Finite Group Databasebeta

← #20273643

Subgroups — #20273643

Order 6784.

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

normal subgroups only

This group has 272 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, complemented
2#16810757C211yesmaximal, characteristic
3#16810537C211yesmaximal, characteristic, complemented
4#16810360C211yesmaximal, characteristic, complemented
5#16810358C211yesmaximal, characteristic, complemented
6#16810348C211yesmaximal, characteristic, complemented
7C53×#450C211yesmaximal, characteristic, complemented
8C2×#11719922—21no—
9#13133212—21no—
10#13133208—21no—
11#13133202—21no—
12#13133201—21no—
13#13133193—21no—
14#13133191C2211yescharacteristic
15#13133177—21no—
16C2×#11719919C2211yescharacteristic
17C2×C22⋊Dic53—21no—
18C2×C22⋊Dic53—21no—
19#13133138—21no—
20C22⋊#88322C2211yescharacteristic
21#13133124C2211yescharacteristic
22C22×D4×C53C2211yescharacteristic
23C2×SD16×C53—21no—
24C22⋊Q8×C53C2211yescharacteristic
25#102×C53—21no—
26C22⋊C8×C53C2211yescharacteristic
27C24×C53—21no—
28C2×Q8×C53D411yescharacteristic
29C2×D4×C53—21no—
30C2×D4×C53—21no—
31C2×D4×C53—42no—
32C22×C4×C53C2311yescharacteristic
33C2×C22⋊D53—21no—
34C22×Dic53—21no—
35C22×Dic53D411yescharacteristic
36C22×Dic53D411yescharacteristic
37C2×D212D411yescharacteristic
38C2×C4×D53—21no—
39SD16×C53—42no—
40C2×C8×C53—21no—
41C4⋊C4×C53—21no—
42C4⋊C4×C53D411yescharacteristic
43C22⋊C4×C53—21no—
44C22⋊Dic53—42no—
45C22⋊Dic53—42no—
46#11719922—42no—
47#11719921—21no—
48#11719921—21no—
49#11719921—21no—
50#11719921—21no—
51C4⋊Dic53—21no—
52C4⋊Dic53—21no—
53C4⋊Dic53D411yescharacteristic
54#11719919—21no—
55#11719919—21no—
56#11719919—21no—
57C4×Dic53—21no—
58C2×#88322—21no—
59C23×C53—22no—
60C23×C53—41no—
61C23×C53—41no—
62C23×C53—41no—
63C23×C53—41no—
64C23×C53—41no—
65C23×C53—41no—
66C23×C53C2×D411yescharacteristic
67C22×D53—21no—
68Q8×C53—41no—
69D4×C53—22no—
70D4×C53—22no—
71D4×C53—41no—
72D4×C53—41no—
73C2×C4×C53—21no—
74C2×C4×C53—21no—
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—21no—
83C2×Dic53—21no—
84C2×Dic53—21no—
85C2×Dic53—41no—
86C2×Dic53—41no—
87C2×Dic53—41no—
88C2×Dic53—41no—
89D212—41no—
90C4×D53—81no—
91C8×C53—41no—
92#88322—41no—
93C22×C53—21no—
94C22×C53—21no—
95C22×C53—42no—
96C22×C53—42no—
97C22×C53—42no—
98C22×C53—81no—
99C22×C53—81no—
100C22×C53—81no—
101C22×C53—81no—
102C22×C53#12011yescharacteristic
103C22×C53#13511yescharacteristic
104C22×C53AGL(1,ℤ/8)11yescharacteristic
105C2×D53—41no—
106C2×D53—41no—
107C2×D53—41no—
108C4×C53—21no—
109C4×C53—21no—
110C4×C53—41no—
111C4×C53—41no—
112Dic53—41no—
113Dic53—41no—
114Dic53—41no—
115Dic53—81no—
116#1597—531nomaximal, Sylow
117C2×C53—21no—
118C2×C53—21no—
119C2×C53—82no—
120C2×C53#44911yescharacteristic
121C2×C53#45411yescharacteristic
122C2×C53B(4,𝔽2)11yescharacteristic
123D53—81no—
124#539—531no—
125#450D5311yescharacteristic, complemented
126#447—531no—
127#394—531no—
128#328—531no—
129#327—531no—
130#324—531no—
131C53#159711yescharacteristic, complemented, Sylow
132C22×D4C2×D5311yescharacteristic
133C2×SD16—21no—
134C2×D8—1061no—
135#126—1061no—
136#125—1061no—
137#123—1061no—
138#123—1061no—
139C22⋊Q8—1061no—
140C22⋊Q8C2×D5311yescharacteristic
141#121—531no—
142C4×D4—1061no—
143C2×C4⋊C4—531no—
144C2×C22⋊C4—1061no—
145C2×C22⋊C4—1061no—
146#102—21no—
147#102—1061no—
148C22⋊C8—531no—
149C22⋊C8C2×D5311yescharacteristic
150#95—531no—
151C24—21no—
152C2×Q8C22⋊D5311yescharacteristic
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—42no—
156C2×D4—531no—
157C2×D4—1061no—
158C22×C4—531no—
159C22×C4—531no—
160C22×C4—1061no—
161C22×C4—1061no—
162C22×C4C22×D5311yescharacteristic
163SD16—42no—
164D8—2122no—
165C2×C8—21no—
166C2×C8—1061no—
167C4⋊C4—21no—
168C4⋊C4—531no—
169C4⋊C4—1061no—
170C4⋊C4—1061no—
171C4⋊C4—1061no—
172C4⋊C4—1061no—
173C4⋊C4—1061no—
174C4⋊C4C22⋊D5311yescharacteristic
175C22⋊C4—21no—
176C22⋊C4—1061no—
177C22⋊C4—1061no—
178C22⋊C4—1061no—
179C22⋊C4—1061no—
180C22⋊C4—2122no—
181C22⋊C4—2122no—
182C42—1061no—
183C23—22no—
184C23—41no—
185C23—41no—
186C23—41no—
187C23—41no—
188C23—41no—
189C23—41no—
190C23—1061no—
191C23D4×D5311yescharacteristic
192Q8—41no—
193D4—22no—
194D4—22no—
195D4—41no—
196D4—41no—
197D4—2121no—
198D4—2121no—
199D4—2121no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—41no—
203C2×C4—1061no—
204C2×C4—1061no—
205C2×C4—1061no—
206C2×C4—1061no—
207C2×C4—1061no—
208C2×C4—4241no—
209C2×C4—2121no—
210C2×C4—2121no—
211C2×C4—2121no—
212C2×C4—2121no—
213C2×C4D4×D5311yescharacteristic
214C2×C4C2×C22⋊D5311yescharacteristic
215C8—41no—
216C8—2121no—
217C22—21no—
218C22—21no—
219C22—42no—
220C22—42no—
221C22—42no—
222C22—81no—
223C22—81no—
224C22—81no—
225C22—81no—
226C22—2121no—
227C22—2121no—
228C22—2121no—
229C22#1313318511yescharacteristic
230C22#1313323811yescharacteristic
231C22#1313324411yescharacteristic
232C4—21no—
233C4—21no—
234C4—41no—
235C4—41no—
236C4—4241no—
237C4—2121no—
238C4—2121no—
239C4—2121no—
240C2—21no—
241C2—21no—
242C2—82no—
243C2—4241no—
244C2#1681076711yescharacteristic
245C2#1681079411yescharacteristic
246C2#1681080911yescharacteristic
subgroupG/H
centerC22#13133185
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#450C2
socleC22×C53#120
hypercenter#450D53
hypocenterC53#1597
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #16811222C2semidirect
3. #16810537C2semidirect
4. #16810360C2semidirect
5. #16810358C2semidirect
6. #16810348C2semidirect
7. C53×#450C2semidirect
125. #450D53semidirect
131. C53#1597semidirect