Finite Group Databasebeta

← #20280653

Subgroups — #20280653

Order 6784.

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

normal subgroups only

This group has 500 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×#13133215C211yesmaximal, characteristic, complemented
2C2×#13133203C211yesmaximal, characteristic
3#16810552C212yesmaximal, complemented
4#16810551C212yesmaximal, complemented
5#16810536C211yesmaximal, characteristic, complemented
6#16810535C211yesmaximal, characteristic, complemented
7#16810534C211yesmaximal, characteristic, complemented
8#16810527C212yesmaximal, complemented
9#16810523C212yesmaximal
10#16810468C211yesmaximal, characteristic, complemented
11C53×#393C211yesmaximal, characteristic, complemented
12C22×Dic106C2211yescharacteristic
13C22×Dic106C2211yescharacteristic
14C22×C22⋊D53C2211yescharacteristic
15#13133215—24no—
16#13133203—24no—
17C2×#11719919C2212yes—
18C2×#11719919C2212yes—
19C2×C22⋊Dic53C2211yescharacteristic
20C2×C22⋊Dic53C2212yes—
21C2×#11719921C2211yescharacteristic
22C2×#11719921C2211yescharacteristic
23C2×#11719921C2212yes—
24C2×#11719921C2212yes—
25C2×C4×Dic53C2212yes—
26C2×C4×Dic53C2211yescharacteristic
27#13133124C2212yes—
28#13133124C2212yes—
29#13133124C2212yes—
30#13133124C2211yescharacteristic
31#13133124C2212yes—
32C22×Q8×C53C2211yescharacteristic
33C2×C22⋊C4×C53C2211yescharacteristic
34C2×C22⋊C4×C53C2212yes—
35#95×C53C2212yes—
36#95×C53C2211yescharacteristic
37C24×C53C2311yescharacteristic
38C23×D53C2311yescharacteristic
39C2×Q8×C53—22no—
40C2×Q8×C53—42no—
41C22×C4×C53C2312yes—
42C22×C4×C53C2311yescharacteristic
43C22×C4×C53C2312yes—
44C22×C4×C53C2311yescharacteristic
45C2×C22⋊D53—22no—
46C2×C22⋊D53—24no—
47C22×Dic53C2312yes—
48C22×Dic53C2311yescharacteristic
49C22×Dic53C2312yes—
50C22×Dic53C2311yescharacteristic
51C22×Dic53C2311yescharacteristic
52C2×Dic106—22no—
53C2×Dic106—22no—
54C2×Dic106—24no—
55C2×Dic106—42no—
56C22⋊C4×C53—22no—
57C22⋊C4×C53—42no—
58C22⋊Dic53—22no—
59C22⋊Dic53—42no—
60#11719921—22no—
61#11719921—22no—
62#11719921—42no—
63#11719921—42no—
64#11719919—24no—
65#11719919—24no—
66C4×Dic53—24no—
67C4×Dic53D414yes—
68C23×C53—21no—
69C23×C53—22no—
70C23×C53—22no—
71C23×C53—21no—
72C23×C53—21no—
73C23×C53C2411yescharacteristic
74C22×D53—21no—
75C22×D53—22no—
76C22×D53—22no—
77C22×D53—21no—
78C22×D53—21no—
79Q8×C53—44no—
80C2×C4×C53—22no—
81C2×C4×C53—21no—
82C2×C4×C53—22no—
83C2×C4×C53—22no—
84C2×C4×C53—22no—
85C2×C4×C53—22no—
86C2×C4×C53—22no—
87C2×C4×C53—22no—
88C2×C4×C53—22no—
89C2×C4×C53C2×D412yes—
90C22⋊D53—44no—
91C2×Dic53—21no—
92C2×Dic53—21no—
93C2×Dic53—22no—
94C2×Dic53—21no—
95C2×Dic53—22no—
96C2×Dic53—22no—
97C2×Dic53—22no—
98C2×Dic53—22no—
99C2×Dic53—22no—
100C2×Dic53—22no—
101C2×Dic53C2×D412yes—
102C2×Dic53C2×D412yes—
103C2×Dic53#4114yes—
104Dic106—44no—
105Dic106—44no—
106C22×C53—41no—
107C22×C53—41no—
108C22×C53—41no—
109C22×C53—42no—
110C22×C53—42no—
111C22×C53C22×D411yescharacteristic
112C22×C53C2×#4112yes—
113C22×C53H(5,𝔽2)12yes—
114C22×C532−1+411yescharacteristic
115C22×C532−1+411yescharacteristic
116C2×D53—41no—
117C2×D53—41no—
118C2×D53—41no—
119C2×D53—42no—
120C2×D53—42no—
121C4×C53—22no—
122C4×C53—41no—
123C4×C53—42no—
124C4×C53—42no—
125Dic53—22no—
126Dic53—41no—
127Dic53—42no—
128Dic53—42no—
129Dic53#12414yes—
130#2640—531nomaximal, Sylow
131C2×C53—81no—
132C2×C53C2×#12411yescharacteristic
133C2×C53#53511yescharacteristic
134C2×C53#53611yescharacteristic
135C2×C53#54212yes—
136C2×C53#54212yes—
137D53—81no—
138C2×C4⋊Q8—531no—
139C2×#124—531no—
140#397—532no—
141#397—532no—
142#393—531no—
143#393—531no—
144#393—531no—
145#393D5311yescharacteristic, complemented
146#391—532no—
147#390—532no—
148#376—531no—
149C53#264011yescharacteristic, complemented, Sylow
150C22×Q8—531no—
151C22×Q8—531no—
152C22×Q8C2×D5311yescharacteristic
153C22×D4—531no—
154C4⋊Q8—1064no—
155#124—1064no—
156C2×C4⋊C4—532no—
157C2×C4⋊C4—532no—
158C2×C22⋊C4—531no—
159C2×C22⋊C4—531no—
160C2×C22⋊C4—531no—
161C2×C22⋊C4—532no—
162C2×C22⋊C4—532no—
163C2×C22⋊C4—532no—
164C2×C22⋊C4C2×D5311yescharacteristic
165C2×C22⋊C4C2×D5312yes—
166C2×C42—531no—
167C2×C42—532no—
168#95—531no—
169#95—532no—
170#95—532no—
171#95—532no—
172#95—532no—
173#95C2×D5311yescharacteristic
174#95C2×D5312yes—
175C24—531no—
176C24C22×D5311yescharacteristic
177C2×Q8—22no—
178C2×Q8—42no—
179C2×Q8—1062no—
180C2×Q8—1064no—
181C2×Q8—1062no—
182C2×Q8—2122no—
183C2×D4—1062no—
184C2×D4—1064no—
185C22×C4—531no—
186C22×C4—531no—
187C22×C4—532no—
188C22×C4—531no—
189C22×C4—532no—
190C22×C4C22×D5312yes—
191C22×C4C22×D5311yescharacteristic
192C22×C4C22×D5312yes—
193C22×C4C22×D5311yescharacteristic
194C4⋊C4—1064no—
195C4⋊C4—1064no—
196C22⋊C4—22no—
197C22⋊C4—42no—
198C22⋊C4—1062no—
199C22⋊C4—1062no—
200C22⋊C4—1062no—
201C22⋊C4—2122no—
202C22⋊C4—2122no—
203C22⋊C4—2122no—
204C42—534no—
205C42—1064no—
206C23—21no—
207C23—22no—
208C23—22no—
209C23—21no—
210C23—21no—
211C23—1061no—
212C23—1062no—
213C23—1062no—
214C23—1061no—
215C23—1061no—
216C23C23×D5311yescharacteristic
217Q8—44no—
218Q8—2124no—
219Q8—2124no—
220D4—2124no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—532no—
231C2×C4—534no—
232C2×C4—532no—
233C2×C4—1061no—
234C2×C4—1061no—
235C2×C4—1061no—
236C2×C4—1062no—
237C2×C4—1062no—
238C2×C4—1062no—
239C2×C4—1062no—
240C2×C4—1062no—
241C2×C4—1062no—
242C2×C4—1062no—
243C2×C4D4×D5312yes—
244C22—41no—
245C22—41no—
246C22—41no—
247C22—42no—
248C22—42no—
249C22—2121no—
250C22—2121no—
251C22—2121no—
252C22—2122no—
253C22—2122no—
254C22C2×D4×D5311yescharacteristic
255C22#41×D5312yes—
256C22#1313326212yes—
257C22#1313326411yescharacteristic
258C22#1313326511yescharacteristic
259C4—22no—
260C4—41no—
261C4—42no—
262C4—42no—
263C4—534no—
264C4—1062no—
265C4—2121no—
266C4—2122no—
267C4—2122no—
268C2—81no—
269C2—4241no—
270C2#124×D5311yescharacteristic
271C2#1681118811yescharacteristic
272C2#1681119611yescharacteristic
273C2#1681123712yes—
274C2#1681124312yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#393C2
hypercenter#393D53
hypocenterC53#2640
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×#13133215C2semidirect
3. #16810552C2semidirect
4. #16810551C2semidirect
5. #16810536C2semidirect
6. #16810535C2semidirect
7. #16810534C2semidirect
8. #16810527C2semidirect
10. #16810468C2semidirect
11. C53×#393C2semidirect
145. #393D53semidirect
149. C53#2640semidirect