Finite Group Databasebeta

← #20280876

Subgroups — #20280876

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×#13133211C211yesmaximal, characteristic
2#16810561C211yesmaximal, characteristic, complemented
3#16810558C211yesmaximal, characteristic
4#16810556C212yesmaximal, complemented
5#16810554C211yesmaximal, characteristic, complemented
6#16810549C211yesmaximal, characteristic
7#16810545C212yesmaximal
8#16810518C212yesmaximal, complemented
9#16810513C212yesmaximal
10C4×C4⋊Dic53C211yesmaximal, characteristic, complemented
11C2×#125×C53C211yesmaximal, characteristic, complemented
12#13133211—24no—
13C2×C4⋊Dic53C2212yes—
14C2×#11719919C2211yescharacteristic
15C2×#11719919C2212yes—
16C2×#11719919C2211yescharacteristic
17C2×#11719919C2211yescharacteristic
18C2×#11719919C2212yes—
19C2×#11719921C2211yescharacteristic
20C2×#11719921C2212yes—
21C2×#11719921C2211yescharacteristic
22C2×#11719921C2211yescharacteristic
23C2×#11719921C2212yes—
24C2×C4×Dic53C2212yes—
25#13133124C2212yes—
26#13133124C2212yes—
27#13133124C2212yes—
28#13133124C2212yes—
29#13133124C2211yescharacteristic
30#13133124C2211yescharacteristic
31#125×C53—24no—
32C2×C4⋊C4×C53C2212yes—
33C2×C4⋊C4×C53C2212yes—
34C2×C4⋊C4×C53C2211yescharacteristic
35C2×C4⋊C4×C53C2211yescharacteristic
36C2×C42×C53C2211yescharacteristic
37C23×D53C2311yescharacteristic
38C22×C4×C53C2312yes—
39C22×C4×C53C2312yes—
40C22×C4×C53C2311yescharacteristic
41C22×C4×C53C2311yescharacteristic
42C22×C4×C53C2311yescharacteristic
43C22×Dic53C2312yes—
44C22×Dic53C2311yescharacteristic
45C22×Dic53C2311yescharacteristic
46C22×Dic53C2311yescharacteristic
47C22×Dic53C2312yes—
48C4⋊C4×C53—24no—
49C4⋊C4×C53—24no—
50C4⋊C4×C53—22no—
51C4⋊C4×C53—22no—
52C42×C53D414yes—
53#11719921—22no—
54#11719921—22no—
55#11719921—22no—
56#11719921—24no—
57#11719921—42no—
58C4⋊Dic53—24no—
59#11719919—22no—
60#11719919—22no—
61#11719919—22no—
62#11719919—24no—
63#11719919—42no—
64C4×Dic53—42no—
65C23×C53C2411yescharacteristic
66C22×D53—21no—
67C22×D53—22no—
68C22×D53—21no—
69C22×D53—21no—
70C22×D53—22no—
71C2×C4×C53—22no—
72C2×C4×C53—21no—
73C2×C4×C53—21no—
74C2×C4×C53—22no—
75C2×C4×C53—22no—
76C2×C4×C53—22no—
77C2×C4×C53—22no—
78C2×C4×C53—22no—
79C2×C4×C53—22no—
80C2×C4×C53—22no—
81C2×C4×C53C2×D412yes—
82C2×C4×C53C2×D412yes—
83C2×C4×C53C2×D412yes—
84C2×Dic53—22no—
85C2×Dic53—22no—
86C2×Dic53—21no—
87C2×Dic53—21no—
88C2×Dic53—21no—
89C2×Dic53—22no—
90C2×Dic53—22no—
91C2×Dic53—22no—
92C2×Dic53—21no—
93C2×Dic53—21no—
94C2×Dic53—22no—
95C2×Dic53—22no—
96C2×Dic53#4114yes—
97C22×C53C22×D411yescharacteristic
98C22×C53C2×#4112yes—
99C22×C53H(5,𝔽2)11yescharacteristic
100C22×C532−1+411yescharacteristic
101C22×C532−1+412yes—
102C2×D53—41no—
103C2×D53—41no—
104C2×D53—41no—
105C2×D53—42no—
106C2×D53—42no—
107C4×C53—22no—
108C4×C53—22no—
109C4×C53—22no—
110C4×C53—42no—
111C4×C53—42no—
112Dic53—24no—
113Dic53—41no—
114Dic53—41no—
115Dic53—41no—
116Dic53—42no—
117#2664—531nomaximal, Sylow
118C2×C53#53211yescharacteristic
119C2×C53#53611yescharacteristic
120C2×C53#53711yescharacteristic
121C2×C53#53912yes—
122C2×C53#54112yes—
123D53—81no—
124C2×#126—531no—
125C2×#125D5311yescharacteristic, complemented
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—1064no—
137#125—24no—
138C2×C4⋊C4—531no—
139C2×C4⋊C4—531no—
140C2×C4⋊C4—531no—
141C2×C4⋊C4—532no—
142C2×C4⋊C4—532no—
143C2×C4⋊C4—532no—
144C2×C4⋊C4C2×D5312yes—
145C2×C4⋊C4C2×D5312yes—
146C2×C4⋊C4C2×D5311yescharacteristic
147C2×C4⋊C4C2×D5311yescharacteristic
148C2×C22⋊C4—531no—
149C2×C22⋊C4—531no—
150C2×C22⋊C4—531no—
151C2×C22⋊C4—532no—
152C2×C22⋊C4—532no—
153C2×C42—532no—
154C2×C42C2×D5311yescharacteristic
155#95—531no—
156#95—531no—
157#95—532no—
158#95—532no—
159#95—532no—
160#95—532no—
161C24—531no—
162C22×C4—532no—
163C22×C4—532no—
164C22×C4—531no—
165C22×C4—531no—
166C22×C4—531no—
167C22×C4C22×D5312yes—
168C22×C4C22×D5312yes—
169C22×C4C22×D5311yescharacteristic
170C22×C4C22×D5311yescharacteristic
171C22×C4C22×D5311yescharacteristic
172C4⋊C4—24no—
173C4⋊C4—24no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—1064no—
177C4⋊C4—1064no—
178C4⋊C4—1062no—
179C4⋊C4—1062no—
180C4⋊C4—1062no—
181C4⋊C4—2122no—
182C22⋊C4—1064no—
183C22⋊C4—1062no—
184C22⋊C4—1062no—
185C22⋊C4—1062no—
186C22⋊C4—2122no—
187C42—2122no—
188C42C22⋊D5314yes—
189C23—1061no—
190C23—1062no—
191C23—1061no—
192C23—1061no—
193C23—1062no—
194C23C23×D5311yescharacteristic
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—534no—
206C2×C4—1062no—
207C2×C4—1062no—
208C2×C4—1061no—
209C2×C4—1061no—
210C2×C4—1061no—
211C2×C4—1062no—
212C2×C4—1062no—
213C2×C4—1062no—
214C2×C4—1061no—
215C2×C4—1061no—
216C2×C4—1062no—
217C2×C4—1062no—
218C2×C4C2×C22⋊D5312yes—
219C2×C4C2×C22⋊D5312yes—
220C2×C4C2×C22⋊D5312yes—
221C22—2121no—
222C22—2121no—
223C22—2121no—
224C22—2122no—
225C22—2122no—
226C22C22×C22⋊D5311yescharacteristic
227C22#41×D5312yes—
228C22#1313326311yescharacteristic
229C22#1313326411yescharacteristic
230C22#1313326512yes—
231C4—22no—
232C4—22no—
233C4—22no—
234C4—42no—
235C4—42no—
236C4—1064no—
237C4—2121no—
238C4—2121no—
239C4—2121no—
240C4—2122no—
241C2—4241no—
242C2#1681116911yescharacteristic
243C2#1681119711yescharacteristic
244C2#1681120111yescharacteristic
245C2#1681122112yes—
246C2#1681123212yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×#125×C53C2
hypercenterC2×#125D53
hypocenterC53#2664
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
2. #16810561C2semidirect
4. #16810556C2semidirect
5. #16810554C2semidirect
8. #16810518C2semidirect
10. C4×C4⋊Dic53C2semidirect
11. C2×#125×C53C2semidirect
125. C2×#125D53semidirect
135. C53#2664semidirect