Finite Group Databasebeta

← #20273644

Subgroups — #20273644

Order 6784.

272 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 53 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#16810773C211yesmaximal, characteristic, complemented
2#16810756C211yesmaximal, characteristic
3#16810540C211yesmaximal, characteristic, complemented
4#16810360C211yesmaximal, characteristic, complemented
5#16810357C211yesmaximal, characteristic, complemented
6#16810349C211yesmaximal, characteristic, complemented
7C53×#539C211yesmaximal, characteristic, complemented
8C22×D212C2211yescharacteristic
9C2×#11719924—21no—
10C2×#11719922—21no—
11C2×#11719921—21no—
12C2×#11719921—21no—
13#13133137—21no—
14#13133137—21no—
15C22⋊#88322C2211yescharacteristic
16C22⋊#88322C2211yescharacteristic
17#13133124C2211yescharacteristic
18#126×C53—21no—
19#125×C53—21no—
20#123×C53—21no—
21#123×C53—21no—
22C22⋊Q8×C53—21no—
23C22⋊Q8×C53C2211yescharacteristic
24#121×C53C2211yescharacteristic
25C4×D4×C53—21no—
26C2×C4⋊C4×C53C2211yescharacteristic
27C23×D53—21no—
28C2×Q8×C53D411yescharacteristic
29C2×D4×C53—21no—
30C2×D4×C53D411yescharacteristic
31C22×C4×C53—21no—
32C22×C4×C53D411yescharacteristic
33C22×C4×C53D411yescharacteristic
34C22×C4×C53C2311yescharacteristic
35C22×Dic53—21no—
36C2×D212—21no—
37C2×D212—21no—
38C2×D212—42no—
39C4⋊C4×C53—21no—
40C4⋊C4×C53—21no—
41C4⋊C4×C53—21no—
42C4⋊C4×C53—21no—
43C4⋊C4×C53—21no—
44C4⋊C4×C53—21no—
45C4⋊C4×C53D411yescharacteristic
46C4⋊C4×C53D411yescharacteristic
47C22⋊C4×C53—21no—
48C22⋊C4×C53—21no—
49C22⋊C4×C53—21no—
50C22⋊C4×C53—21no—
51C22⋊C4×C53—21no—
52C42×C53—21no—
53#11719924—42no—
54#11719922—42no—
55#11719921—42no—
56#11719921—42no—
57C2×#88322—21no—
58C2×#88322—21no—
59C23×C53—21no—
60C23×C53C2×D411yescharacteristic
61C22×D53—22no—
62C22×D53—41no—
63C22×D53—41no—
64C22×D53—41no—
65C22×D53—41no—
66C22×D53—41no—
67C22×D53—41no—
68Q8×C53—41no—
69D4×C53—41no—
70D4×C53—41no—
71D4×C53—41no—
72C2×C4×C53—21no—
73C2×C4×C53—21no—
74C2×C4×C53—21no—
75C2×C4×C53—21no—
76C2×C4×C53—21no—
77C2×C4×C53—21no—
78C2×C4×C53—21no—
79C2×C4×C53—41no—
80C2×C4×C53—41no—
81C2×C4×C53—81no—
82C2×C4×C53C2×D411yescharacteristic
83C2×C4×C53C2×D411yescharacteristic
84C2×Dic53—41no—
85C2×Dic53—41no—
86C2×Dic53—41no—
87D212—22no—
88D212—22no—
89D212—41no—
90D212—41no—
91#88322—41no—
92#88322—41no—
93C22×C53—21no—
94C22×C53—21no—
95C22×C53—41no—
96C22×C53—41no—
97C22×C53—41no—
98C22×C53#12011yescharacteristic
99C22×C53#13511yescharacteristic
100C22×C53AGL(1,ℤ/8)11yescharacteristic
101C2×D53—42no—
102C2×D53—42no—
103C2×D53—42no—
104C2×D53—81no—
105C2×D53—81no—
106C2×D53—81no—
107C2×D53—81no—
108C4×C53—21no—
109C4×C53—21no—
110C4×C53—41no—
111C4×C53—41no—
112C4×C53—41no—
113C4×C53—41no—
114C4×C53—41no—
115Dic53—81no—
116#1597—531nomaximal, Sylow
117C2×C53—21no—
118C2×C53—21no—
119C2×C53—81no—
120C2×C53#44911yescharacteristic
121C2×C53#45411yescharacteristic
122C2×C53B(4,𝔽2)11yescharacteristic
123D53—82no—
124#539D5311yescharacteristic, complemented
125#450—531no—
126#447—531no—
127#394—531no—
128#328—531no—
129#327—531no—
130#324—531no—
131C53#159711yescharacteristic, complemented, Sylow
132C22×D4—531no—
133C2×SD16—1061no—
134C2×D8—1061no—
135#126—21no—
136#125—21no—
137#123—21no—
138#123—21no—
139C22⋊Q8—21no—
140C22⋊Q8C2×D5311yescharacteristic
141#121C2×D5311yescharacteristic
142C4×D4—21no—
143C2×C4⋊C4C2×D5311yescharacteristic
144C2×C22⋊C4—1061no—
145C2×C22⋊C4—1061no—
146#102—1061no—
147#102—1061no—
148C22⋊C8—531no—
149C22⋊C8—531no—
150#95—531no—
151C24—1061no—
152C2×Q8C22⋊D5311yescharacteristic
153C2×D4—21no—
154C2×D4—1061no—
155C2×D4—1061no—
156C2×D4—2122no—
157C2×D4C22⋊D5311yescharacteristic
158C22×C4—21no—
159C22×C4—1061no—
160C22×C4C22⋊D5311yescharacteristic
161C22×C4C22⋊D5311yescharacteristic
162C22×C4C22×D5311yescharacteristic
163SD16—2122no—
164D8—2122no—
165C2×C8—1061no—
166C2×C8—1061no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4C22⋊D5311yescharacteristic
174C4⋊C4C22⋊D5311yescharacteristic
175C22⋊C4—21no—
176C22⋊C4—21no—
177C22⋊C4—21no—
178C22⋊C4—21no—
179C22⋊C4—21no—
180C22⋊C4—2122no—
181C22⋊C4—2122no—
182C42—21no—
183C23—21no—
184C23—1062no—
185C23—2121no—
186C23—2121no—
187C23—2121no—
188C23—2121no—
189C23—2121no—
190C23—2121no—
191C23C2×C22⋊D5311yescharacteristic
192Q8—41no—
193D4—41no—
194D4—41no—
195D4—41no—
196D4—1062no—
197D4—1062no—
198D4—2121no—
199D4—2121no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—81no—
210C2×C4—2121no—
211C2×C4—2121no—
212C2×C4—2121no—
213C2×C4C2×C22⋊D5311yescharacteristic
214C2×C4C2×C22⋊D5311yescharacteristic
215C8—2121no—
216C8—2121no—
217C22—21no—
218C22—21no—
219C22—41no—
220C22—41no—
221C22—41no—
222C22—4241no—
223C22—4241no—
224C22—4241no—
225C22—4241no—
226C22—2122no—
227C22—2122no—
228C22—2122no—
229C22#1313318611yescharacteristic
230C22#1313323711yescharacteristic
231C22#1313324211yescharacteristic
232C4—21no—
233C4—21no—
234C4—41no—
235C4—41no—
236C4—41no—
237C4—41no—
238C4—41no—
239C4—4241no—
240C2—21no—
241C2—21no—
242C2—81no—
243C2—4242no—
244C2#1681076811yescharacteristic
245C2#1681079611yescharacteristic
246C2#1681081011yescharacteristic
subgroupG/H
centerC22#13133186
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#539C2
socleC22×C53#120
hypercenter#539D53
hypocenterC53#1597
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #539

Complements

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

NH ≅ G/Ntype
1. #16810773C2semidirect
3. #16810540C2semidirect
4. #16810360C2semidirect
5. #16810357C2semidirect
6. #16810349C2semidirect
7. C53×#539C2semidirect
124. #539D53semidirect
131. C53#1597semidirect