Finite Group Databasebeta

← #20273694

Subgroups — #20273694

Order 6784.

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

normal subgroups only

This group has 260 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#16811213C211yesmaximal, characteristic
2#16810762C211yesmaximal, characteristic, complemented
3#16810534C211yesmaximal, characteristic, complemented
4#16810360C211yesmaximal, characteristic, complemented
5#16810358C211yesmaximal, characteristic, complemented
6#16810346C211yesmaximal, characteristic, complemented
7C53×#451C211yesmaximal, characteristic, complemented
8C2×#11719924—21no—
9#13133212—21no—
10#13133206—21no—
11#13133202—21no—
12#13133192—21no—
13#13133191C2211yescharacteristic
14#13133187—21no—
15#13133177—21no—
16C2×#11719921—21no—
17C2×#11719921C2211yescharacteristic
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
27C23×D53D411yescharacteristic
28C2×Q8×C53—21no—
29C2×Q8×C53—21no—
30C2×Q8×C53—42no—
31C2×Q8×C53D411yescharacteristic
32C22×C4×C53—21no—
33C22×C4×C53C2311yescharacteristic
34C2×C22⋊D53—21no—
35C22×Dic53—21no—
36C22×Dic53D411yescharacteristic
37C2×D212—21no—
38C2×D212—21no—
39C2×D212D411yescharacteristic
40C2×C4×D53—21no—
41Q16×C53—42no—
42C2×C8×C53—21no—
43C4⋊C4×C53—21no—
44C4⋊C4×C53D411yescharacteristic
45C22⋊C4×C53—21no—
46#11719924—42no—
47#11719921—21no—
48#11719921—21no—
49#11719921—21no—
50#11719921—21no—
51#11719921—21no—
52#11719921—21no—
53#11719921—42no—
54C4⋊Dic53D411yescharacteristic
55#11719919—21no—
56C4×Dic53—21no—
57C2×#88322—21no—
58C23×C53C2×D411yescharacteristic
59C22×D53—21no—
60C22×D53—21no—
61C22×D53—21no—
62C22×D53—21no—
63C22×D53—41no—
64C22×D53—41no—
65Q8×C53—22no—
66Q8×C53—22no—
67Q8×C53—41no—
68Q8×C53—41no—
69Q8×C53—41no—
70C2×C4×C53—22no—
71C2×C4×C53—21no—
72C2×C4×C53—21no—
73C2×C4×C53—41no—
74C2×C4×C53—41no—
75C2×C4×C53—41no—
76C2×C4×C53C2×D411yescharacteristic
77C2×C4×C53C2×D411yescharacteristic
78C22⋊D53—41no—
79C22⋊D53—41no—
80C2×Dic53—21no—
81C2×Dic53—21no—
82C2×Dic53—41no—
83C2×Dic53—41no—
84C2×Dic53—41no—
85C2×Dic53—41no—
86D212—41no—
87D212—81no—
88D212—81no—
89C4×D53—81no—
90C8×C53—41no—
91#88322—41no—
92C22×C53—21no—
93C22×C53—21no—
94C22×C53#12011yescharacteristic
95C22×C53#13511yescharacteristic
96C22×C53#13711yescharacteristic
97C2×D53—42no—
98C2×D53—41no—
99C2×D53—41no—
100C2×D53—41no—
101C2×D53—41no—
102C2×D53—41no—
103C2×D53—41no—
104C2×D53—81no—
105C4×C53—21no—
106C4×C53—21no—
107C4×C53—41no—
108C4×C53—41no—
109C4×C53—42no—
110Dic53—41no—
111Dic53—41no—
112Dic53—81no—
113#1606—531nomaximal, Sylow
114C2×C53—21no—
115C2×C53—21no—
116C2×C53#45211yescharacteristic
117C2×C53#45511yescharacteristic
118C2×C53B(4,𝔽2)11yescharacteristic
119D53—81no—
120D53—81no—
121#538—531no—
122#451D5311yescharacteristic, complemented
123#448—531no—
124#393—531no—
125#328—531no—
126#327—531no—
127#323—531no—
128C53#160611yescharacteristic, complemented, Sylow
129C22×Q8C2×D5311yescharacteristic
130C2×Q16—21no—
131C2×SD16—1061no—
132#126—1061no—
133#124—1061no—
134#123—1061no—
135C22⋊Q8C2×D5311yescharacteristic
136#121—531no—
137#121—1061no—
138#120—1061no—
139C4×D4—1061no—
140C2×C22⋊C4—531no—
141C2×C22⋊C4—1061no—
142#103—21no—
143#103—1061no—
144C22⋊C8—531no—
145C22⋊C8C2×D5311yescharacteristic
146#95—531no—
147#95—1061no—
148C24—531no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—42no—
152C2×Q8C22⋊D5311yescharacteristic
153C2×D4—531no—
154C2×D4—1061no—
155C2×D4—1061no—
156C2×D4—1061no—
157C22×C4—21no—
158C22×C4—531no—
159C22×C4—1061no—
160C22×C4—1061no—
161C22×C4C22×D5311yescharacteristic
162Q16—42no—
163SD16—2122no—
164C2×C8—21no—
165C2×C8—1061no—
166C4⋊C4—21no—
167C4⋊C4—531no—
168C4⋊C4—1061no—
169C4⋊C4C22⋊D5311yescharacteristic
170C22⋊C4—21no—
171C22⋊C4—1061no—
172C22⋊C4—1061no—
173C22⋊C4—1061no—
174C22⋊C4—1061no—
175C22⋊C4—1061no—
176C22⋊C4—1061no—
177C22⋊C4—2122no—
178C42—1061no—
179C23—1061no—
180C23—1061no—
181C23—1061no—
182C23—1061no—
183C23—2121no—
184C23—2121no—
185C23D4×D5311yescharacteristic
186Q8—22no—
187Q8—22no—
188Q8—41no—
189Q8—41no—
190Q8—41no—
191D4—4241no—
192D4—4241no—
193D4—2121no—
194D4—2121no—
195D4—2121no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—1061no—
203C2×C4—1061no—
204C2×C4—4241no—
205C2×C4—2121no—
206C2×C4—2121no—
207C2×C4—2121no—
208C2×C4—2121no—
209C2×C4D4×D5311yescharacteristic
210C2×C4C2×C22⋊D5311yescharacteristic
211C8—41no—
212C8—2121no—
213C22—21no—
214C22—21no—
215C22—4241no—
216C22—2122no—
217C22—2121no—
218C22—2121no—
219C22—2121no—
220C22—2121no—
221C22—2121no—
222C22—2121no—
223C22#1313318511yescharacteristic
224C22#1313324011yescharacteristic
225C22#1313324911yescharacteristic
226C4—21no—
227C4—21no—
228C4—41no—
229C4—42no—
230C4—41no—
231C4—4241no—
232C4—2121no—
233C4—2121no—
234C2—21no—
235C2—21no—
236C2—4241no—
237C2—4241no—
238C2#1681078411yescharacteristic
239C2#1681079911yescharacteristic
240C2#1681080911yescharacteristic
subgroupG/H
centerC22#13133185
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#451C2
socleC22×C53#120
hypercenter#451D53
hypocenterC53#1606
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
2. #16810762C2semidirect
3. #16810534C2semidirect
4. #16810360C2semidirect
5. #16810358C2semidirect
6. #16810346C2semidirect
7. C53×#451C2semidirect
122. #451D53semidirect
128. C53#1606semidirect