Finite Group Databasebeta

← #20280516

Subgroups — #20280516

Order 6784.

464 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 133 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×#13133195C211yesmaximal, characteristic
2C2×#13133175C211yesmaximal, characteristic
3#16810561C211yesmaximal, characteristic, complemented
4#16810555C211yesmaximal, characteristic, complemented
5#16810553C211yesmaximal, characteristic, complemented
6#16810552C212yesmaximal
7#16810550C212yesmaximal
8#16810527C212yesmaximal, complemented
9#16810520C212yesmaximal, complemented
10#16810482C211yesmaximal, characteristic
11C53×#400C211yesmaximal, characteristic, complemented
12C22×Dic106C2211yescharacteristic
13C22×C4×D53C2211yescharacteristic
14#13133195—24no—
15#13133175—24no—
16C2×C4⋊Dic53C2211yescharacteristic
17C2×C4⋊Dic53C2212yes—
18C2×#11719919C2211yescharacteristic
19C2×#11719919C2212yes—
20C2×#11719919C2211yescharacteristic
21C2×#11719919C2212yes—
22C2×#11719921C2211yescharacteristic
23C2×#11719921C2212yes—
24C2×#11719921C2211yescharacteristic
25C2×#11719921C2212yes—
26C2×C4×Dic53C2212yes—
27#13133124C2212yes—
28#13133124C2212yes—
29#13133124C2212yes—
30#13133124C2212yes—
31#13133124C2211yescharacteristic
32C2×C4⋊C4×C53C2211yescharacteristic
33C2×C4⋊C4×C53C2211yescharacteristic
34#95×C53C2212yes—
35#95×C53C2212yes—
36#95×C53C2211yescharacteristic
37C23×D53C2311yescharacteristic
38C22×C4×C53C2312yes—
39C22×C4×C53C2312yes—
40C22×C4×C53C2311yescharacteristic
41C22×C4×C53C2311yescharacteristic
42C22×C4×C53C2311yescharacteristic
43C22×Dic53C2312yes—
44C22×Dic53C2311yescharacteristic
45C22×Dic53C2312yes—
46C22×Dic53C2311yescharacteristic
47C22×Dic53C2311yescharacteristic
48C2×C4×D53—24no—
49C2×C4×D53D414yes—
50C2×Dic106—22no—
51C2×Dic106—24no—
52C4⋊C4×C53—22no—
53C4⋊C4×C53—22no—
54#11719921—24no—
55#11719921—22no—
56#11719921—22no—
57#11719921—42no—
58C4⋊Dic53—22no—
59C4⋊Dic53—42no—
60#11719919—22no—
61#11719919—22no—
62#11719919—24no—
63#11719919—42no—
64C4×Dic53—24no—
65C23×C53C2411yescharacteristic
66C22×D53—22no—
67C22×D53—21no—
68C22×D53—22no—
69C22×D53—21no—
70C22×D53C2×D412yes—
71C2×C4×C53—22no—
72C2×C4×C53—21no—
73C2×C4×C53—22no—
74C2×C4×C53—21no—
75C2×C4×C53—22no—
76C2×C4×C53—22no—
77C2×C4×C53—21no—
78C2×C4×C53—22no—
79C2×C4×C53—24no—
80C2×C4×C53—22no—
81C2×C4×C53—21no—
82C2×C4×C53C2×D412yes—
83C2×Dic53—21no—
84C2×Dic53—22no—
85C2×Dic53—21no—
86C2×Dic53—22no—
87C2×Dic53—22no—
88C2×Dic53—22no—
89C2×Dic53—22no—
90C2×Dic53—21no—
91C2×Dic53—22no—
92C2×Dic53—22no—
93C2×Dic53—21no—
94C2×Dic53C2×D412yes—
95C2×Dic53#4114yes—
96C4×D53—28no—
97Dic106—44no—
98C22×C53C22×D411yescharacteristic
99C22×C53C2×#4112yes—
100C22×C53H(5,𝔽2)11yescharacteristic
101C22×C532−1+412yes—
102C22×C532−1+411yescharacteristic
103C2×D53—22no—
104C2×D53—22no—
105C2×D53—22no—
106C2×D53—42no—
107C2×D53—42no—
108C4×C53—22no—
109C4×C53—41no—
110C4×C53—42no—
111C4×C53—41no—
112C4×C53—42no—
113Dic53—41no—
114Dic53—42no—
115Dic53—41no—
116Dic53—42no—
117Dic53#12314yes—
118#2625—531nomaximal, Sylow
119C2×C53C2×#12311yescharacteristic
120C2×C53#53611yescharacteristic
121C2×C53#53711yescharacteristic
122C2×C53#54112yes—
123C2×C53#54212yes—
124D53—42no—
125C2×C22⋊Q8—531no—
126C2×#117—531no—
127#400—531no—
128#400D5311yescharacteristic, complemented
129#398—531no—
130#398—531no—
131#397—532no—
132#397—532no—
133#391—532no—
134#389—532no—
135#380—531no—
136C53#262511yescharacteristic, complemented, Sylow
137C22×Q8—531no—
138C23×C4—531no—
139C22⋊Q8—1064no—
140#117—1064no—
141C2×C4⋊C4—531no—
142C2×C4⋊C4—531no—
143C2×C4⋊C4—531no—
144C2×C4⋊C4—532no—
145C2×C4⋊C4—532no—
146C2×C4⋊C4—532no—
147C2×C4⋊C4C2×D5311yescharacteristic
148C2×C4⋊C4C2×D5311yescharacteristic
149C2×C22⋊C4—532no—
150C2×C22⋊C4—531no—
151C2×C22⋊C4—531no—
152C2×C22⋊C4—532no—
153C2×C42—532no—
154#95—531no—
155#95—532no—
156#95—532no—
157#95—532no—
158#95—532no—
159#95C2×D5311yescharacteristic
160#95C2×D5312yes—
161#95C2×D5312yes—
162C24—531no—
163C2×Q8—1062no—
164C2×Q8—1064no—
165C22×C4—531no—
166C22×C4—534no—
167C22×C4—532no—
168C22×C4—531no—
169C22×C4—532no—
170C22×C4—531no—
171C22×C4—1064no—
172C22×C4C22×D5312yes—
173C22×C4C22×D5311yescharacteristic
174C22×C4C22×D5312yes—
175C22×C4C22×D5311yescharacteristic
176C22×C4C22×D5311yescharacteristic
177C4⋊C4—22no—
178C4⋊C4—22no—
179C4⋊C4—1062no—
180C4⋊C4—1064no—
181C4⋊C4—1062no—
182C4⋊C4—1062no—
183C4⋊C4—2122no—
184C4⋊C4—2122no—
185C22⋊C4—1064no—
186C22⋊C4—1062no—
187C22⋊C4—1062no—
188C22⋊C4—2122no—
189C42—1064no—
190C23—532no—
191C23—1062no—
192C23—1061no—
193C23—1062no—
194C23—1061no—
195C23C23×D5311yescharacteristic
196Q8—2124no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—24no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—532no—
209C2×C4—534no—
210C2×C4—1068no—
211C2×C4—1062no—
212C2×C4—1061no—
213C2×C4—1061no—
214C2×C4—1062no—
215C2×C4—1062no—
216C2×C4—1062no—
217C2×C4—1062no—
218C2×C4—1061no—
219C2×C4—1062no—
220C2×C4—1062no—
221C2×C4—1061no—
222C2×C4D4×D5312yes—
223C22—1062no—
224C22—1062no—
225C22—1062no—
226C22—2122no—
227C22—2122no—
228C22C2×D4×D5311yescharacteristic
229C22#41×D5312yes—
230C22#1313326311yescharacteristic
231C22#1313326412yes—
232C22#1313326511yescharacteristic
233C4—22no—
234C4—41no—
235C4—42no—
236C4—41no—
237C4—42no—
238C4—534no—
239C4—2121no—
240C4—2122no—
241C4—2121no—
242C4—2122no—
243C2—2122no—
244C2#123×D5311yescharacteristic
245C2#1681119311yescharacteristic
246C2#1681120311yescharacteristic
247C2#1681123212yes—
248C2#1681123912yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#400C2
hypercenter#400D53
hypocenterC53#2625
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
3. #16810561C2semidirect
4. #16810555C2semidirect
5. #16810553C2semidirect
8. #16810527C2semidirect
9. #16810520C2semidirect
11. C53×#400C2semidirect
128. #400D53semidirect
136. C53#2625semidirect