Finite Group Databasebeta

← #20280877

Subgroups — #20280877

Order 6784.

434 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 137 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
1C2×#13133209C211yesmaximal, characteristic, complemented
2#16810562C211yesmaximal, characteristic, complemented
3#16810559C211yesmaximal, characteristic
4#16810556C212yesmaximal, complemented
5#16810553C211yesmaximal, characteristic, complemented
6#16810548C211yesmaximal, characteristic
7#16810546C212yesmaximal
8#16810518C212yesmaximal, complemented
9#16810517C212yesmaximal
10C4×C4⋊Dic53C211yesmaximal, characteristic, complemented
11C2×#126×C53C211yesmaximal, characteristic
12#13133209—24no—
13C2×C4⋊Dic53C2212yes—
14C2×C4⋊Dic53C2211yescharacteristic
15C2×C4⋊Dic53C2211yescharacteristic
16C2×#11719919C2212yes—
17C2×#11719919C2212yes—
18C2×#11719919C2212yes—
19C2×#11719919C2212yes—
20C2×C22⋊Dic53C2212yes—
21C2×C22⋊Dic53C2212yes—
22C2×C4×Dic53C2212yes—
23#13133124C2212yes—
24#13133124C2212yes—
25#13133124C2212yes—
26#13133124C2212yes—
27#13133124C2211yescharacteristic
28#13133124C2211yescharacteristic
29#126×C53—24no—
30C2×C4⋊C4×C53C2211yescharacteristic
31C2×C4⋊C4×C53C2211yescharacteristic
32C2×C4⋊C4×C53C2211yescharacteristic
33C2×C22⋊C4×C53C2211yescharacteristic
34C2×C22⋊C4×C53C2211yescharacteristic
35C2×C22⋊C4×C53C2211yescharacteristic
36C2×C42×C53C2211yescharacteristic
37C24×C53C2311yescharacteristic
38C22×C4×C53C2311yescharacteristic
39C22×C4×C53C2311yescharacteristic
40C22×C4×C53C2311yescharacteristic
41C22×C4×C53C2311yescharacteristic
42C22×C4×C53C2311yescharacteristic
43C22×C4×C53C2311yescharacteristic
44C22×Dic53C2312yes—
45C22×Dic53C2312yes—
46C22×Dic53C2312yes—
47C22×Dic53C2312yes—
48C4⋊C4×C53—22no—
49C4⋊C4×C53—22no—
50C4⋊C4×C53—22no—
51C22⋊C4×C53—22no—
52C22⋊C4×C53—22no—
53C22⋊C4×C53—22no—
54C42×C53D414yes—
55C22⋊Dic53—24no—
56C22⋊Dic53—42no—
57C4⋊Dic53—22no—
58C4⋊Dic53—22no—
59C4⋊Dic53—24no—
60#11719919—24no—
61#11719919—24no—
62#11719919—24no—
63#11719919—42no—
64C4×Dic53—42no—
65C23×C53—21no—
66C23×C53—22no—
67C23×C53—21no—
68C23×C53—22no—
69C23×C53—21no—
70C23×C53C2411yescharacteristic
71C2×C4×C53—22no—
72C2×C4×C53—21no—
73C2×C4×C53—22no—
74C2×C4×C53—21no—
75C2×C4×C53—21no—
76C2×C4×C53—21no—
77C2×C4×C53—21no—
78C2×C4×C53—22no—
79C2×C4×C53—21no—
80C2×C4×C53—22no—
81C2×C4×C53—21no—
82C2×C4×C53C2×D412yes—
83C2×C4×C53C2×D412yes—
84C2×C4×C53C2×D412yes—
85C2×Dic53—22no—
86C2×Dic53—22no—
87C2×Dic53—22no—
88C2×Dic53—22no—
89C2×Dic53—22no—
90C2×Dic53—22no—
91C2×Dic53—22no—
92C2×Dic53—22no—
93C2×Dic53—22no—
94C2×Dic53—22no—
95C2×Dic53—22no—
96C2×Dic53#4114yes—
97C22×C53—41no—
98C22×C53—41no—
99C22×C53—41no—
100C22×C53—42no—
101C22×C53—42no—
102C22×C53C22×D411yescharacteristic
103C22×C53C2×#4112yes—
104C22×C53H(5,𝔽2)11yescharacteristic
105C22×C532−1+412yes—
106C22×C532−1+411yescharacteristic
107C4×C53—22no—
108C4×C53—22no—
109C4×C53—22no—
110C4×C53—41no—
111C4×C53—41no—
112C4×C53—41no—
113Dic53—24no—
114Dic53—42no—
115Dic53—42no—
116Dic53—42no—
117#2664—531nomaximal, Sylow
118C2×C53—81no—
119C2×C53#53211yescharacteristic
120C2×C53#53611yescharacteristic
121C2×C53#53711yescharacteristic
122C2×C53#53912yes—
123C2×C53#54112yes—
124C2×#126D5311yescharacteristic
125C2×#125—531no—
126#400—531no—
127#399—531no—
128#398—531no—
129#398—532no—
130#397—531no—
131#396—532no—
132#389—532no—
133#388—532no—
134C4×C4⋊C4—531no—
135C53#266411yescharacteristic, complemented, Sylow
136#126—24no—
137#125—1064no—
138C2×C4⋊C4—532no—
139C2×C4⋊C4—532no—
140C2×C4⋊C4—531no—
141C2×C4⋊C4—531no—
142C2×C4⋊C4—532no—
143C2×C4⋊C4—532no—
144C2×C4⋊C4—532no—
145C2×C4⋊C4C2×D5311yescharacteristic
146C2×C4⋊C4C2×D5311yescharacteristic
147C2×C4⋊C4C2×D5311yescharacteristic
148C2×C22⋊C4—532no—
149C2×C22⋊C4—532no—
150C2×C22⋊C4C2×D5311yescharacteristic
151C2×C22⋊C4C2×D5311yescharacteristic
152C2×C22⋊C4C2×D5311yescharacteristic
153C2×C42—532no—
154C2×C42C2×D5311yescharacteristic
155#95—531no—
156#95—531no—
157#95—532no—
158#95—532no—
159#95—532no—
160#95—532no—
161C24C22×D5311yescharacteristic
162C22×C4—532no—
163C22×C4—532no—
164C22×C4—532no—
165C22×C4—532no—
166C22×C4C22×D5311yescharacteristic
167C22×C4C22×D5311yescharacteristic
168C22×C4C22×D5311yescharacteristic
169C22×C4C22×D5311yescharacteristic
170C22×C4C22×D5311yescharacteristic
171C22×C4C22×D5311yescharacteristic
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—1064no—
176C4⋊C4—1064no—
177C4⋊C4—1064no—
178C4⋊C4—1064no—
179C4⋊C4—1062no—
180C4⋊C4—1062no—
181C4⋊C4—2122no—
182C22⋊C4—22no—
183C22⋊C4—22no—
184C22⋊C4—22no—
185C22⋊C4—1064no—
186C22⋊C4—2122no—
187C42—2122no—
188C42C22⋊D5314yes—
189C23—21no—
190C23—22no—
191C23—21no—
192C23—22no—
193C23—21no—
194C23C23×D5311yescharacteristic
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—534no—
207C2×C4—1062no—
208C2×C4—1062no—
209C2×C4—1062no—
210C2×C4—1062no—
211C2×C4—1062no—
212C2×C4—1062no—
213C2×C4—1062no—
214C2×C4—1062no—
215C2×C4—1062no—
216C2×C4—1062no—
217C2×C4—1062no—
218C2×C4C2×C22⋊D5312yes—
219C2×C4C2×C22⋊D5312yes—
220C2×C4C2×C22⋊D5312yes—
221C22—41no—
222C22—41no—
223C22—41no—
224C22—42no—
225C22—42no—
226C22C22×C22⋊D5311yescharacteristic
227C22#41×D5312yes—
228C22#1313326311yescharacteristic
229C22#1313326412yes—
230C22#1313326411yescharacteristic
231C4—22no—
232C4—22no—
233C4—22no—
234C4—41no—
235C4—41no—
236C4—41no—
237C4—1064no—
238C4—2122no—
239C4—2122no—
240C4—2122no—
241C2—81no—
242C2#1681116911yescharacteristic
243C2#1681119411yescharacteristic
244C2#1681120111yescharacteristic
245C2#1681121412yes—
246C2#1681123012yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×#126×C53C2
hypercenterC2×#126D53
hypocenterC53#2664
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. C2×#13133209C2semidirect
2. #16810562C2semidirect
4. #16810556C2semidirect
5. #16810553C2semidirect
8. #16810518C2semidirect
10. C4×C4⋊Dic53C2semidirect
135. C53#2664semidirect