Finite Group Databasebeta

← #20277298

Subgroups — #20277298

Order 6784.

784 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 229 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
2C2×#13133175C211yesmaximal, characteristic
3C2×#13133174C211yesmaximal, characteristic
4#16810530C211yesmaximal, characteristic, complemented
5#16810529C211yesmaximal, characteristic, complemented
6#16810528C211yesmaximal, characteristic, complemented
7#16810527C214yesmaximal, complemented
8#16810485C214yesmaximal
9C53×#391C211yesmaximal, characteristic, complemented
10C22×Dic106C2212yes—
11C22×Dic106C2211yescharacteristic
12C2×Q8×D53—24no—
13C2×Q8×D53—24no—
14C2×Q8×D53C418yescomplemented
15C22×C4×D53C2212yes—
16C22×C4×D53C2211yescharacteristic
17#13133175—24no—
18#13133174—24no—
19C2×C4⋊Dic53C2211yescharacteristic
20C2×#11719919C2212yes—
21C2×#11719921C2212yes—
22C2×#11719921C2212yes—
23C2×C4×Dic53C2212yes—
24C2×C4×Dic53C2211yescharacteristic
25#13133124C2214yes—
26#13133124C2214yes—
27#13133124C2214yes—
28C22×Q8×C53C2211yescharacteristic
29C2×C4⋊C4×C53C2211yescharacteristic
30C2×C42×C53C2211yescharacteristic
31#95×C53C2214yes—
32C23×D53C2311yescharacteristic
33C2×Q8×C53—22no—
34C2×Q8×C53—22no—
35C2×Q8×C53C2×C414yes—
36C22×C4×C53C2312yes—
37C22×C4×C53C2312yes—
38C22×C4×C53C2312yes—
39C22×C4×C53C2311yescharacteristic
40C22×Dic53C2312yes—
41C22×Dic53C2312yes—
42C22×Dic53C2312yes—
43C22×Dic53C2311yescharacteristic
44Q8×D53—232no—
45C2×C4×D53—24no—
46C2×C4×D53—24no—
47C2×C4×D53C2×C418yes—
48C2×C4×D53C2×C414yes—
49C2×C4×D53D414yes—
50C2×C4×D53D414yes—
51C2×Dic106—22no—
52C2×Dic106—22no—
53C2×Dic106—24no—
54C2×Dic106—24no—
55C2×Dic106C2×C414yes—
56C2×Dic106C2×C418yes—
57C4⋊C4×C53—22no—
58C42×C53—22no—
59#11719921—24no—
60#11719921—24no—
61C4⋊Dic53—22no—
62#11719919—24no—
63C4×Dic53—22no—
64C4×Dic53—24no—
65C23×C53C2411yescharacteristic
66C22×D53—22no—
67C22×D53—22no—
68C22×D53C22×C412yes—
69C22×D53C2×D412yes—
70C22×D53C2×D412yes—
71Q8×C53—28no—
72C2×C4×C53—22no—
73C2×C4×C53—22no—
74C2×C4×C53—22no—
75C2×C4×C53—22no—
76C2×C4×C53—22no—
77C2×C4×C53—24no—
78C2×C4×C53—22no—
79C2×C4×C53C22×C414yes—
80C2×C4×C53C22×C412yes—
81C2×C4×C53C2×D412yes—
82C2×C4×C53C2×D412yes—
83C2×Dic53—22no—
84C2×Dic53—22no—
85C2×Dic53—22no—
86C2×Dic53—22no—
87C2×Dic53—22no—
88C2×Dic53—22no—
89C2×Dic53—24no—
90C2×Dic53C22×C414yes—
91C2×Dic53C22×C412yes—
92C2×Dic53C2×D412yes—
93C2×Dic53C2×D412yes—
94C4×D53—216no—
95C4×D53C22⋊C4116yes—
96Dic106—28no—
97Dic106—216no—
98C22×C53C23×C411yescharacteristic
99C22×C53C22×D411yescharacteristic
100C22×C53C22×D411yescharacteristic
101C22×C532−1+412yes—
102C22×C532−1+412yes—
103C2×D53—24no—
104C2×D53—22no—
105C2×D53—24no—
106C2×D53—22no—
107C2×D53C2×C22⋊C414yes—
108C4×C53—24no—
109C4×C53—42no—
110C4×C53—42no—
111C4×C53C2×C22⋊C414yes—
112Dic53—24no—
113Dic53—42no—
114Dic53—42no—
115Dic53C2×C22⋊C414yes—
116#2310—531nomaximal, Sylow
117C2×C53C22×C22⋊C411yescharacteristic
118C2×C53#51912yes—
119C2×C53#53611yescharacteristic
120C2×C53#53612yes—
121C2×C53#53611yescharacteristic
122D53—24no—
123C23×Q8—531no—
124C2×#117—531no—
125C2×#117—531no—
126#391—531no—
127#391—531no—
128#391—531no—
129#391—534no—
130#391D5311yescharacteristic, complemented
131#381—534no—
132C53#231011yescharacteristic, complemented, Sylow
133C22×Q8—538no—
134C22×Q8—532no—
135C22×Q8—531no—
136C22×Q8—1064no—
137C22×Q8—1064no—
138C22×Q8C2×D5311yescharacteristic
139C23×C4—531no—
140C23×C4—532no—
141#117—1064no—
142#117—1064no—
143C2×C4⋊C4—532no—
144C2×C4⋊C4—531no—
145C2×C4⋊C4C2×D5311yescharacteristic
146C2×C22⋊C4—532no—
147C2×C22⋊C4—532no—
148C2×C42—532no—
149C2×C42—531no—
150C2×C42C2×D5311yescharacteristic
151#95—534no—
152#95—534no—
153#95—534no—
154#95C2×D5314yes—
155C24—531no—
156C2×Q8—22no—
157C2×Q8—22no—
158C2×Q8—538no—
159C2×Q8—534no—
160C2×Q8—1064no—
161C2×Q8—1064no—
162C2×Q8—10632no—
163C2×Q8—1062no—
164C2×Q8—1062no—
165C2×Q8C4×D5314yes—
166C22×C4—531no—
167C22×C4—534no—
168C22×C4—534no—
169C22×C4—534no—
170C22×C4—532no—
171C22×C4—538no—
172C22×C4—532no—
173C22×C4—532no—
174C22×C4—1064no—
175C22×C4—1064no—
176C22×C4C22×D5312yes—
177C22×C4C22×D5312yes—
178C22×C4C22×D5312yes—
179C22×C4C22×D5311yescharacteristic
180C4⋊C4—22no—
181C4⋊C4—1064no—
182C4⋊C4—1062no—
183C22⋊C4—1064no—
184C22⋊C4—1064no—
185C42—22no—
186C42—1064no—
187C42—1062no—
188C23—532no—
189C23—532no—
190C23—532no—
191C23—1062no—
192C23—1062no—
193C23C23×D5311yescharacteristic
194Q8—28no—
195Q8—10616no—
196Q8—1068no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—532no—
205C2×C4—532no—
206C2×C4—532no—
207C2×C4—5316no—
208C2×C4—534no—
209C2×C4—10616no—
210C2×C4—1062no—
211C2×C4—1062no—
212C2×C4—1062no—
213C2×C4—1062no—
214C2×C4—1062no—
215C2×C4—1062no—
216C2×C4—1064no—
217C2×C4C2×C4×D5314yes—
218C2×C4C2×C4×D5312yes—
219C2×C4D4×D5312yes—
220C2×C4D4×D5312yes—
221C22—534no—
222C22—1064no—
223C22—1062no—
224C22—1064no—
225C22—1062no—
226C22C22×C4×D5311yescharacteristic
227C22C2×D4×D5311yescharacteristic
228C22C2×D4×D5311yescharacteristic
229C22#1313326412yes—
230C22#1313326512yes—
231C4—24no—
232C4—42no—
233C4—42no—
234C4—534no—
235C4—1064no—
236C4—2122no—
237C4—2122no—
238C4C22⋊C4×D5314yes—
239C2—1064no—
240C2C2×C22⋊C4×D5311yescharacteristic
241C2#1681109912yes—
242C2#1681119311yescharacteristic
243C2#1681119612yes—
244C2#1681119811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutatorC22×C53C23×C4
FittingC53×#391C2
socleC23×C53C24
hypercenter#391D53
hypocenterC53#2310
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
4. #16810530C2semidirect
5. #16810529C2semidirect
6. #16810528C2semidirect
7. #16810527C2semidirect
9. C53×#391C2semidirect
14. C2×Q8×D53C4semidirect
130. #391D53semidirect
132. C53#2310semidirect