—
163C4⋊C4C22⋊D5311yescharacteristic
164C22⋊C4—21no—
165C22⋊C4—1061no—
166C22⋊C4—1061no—
167C22⋊C4—1061no—
168C22⋊C4—1061no—
169C42—1061no—
170C23—1061no—
171C23D4×D5311yescharacteristic
172Q8—22no—
173Q8—22no—
174Q8—41no—
175Q8—41no—
176Q8—41no—
177D4—2121no—
178D4—2121no—
179D4—2121no—
180C2×C4—22no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—1062no—
187C2×C4—1061no—
188C2×C4—1061no—
189C2×C4—1061no—
190C2×C4—1061no—
191C2×C4—1061no—
192C2×C4—4241no—
193C2×C4—2121no—
194C2×C4—2121no—
195C2×C4—2121no—
196C2×C4D4×D5311yescharacteristic
197C2×C4C2×C22⋊D5311yescharacteristic
198C8—41no—
199C8—2121no—
200C22—21no—
201C22—21no—
202C22—2121no—
203C22—2121no—
204C22—2121no—
205C22#1313318511yescharacteristic
206C22#1313324011yescharacteristic
207C22#1313324911yescharacteristic
208C4—21no—
209C4—21no—
210C4—41no—
211C4—42no—
212C4—41no—
213C4—2122no—
214C4—2121no—
215C4—2121no—
216C4—2121no—
217C2—21no—
218C2—21no—
219C2—4241no—
220C2#1681078411yescharacteristic
221C2#1681080411yescharacteristic
222C2#1681080911yescharacteristic
subgroupG/H
centerC22#13133185
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#451C2
socleC22×C53#120
hypercenter#451D53
hypocenterC53#1603
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
3. #16810542C2semidirect
4. #16810360C2semidirect
6. #16810348C2semidirect
7. C53×#451C2semidirect
113. #451D53semidirect
119. C53#1603semidirect