Finite Group Databasebeta

← #20280621

Subgroups — #20280621

Order 6784.

450 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 141 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#16810556C212yesmaximal, complemented
3#16810554C211yesmaximal, characteristic, complemented
4#16810549C211yesmaximal, characteristic
5#16810545C212yesmaximal
6#16810542C211yesmaximal, characteristic, complemented
7#16810534C211yesmaximal, characteristic
8#16810514C212yesmaximal
9#16810507C212yesmaximal, complemented
10#16810489C211yesmaximal, characteristic, complemented
11C2×C4×Q8×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—
24#13133124C2212yes—
25#13133124C2212yes—
26#13133124C2212yes—
27#13133124C2212yes—
28#13133124C2212yes—
29#13133124C2211yescharacteristic
30#13133124C2211yescharacteristic
31C22×Q8×C53C2211yescharacteristic
32C4×Q8×C53—24no—
33C2×C4⋊C4×C53C2212yes—
34C2×C4⋊C4×C53C2211yescharacteristic
35C2×C42×C53C2212yes—
36C2×C42×C53C2211yescharacteristic
37C23×D53C2311yescharacteristic
38C2×Q8×C53—22no—
39C2×Q8×C53—24no—
40C22×C4×C53C2312yes—
41C22×C4×C53C2311yescharacteristic
42C22×C4×C53C2311yescharacteristic
43C22×C4×C53C2311yescharacteristic
44C22×C4×C53C2312yes—
45C22×Dic53C2311yescharacteristic
46C22×Dic53C2312yes—
47C22×Dic53C2312yes—
48C22×Dic53C2311yescharacteristic
49C22×Dic53C2311yescharacteristic
50C4⋊C4×C53—24no—
51C4⋊C4×C53—22no—
52C42×C53—24no—
53C42×C53D414yes—
54#11719921—22no—
55#11719921—22no—
56#11719921—22no—
57#11719921—24no—
58#11719921—42no—
59C4⋊Dic53—42no—
60#11719919—22no—
61#11719919—22no—
62#11719919—22no—
63#11719919—24no—
64#11719919—42no—
65C23×C53C2411yescharacteristic
66C22×D53—21no—
67C22×D53—22no—
68C22×D53—21no—
69C22×D53—21no—
70C22×D53—22no—
71Q8×C53—28no—
72C2×C4×C53—22no—
73C2×C4×C53—22no—
74C2×C4×C53—22no—
75C2×C4×C53—21no—
76C2×C4×C53—21no—
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×C4×C53#4114yes—
85C2×Dic53—21no—
86C2×Dic53—21no—
87C2×Dic53—21no—
88C2×Dic53—22no—
89C2×Dic53—22no—
90C2×Dic53—22no—
91C2×Dic53—21no—
92C2×Dic53—22no—
93C2×Dic53—22no—
94C2×Dic53—21no—
95C2×Dic53—22no—
96C2×Dic53—22no—
97C2×Dic53—22no—
98C22×C53C22×D411yescharacteristic
99C22×C53C2×#4112yes—
100C22×C53H(5,𝔽2)11yescharacteristic
101C22×C532−1+411yescharacteristic
102C22×C532−1+412yes—
103C2×D53—41no—
104C2×D53—41no—
105C2×D53—41no—
106C2×D53—42no—
107C2×D53—42no—
108C4×C53—22no—
109C4×C53—24no—
110C4×C53—22no—
111C4×C53—22no—
112C4×C53—42no—
113Dic53—41no—
114Dic53—41no—
115Dic53—41no—
116Dic53—42no—
117Dic53—42no—
118#2637—531nomaximal, Sylow
119C2×C53#52511yescharacteristic
120C2×C53#53611yescharacteristic
121C2×C53#53711yescharacteristic
122C2×C53#54112yes—
123C2×C53#54212yes—
124D53—81no—
125C2×#126—531no—
126C2×C4×Q8D5311yescharacteristic, complemented
127#398—531no—
128#398—532no—
129#397—531no—
130#396—532no—
131#395—531no—
132#393—531no—
133#388—532no—
134#387—532no—
135#382—531no—
136C53#263711yescharacteristic, complemented, Sylow
137C22×Q8C2×D5311yescharacteristic
138#126—1064no—
139C4×Q8—24no—
140C2×C4⋊C4—531no—
141C2×C4⋊C4—531no—
142C2×C4⋊C4—531no—
143C2×C4⋊C4—532no—
144C2×C4⋊C4—532no—
145C2×C4⋊C4—532no—
146C2×C4⋊C4C2×D5312yes—
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×C42C2×D5312yes—
154C2×C42C2×D5311yescharacteristic
155#95—532no—
156#95—531no—
157#95—531no—
158#95—532no—
159#95—532no—
160#95—532no—
161#95—532no—
162C24—531no—
163C2×Q8—24no—
164C2×Q8—22no—
165C22×C4—531no—
166C22×C4—532no—
167C22×C4—532no—
168C22×C4—531no—
169C22×C4—531no—
170C22×C4C22×D5312yes—
171C22×C4C22×D5312yes—
172C22×C4C22×D5311yescharacteristic
173C22×C4C22×D5311yescharacteristic
174C22×C4C22×D5311yescharacteristic
175C4⋊C4—24no—
176C4⋊C4—22no—
177C4⋊C4—1064no—
178C4⋊C4—1062no—
179C4⋊C4—1062no—
180C4⋊C4—1062no—
181C4⋊C4—2122no—
182C4⋊C4—2122no—
183C22⋊C4—1064no—
184C22⋊C4—1062no—
185C22⋊C4—1062no—
186C22⋊C4—1062no—
187C22⋊C4—2122no—
188C42—24no—
189C42C22⋊D5314yes—
190C23—1061no—
191C23—1062no—
192C23—1061no—
193C23—1061no—
194C23—1062no—
195C23C23×D5311yescharacteristic
196Q8—28no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—1061no—
207C2×C4—1061no—
208C2×C4—1061no—
209C2×C4—1062no—
210C2×C4—1062no—
211C2×C4—1062no—
212C2×C4—1062no—
213C2×C4—1061no—
214C2×C4—1062no—
215C2×C4—1061no—
216C2×C4—1062no—
217C2×C4—1062no—
218C2×C4—1062no—
219C2×C4#1171994514yes—
220C2×C4C2×C22⋊D5312yes—
221C2×C4C2×C22⋊D5312yes—
222C2×C4C2×C22⋊D5312yes—
223C22—2121no—
224C22—2121no—
225C22—2121no—
226C22—2122no—
227C22—2122no—
228C22C22×C22⋊D5311yescharacteristic
229C22C2×#1171994512yes—
230C22#1313326311yescharacteristic
231C22#1313326411yescharacteristic
232C22#1313326512yes—
233C4—24no—
234C4—22no—
235C4—22no—
236C4—22no—
237C4—42no—
238C4—2121no—
239C4—2121no—
240C4—2121no—
241C4—2122no—
242C4—2122no—
243C2—4241no—
244C2#1681113411yescharacteristic
245C2#1681119711yescharacteristic
246C2#1681120111yescharacteristic
247C2#1681123112yes—
248C2#1681124512yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×C4×Q8×C53C2
hypercenterC2×C4×Q8D53
hypocenterC53#2637
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
2. #16810556C2semidirect
3. #16810554C2semidirect
6. #16810542C2semidirect
9. #16810507C2semidirect
10. #16810489C2semidirect
11. C2×C4×Q8×C53C2semidirect
126. C2×C4×Q8D53semidirect
136. C53#2637semidirect