Finite Group Databasebeta

← #20282744

Subgroups — #20282744

Order 6784.

528 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 125 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×#13133197C211yesmaximal, characteristic, complemented
2C2×#13133190C211yesmaximal, characteristic
3#16810558C211yesmaximal, characteristic, complemented
4#16810556C212yesmaximal, complemented
5#16810540C212yesmaximal, complemented
6#16810539C212yesmaximal, complemented
7#16810538C212yesmaximal, complemented
8#16810536C211yesmaximal, characteristic, complemented
9#16810535C211yesmaximal, characteristic, complemented
10#16810485C211yesmaximal, characteristic, complemented
11C53×#399C211yesmaximal, characteristic, complemented
12C22×Dic106C2211yescharacteristic
13C22×D212C2211yescharacteristic
14C22×C22⋊D53C2211yescharacteristic
15C22×C22⋊D53C2211yescharacteristic
16C22×C4×D53C2211yescharacteristic
17#13133197—24no—
18#13133190—24no—
19C2×C4⋊Dic53C2212yes—
20C2×#11719919C2211yescharacteristic
21C2×#11719919C2212yes—
22C2×C22⋊Dic53C2212yes—
23C2×#11719921C2211yescharacteristic
24C2×#11719921C2212yes—
25C2×#11719921C2212yes—
26C2×#11719921C2212yes—
27C2×#11719921C2212yes—
28#13133124C2212yes—
29#13133124C2211yescharacteristic
30#13133124C2212yes—
31#13133124C2212yes—
32C2×C4⋊C4×C53C2211yescharacteristic
33C2×C4⋊C4×C53C2212yes—
34C2×C22⋊C4×C53C2211yescharacteristic
35C2×C22⋊C4×C53C2212yes—
36#95×C53C2211yescharacteristic
37C24×C53C2311yescharacteristic
38C23×D53C2311yescharacteristic
39C23×D53C2311yescharacteristic
40C22×C4×C53C2312yes—
41C22×C4×C53C2311yescharacteristic
42C22×C4×C53C2312yes—
43C22×C4×C53C2311yescharacteristic
44C2×C22⋊D53—22no—
45C2×C22⋊D53—22no—
46C2×C22⋊D53—42no—
47C2×C22⋊D53—42no—
48C22×Dic53C2312yes—
49C22×Dic53C2312yes—
50C22×Dic53C2311yescharacteristic
51C22×Dic53C2311yescharacteristic
52C2×D212—22no—
53C2×D212—42no—
54C2×C4×D53—42no—
55C2×C4×D53D414yes—
56C2×Dic106—22no—
57C2×Dic106—42no—
58C4⋊C4×C53—22no—
59C4⋊C4×C53—42no—
60C22⋊C4×C53—22no—
61C22⋊C4×C53—42no—
62C22⋊Dic53—42no—
63#11719921—24no—
64#11719921—22no—
65#11719921—42no—
66#11719921—42no—
67#11719921—42no—
68C4⋊Dic53—24no—
69#11719919—22no—
70#11719919—42no—
71C23×C53—21no—
72C23×C53—22no—
73C23×C53—22no—
74C23×C53—21no—
75C23×C53—21no—
76C23×C53C2411yescharacteristic
77C22×D53—22no—
78C22×D53—22no—
79C22×D53—21no—
80C22×D53—21no—
81C22×D53—21no—
82C22×D53—22no—
83C22×D53—22no—
84C22×D53—21no—
85C22×D53—21no—
86C22×D53C2×D412yes—
87C2×C4×C53—22no—
88C2×C4×C53—21no—
89C2×C4×C53—22no—
90C2×C4×C53—22no—
91C2×C4×C53—22no—
92C2×C4×C53—22no—
93C2×C4×C53—22no—
94C2×C4×C53—22no—
95C2×C4×C53—22no—
96C2×C4×C53C2×D412yes—
97C22⋊D53—44no—
98C22⋊D53—44no—
99C2×Dic53—22no—
100C2×Dic53—22no—
101C2×Dic53—21no—
102C2×Dic53—22no—
103C2×Dic53—22no—
104C2×Dic53—22no—
105C2×Dic53—22no—
106C2×Dic53—22no—
107C2×Dic53—22no—
108C2×Dic53C2×D412yes—
109D212—44no—
110C4×D53—44no—
111Dic106—44no—
112C22×C53—41no—
113C22×C53—41no—
114C22×C53—41no—
115C22×C53—42no—
116C22×C53—42no—
117C22×C53C22×D411yescharacteristic
118C22×C53H(5,𝔽2)12yes—
119C22×C53H(5,𝔽2)12yes—
120C22×C532−1+411yescharacteristic
121C22×C532−1+411yescharacteristic
122C2×D53—22no—
123C2×D53—22no—
124C2×D53—22no—
125C2×D53—42no—
126C2×D53—42no—
127C2×D53—41no—
128C2×D53—41no—
129C2×D53—41no—
130C2×D53—42no—
131C2×D53—42no—
132C4×C53—22no—
133C4×C53—41no—
134C4×C53—42no—
135C4×C53—42no—
136Dic53—22no—
137Dic53—42no—
138Dic53—41no—
139Dic53—42no—
140#2816—531nomaximal, Sylow
141C2×C53—81no—
142C2×C53#53411yescharacteristic
143C2×C53#53511yescharacteristic
144C2×C53#53611yescharacteristic
145C2×C53#56212yes—
146C2×C53#56212yes—
147D53—42no—
148D53—81no—
149C2×C22⋊Q8—531no—
150C2×#121—531no—
151#399—531no—
152#399D5311yescharacteristic, complemented
153#398—532no—
154#394—532no—
155#394—532no—
156#394—532no—
157#393—531no—
158#393—531no—
159#381—531no—
160C53#281611yescharacteristic, complemented, Sylow
161C22×Q8—531no—
162C22×D4—531no—
163C22×D4—531no—
164C22×D4—531no—
165C23×C4—531no—
166C22⋊Q8—1064no—
167#121—1064no—
168C2×C4⋊C4—531no—
169C2×C4⋊C4—532no—
170C2×C4⋊C4—532no—
171C2×C4⋊C4C2×D5311yescharacteristic
172C2×C4⋊C4C2×D5312yes—
173C2×C22⋊C4—531no—
174C2×C22⋊C4—532no—
175C2×C22⋊C4—532no—
176C2×C22⋊C4—532no—
177C2×C22⋊C4—532no—
178C2×C22⋊C4—532no—
179C2×C22⋊C4C2×D5311yescharacteristic
180C2×C22⋊C4C2×D5312yes—
181#95—531no—
182#95—532no—
183#95—532no—
184#95—532no—
185#95C2×D5311yescharacteristic
186C24—531no—
187C24—531no—
188C24C22×D5311yescharacteristic
189C2×Q8—1062no—
190C2×Q8—2122no—
191C2×D4—1062no—
192C2×D4—1062no—
193C2×D4—1062no—
194C2×D4—2122no—
195C2×D4—2122no—
196C2×D4—2122no—
197C22×C4—531no—
198C22×C4—534no—
199C22×C4—532no—
200C22×C4—532no—
201C22×C4—531no—
202C22×C4—2122no—
203C22×C4C22×D5312yes—
204C22×C4C22×D5311yescharacteristic
205C22×C4C22×D5312yes—
206C22×C4C22×D5311yescharacteristic
207C4⋊C4—22no—
208C4⋊C4—42no—
209C4⋊C4—1062no—
210C4⋊C4—1064no—
211C4⋊C4—2122no—
212C22⋊C4—22no—
213C22⋊C4—42no—
214C22⋊C4—1064no—
215C22⋊C4—1062no—
216C22⋊C4—2122no—
217C22⋊C4—2122no—
218C22⋊C4—2122no—
219C22⋊C4—2122no—
220C23—21no—
221C23—22no—
222C23—22no—
223C23—21no—
224C23—21no—
225C23—532no—
226C23—1062no—
227C23—1062no—
228C23—1061no—
229C23—1061no—
230C23—1061no—
231C23—1062no—
232C23—1062no—
233C23—1061no—
234C23—1061no—
235C23C23×D5311yescharacteristic
236Q8—2124no—
237D4—2124no—
238D4—2124no—
239D4—2124no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—532no—
250C2×C4—1062no—
251C2×C4—1061no—
252C2×C4—1062no—
253C2×C4—1062no—
254C2×C4—1062no—
255C2×C4—1062no—
256C2×C4—1062no—
257C2×C4—1062no—
258C2×C4—1062no—
259C2×C4—2124no—
260C2×C4D4×D5312yes—
261C22—42no—
262C22—42no—
263C22—41no—
264C22—41no—
265C22—41no—
266C22—1062no—
267C22—1062no—
268C22—1062no—
269C22—2122no—
270C22—2122no—
271C22—2121no—
272C22—2121no—
273C22—2121no—
274C22—2122no—
275C22—2122no—
276C22C2×D4×D5311yescharacteristic
277C22#1313326212yes—
278C22#1313326312yes—
279C22#1313326411yescharacteristic
280C22#1313326511yescharacteristic
281C4—22no—
282C4—42no—
283C4—41no—
284C4—42no—
285C4—1062no—
286C4—2122no—
287C4—2122no—
288C4—2121no—
289C2—81no—
290C2—4241no—
291C2—2122no—
292C2#1681118411yescharacteristic
293C2#1681118611yescharacteristic
294C2#1681119611yescharacteristic
295C2#1681136912yes—
296C2#1681137012yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#399C2
hypercenter#399D53
hypocenterC53#2816
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#13133197C2semidirect
3. #16810558C2semidirect
4. #16810556C2semidirect
5. #16810540C2semidirect
6. #16810539C2semidirect
7. #16810538C2semidirect
8. #16810536C2semidirect
9. #16810535C2semidirect
10. #16810485C2semidirect
11. C53×#399C2semidirect
152. #399D53semidirect
160. C53#2816semidirect