Finite Group Databasebeta

← #328×D53 #20273194

Subgroups — #328×D53 #20273194

Order 6784.

336 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 77 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
1C22⋊Q8×D53C211yesmaximal, characteristic
2C22⋊C8×D53C211yesmaximal, characteristic
3#95×D53C211yesmaximal, characteristic
4#16810361C211yesmaximal, characteristic, complemented
5#16810360C211yesmaximal, characteristic, complemented
6#16810359C211yesmaximal, characteristic, complemented
7C53×#328C211yesmaximal, characteristic, complemented
8C2×Q8×D53C411yescharacteristic, complemented
9C22×C4×D53—21no—
10C22×C4×D53C2211yescharacteristic
11C2×C8×D53—21no—
12#13133197C411yescharacteristic, complemented
13#13133196C2211yescharacteristic
14#13133195—21no—
15#13133194—21no—
16#13133193C411yescharacteristic, complemented
17C4⋊C4×D53—21no—
18C4⋊C4×D53C411yescharacteristic, complemented
19C22⋊C4×D53—21no—
20C22⋊#88322C2211yescharacteristic
21#13133129—21no—
22#13133124—21no—
23#13133124C2211yescharacteristic
24C22⋊Q8×C53C2211yescharacteristic
25C22⋊C8×C53C2211yescharacteristic
26#95×C53C2211yescharacteristic
27C23×D53D411yescharacteristic
28C2×Q8×C53C2×C411yescharacteristic, complemented
29C22×C4×C53—21no—
30C22×C4×C53C2311yescharacteristic
31C22×Dic53—21no—
32C22×Dic53D411yescharacteristic
33Q8×D53—42no—
34C2×C4×D53—24no—
35C2×C4×D53—21no—
36C2×C4×D53—21no—
37C2×C4×D53—42no—
38C2×C4×D53—42no—
39C2×C4×D53—42no—
40C2×C4×D53C2×C411yescharacteristic
41C2×C4×D53C2×C411yescharacteristic
42C2×C4×D53D411yescharacteristic
43C2×C4×D53D411yescharacteristic
44C2×Dic106—21no—
45C2×Dic106C2×C411yescharacteristic, complemented
46C2×C8×C53—21no—
47C4⋊C4×C53—21no—
48C4⋊C4×C53C2×C411yescharacteristic, complemented
49C22⋊C4×C53—21no—
50C22⋊Dic53—21no—
51#11719921—21no—
52#11719921—21no—
53C4⋊Dic53—21no—
54C4⋊Dic53C2×C411yescharacteristic, complemented
55#11719919—21no—
56#11719919—21no—
57#11719919—21no—
58C2×#88322—21no—
59C8×D53—42no—
60C23×C53C2×D411yescharacteristic
61C22×D53—22no—
62C22×D53—22no—
63C22×D53C22⋊C411yescharacteristic
64C22×D53C22⋊C411yescharacteristic
65C22×D53SD1612yes—
66C22×D53Q1612yes—
67Q8×C53—41no—
68C2×C4×C53—22no—
69C2×C4×C53—21no—
70C2×C4×C53—21no—
71C2×C4×C53—41no—
72C2×C4×C53—41no—
73C2×C4×C53—41no—
74C2×C4×C53C22×C411yescharacteristic
75C2×C4×C53C2×D411yescharacteristic
76C2×Dic53—22no—
77C2×Dic53—21no—
78C2×Dic53—21no—
79C2×Dic53—41no—
80C2×Dic53—41no—
81C2×Dic53—41no—
82C2×Dic53C22⋊C411yescharacteristic
83C2×Dic53C22⋊C411yescharacteristic
84C4×D53—22no—
85C4×D53—22no—
86C4×D53—41no—
87C4×D53—41no—
88C4×D53—48no—
89C4×D53—42no—
90C4×D53—42no—
91Dic106—42no—
92Dic106—41no—
93C8×C53—41no—
94#88322—41no—
95C22×C53—21no—
96C22×C53—21no—
97C22×C53C2×C22⋊C411yescharacteristic
98C22×C53C2×SD1611yescharacteristic
99C22×C53C2×Q1611yescharacteristic
100C2×D53—21no—
101C2×D53—21no—
102C2×D53—24no—
103C2×D53—24no—
104C2×D53#9912yes—
105C2×D53#10312yes—
106C2×D53#10312yes—
107C2×D53#10412yes—
108C4×C53—21no—
109C4×C53—21no—
110C4×C53—42no—
111C4×C53—41no—
112C4×C53—41no—
113Dic53—21no—
114Dic53—21no—
115Dic53—42no—
116Dic53—41no—
117Dic53—41no—
118C2×#328—531nomaximal, Sylow
119C2×C53—21no—
120C2×C53—21no—
121C2×C53C2×#9911yescharacteristic
122C2×C53C2×#10311yescharacteristic
123C2×C53C2×#10411yescharacteristic
124D53—22no—
125D53#32814yesdirect product factor
126C2×C22⋊Q8—531no—
127C2×C22⋊C8—531no—
128C2×#95—531no—
129#328—531no—
130#328—531no—
131#328—531no—
132#328D5311yescharacteristic, direct product factor
133C53C2×#32811yescharacteristic, complemented, Sylow
134C22×Q8—531no—
135C23×C4—531no—
136C23×C4—1061no—
137C22×C8—1061no—
138C22⋊Q8—531no—
139C22⋊Q8—531no—
140C22⋊Q8—531no—
141C22⋊Q8—1061no—
142C22⋊Q8—1061no—
143C22⋊Q8C2×D5311yescharacteristic
144C2×C4⋊C4—531no—
145C2×C4⋊C4—1061no—
146C2×C22⋊C4—1061no—
147C22⋊C8—531no—
148C22⋊C8—1061no—
149C22⋊C8C2×D5311yescharacteristic
150#95—531no—
151#95—1061no—
152#95C2×D5311yescharacteristic
153C24—531no—
154C2×Q8—531no—
155C2×Q8—1061no—
156C2×Q8—2122no—
157C2×Q8C4×D5311yescharacteristic, complemented
158C22×C4—21no—
159C22×C4—531no—
160C22×C4—531no—
161C22×C4—531no—
162C22×C4—531no—
163C22×C4—531no—
164C22×C4—1061no—
165C22×C4—1061no—
166C22×C4—1064no—
167C22×C4—1061no—
168C22×C4—2122no—
169C22×C4—2122no—
170C22×C4—2122no—
171C22×C4C22×D5311yescharacteristic
172C2×C8—21no—
173C2×C8—1061no—
174C2×C8—2122no—
175C4⋊C4—21no—
176C4⋊C4—531no—
177C4⋊C4—1061no—
178C4⋊C4—1061no—
179C4⋊C4—1061no—
180C4⋊C4—1061no—
181C4⋊C4C4×D5311yescharacteristic, complemented
182C22⋊C4—21no—
183C22⋊C4—1061no—
184C22⋊C4—1061no—
185C22⋊C4—1061no—
186C23—531no—
187C23—531no—
188C23—532no—
189C23—532no—
190C23—1062no—
191C23—1062no—
192C23D4×D5311yescharacteristic
193Q8—41no—
194Q8—2121no—
195Q8—2122no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—531no—
203C2×C4—531no—
204C2×C4—1062no—
205C2×C4—1062no—
206C2×C4—1062no—
207C2×C4—1061no—
208C2×C4—1061no—
209C2×C4—2121no—
210C2×C4—2121no—
211C2×C4—2122no—
212C2×C4—2122no—
213C2×C4—2128no—
214C2×C4—2121no—
215C2×C4—2121no—
216C2×C4—2121no—
217C2×C4C2×C4×D5311yescharacteristic
218C2×C4D4×D5311yescharacteristic
219C8—41no—
220C8—2121no—
221C22—21no—
222C22—21no—
223C22—532no—
224C22—532no—
225C22—532no—
226C22—532no—
227C22—1064no—
228C22—1064no—
229C22—1061no—
230C22—1061no—
231C22C22⋊C4×D5311yescharacteristic
232C22SD16×D5311yescharacteristic
233C22Q16×D5311yescharacteristic
234C4—21no—
235C4—21no—
236C4—42no—
237C4—41no—
238C4—41no—
239C4—1061no—
240C4—1061no—
241C4—2122no—
242C4—2121no—
243C4—2121no—
244C2—21no—
245C2—21no—
246C2—534no—
247C2—1062no—
248C2#99×D5311yescharacteristic
249C2#103×D5311yescharacteristic
250C2#104×D5311yescharacteristic
subgroupG/H
centerC22C22⋊C4×D53
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#328C2
socleC22×C53C2×C22⋊C4
hypercenter#328D53
hypocenterC53C2×#328
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
4. #16810361C2semidirect
5. #16810360C2semidirect
6. #16810359C2semidirect
7. C53×#328C2semidirect
8. C2×Q8×D53C4semidirect
12. #13133197C4semidirect
16. #13133193C4semidirect
18. C4⋊C4×D53C4semidirect
28. C2×Q8×C53C2×C4semidirect
45. C2×Dic106C2×C4semidirect
48. C4⋊C4×C53C2×C4semidirect
54. C4⋊Dic53C2×C4semidirect
125. D53#328direct (N × H)
132. #328D53direct (N × H)
133. C53C2×#328semidirect
157. C2×Q8C4×D53semidirect
181. C4⋊C4C4×D53semidirect