Finite Group Databasebeta

← #20273659

Subgroups — #20273659

Order 6784.

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

normal subgroups only

This group has 320 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#16811211C211yesmaximal, characteristic, complemented
2#16810774C211yesmaximal, characteristic, complemented
3#16810757C211yesmaximal, characteristic, complemented
4#16810359C211yesmaximal, characteristic, complemented
5#16810356C211yesmaximal, characteristic, complemented
6#16810347C211yesmaximal, characteristic, complemented
7C53×#392C211yesmaximal, characteristic, complemented
8C2×#11719923—21no—
9C2×#11719922—21no—
10#13133210—21no—
11#13133203—21no—
12#13133199—21no—
13#13133196C2211yescharacteristic
14#13133191C2211yescharacteristic
15#13133190—21no—
16#13133186—21no—
17C4×C22⋊D53—21no—
18#13133138—21no—
19#13133138—21no—
20C22⋊#88322C2211yescharacteristic
21C22⋊#88322C2211yescharacteristic
22C22×D4×C53—21no—
23C22×D4×C53C2211yescharacteristic
24C2×C22⋊C4×C53—21no—
25C2×C22⋊C4×C53C2211yescharacteristic
26#95×C53C2211yescharacteristic
27C24×C53—21no—
28C24×C53D411yescharacteristic
29C2×D4×C53—21no—
30C2×D4×C53—21no—
31C2×D4×C53—42no—
32C2×D4×C53—42no—
33C2×D4×C53—42no—
34C2×D4×C53—42no—
35C22×C4×C53—21no—
36C22×C4×C53D411yescharacteristic
37C22×C4×C53C2311yescharacteristic
38C2×C22⋊D53—21no—
39C2×C22⋊D53—21no—
40C2×C22⋊D53—21no—
41C2×D212D411yescharacteristic, complemented
42C2×C4×D53—21no—
43C2×Dic106D411yescharacteristic, complemented
44C22⋊C4×C53—21no—
45C22⋊C4×C53—21no—
46C22⋊C4×C53—42no—
47C22⋊Dic53—21no—
48C22⋊Dic53—21no—
49C22⋊Dic53—21no—
50#11719923—42no—
51#11719922—42no—
52#11719921—21no—
53#11719921—21no—
54C4⋊Dic53D411yescharacteristic, complemented
55C4⋊Dic53D411yescharacteristic, complemented
56#11719919—21no—
57#11719919—21no—
58C4×Dic53—21no—
59C2×#88322—21no—
60C2×#88322—21no—
61C23×C53—22no—
62C23×C53—21no—
63C23×C53—21no—
64C23×C53—21no—
65C23×C53—41no—
66C23×C53—41no—
67C23×C53—41no—
68C23×C53—41no—
69C23×C53—41no—
70C23×C53—41no—
71C23×C53—41no—
72C23×C53—41no—
73C23×C53C2×D411yescharacteristic
74C22×D53—21no—
75D4×C53—22no—
76D4×C53—22no—
77D4×C53—41no—
78D4×C53—41no—
79D4×C53—48no—
80C2×C4×C53—22no—
81C2×C4×C53—21no—
82C2×C4×C53—41no—
83C2×C4×C53—41no—
84C2×C4×C53—41no—
85C2×C4×C53—41no—
86C2×C4×C53C2×D411yescharacteristic
87C2×C4×C53C2×D411yescharacteristic
88C22⋊D53—41no—
89C22⋊D53—41no—
90C22⋊D53—81no—
91C22⋊D53—81no—
92C2×Dic53—21no—
93C2×Dic53—21no—
94C2×Dic53—21no—
95D212—41no—
96C4×D53—81no—
97Dic106—41no—
98#88322—41no—
99#88322—41no—
100C22×C53—21no—
101C22×C53—21no—
102C22×C53—42no—
103C22×C53—42no—
104C22×C53—42no—
105C22×C53—41no—
106C22×C53—41no—
107C22×C53—41no—
108C22×C53—42no—
109C22×C53—81no—
110C22×C53—81no—
111C22×C53—81no—
112C22×C53—81no—
113C22×C53—81no—
114C22×C53#12011yescharacteristic
115C22×C53#13511yescharacteristic
116C22×C53AGL(1,ℤ/8)11yescharacteristic
117C2×D53—41no—
118C2×D53—41no—
119C2×D53—41no—
120C4×C53—21no—
121C4×C53—21no—
122C4×C53—42no—
123C4×C53—41no—
124Dic53—41no—
125Dic53—41no—
126Dic53—41no—
127#1600—531nomaximal, Sylow
128C2×C53—21no—
129C2×C53—21no—
130C2×C53—81no—
131C2×C53—82no—
132C2×C53#44911yescharacteristic
133C2×C53#45311yescharacteristic
134C2×C53B(4,𝔽2)11yescharacteristic
135D53—81no—
136#538—531no—
137#450—531no—
138#447—531no—
139#392D5311yescharacteristic, complemented
140#328—531no—
141#327—531no—
142#323—531no—
143C53#160011yescharacteristic, complemented, Sylow
144C22×D4—21no—
145C22×D4C2×D5311yescharacteristic
146C2×SD16—1061no—
147C2×D8—1061no—
148#126—1061no—
149#124—1061no—
150#123—1061no—
151C22⋊Q8—531no—
152#121—531no—
153#121—1061no—
154#120—1061no—
155C4×D4—1061no—
156C2×C22⋊C4—21no—
157C2×C22⋊C4C2×D5311yescharacteristic
158#102—1061no—
159#102—1061no—
160C22⋊C8—531no—
161C22⋊C8—531no—
162#95C2×D5311yescharacteristic
163C24—21no—
164C24C22⋊D5311yescharacteristic
165C2×Q8—531no—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—42no—
169C2×D4—42no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—531no—
173C2×D4—1061no—
174C2×D4—1061no—
175C2×D4—1061no—
176C22×C4—21no—
177C22×C4—1061no—
178C22×C4C22⋊D5311yescharacteristic
179C22×C4C22×D5311yescharacteristic
180SD16—2122no—
181D8—2122no—
182C2×C8—1061no—
183C2×C8—1061no—
184C4⋊C4—531no—
185C4⋊C4—531no—
186C4⋊C4—1061no—
187C4⋊C4—1061no—
188C22⋊C4—21no—
189C22⋊C4—21no—
190C22⋊C4—42no—
191C22⋊C4—1061no—
192C22⋊C4—1061no—
193C22⋊C4—1061no—
194C22⋊C4—1061no—
195C22⋊C4—1061no—
196C42—1061no—
197C23—22no—
198C23—21no—
199C23—21no—
200C23—21no—
201C23—41no—
202C23—41no—
203C23—41no—
204C23—41no—
205C23—41no—
206C23—41no—
207C23—41no—
208C23—41no—
209C23—1061no—
210C23C2×C22⋊D5311yescharacteristic
211Q8—2121no—
212D4—22no—
213D4—22no—
214D4—41no—
215D4—41no—
216D4—48no—
217D4—4241no—
218D4—4241no—
219D4—2121no—
220D4—2121no—
221D4—2121no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—1061no—
229C2×C4—1061no—
230C2×C4—1061no—
231C2×C4—4241no—
232C2×C4D4×D5311yescharacteristic
233C2×C4D4×D5311yescharacteristic
234C8—2121no—
235C8—2121no—
236C22—21no—
237C22—21no—
238C22—42no—
239C22—42no—
240C22—42no—
241C22—41no—
242C22—41no—
243C22—42no—
244C22—41no—
245C22—81no—
246C22—81no—
247C22—81no—
248C22—81no—
249C22—81no—
250C22—2121no—
251C22—2121no—
252C22—2121no—
253C22#1313318511yescharacteristic
254C22#1313323711yescharacteristic
255C22#1313324311yescharacteristic
256C4—21no—
257C4—21no—
258C4—42no—
259C4—41no—
260C4—2121no—
261C4—2121no—
262C4—2121no—
263C2—21no—
264C2—21no—
265C2—81no—
266C2—82no—
267C2—4241no—
268C2#1681076611yescharacteristic
269C2#1681079011yescharacteristic
270C2#1681080911yescharacteristic
subgroupG/H
centerC22#13133185
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#392C2
socleC22×C53#120
hypercenter#392D53
hypocenterC53#1600
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #392

Complements

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

NH ≅ G/Ntype
1. #16811211C2semidirect
2. #16810774C2semidirect
3. #16810757C2semidirect
4. #16810359C2semidirect
5. #16810356C2semidirect
6. #16810347C2semidirect
7. C53×#392C2semidirect
41. C2×D212D4semidirect
43. C2×Dic106D4semidirect
54. C4⋊Dic53D4semidirect
55. C4⋊Dic53D4semidirect
139. #392D53semidirect
143. C53#1600semidirect