Finite Group Databasebeta

← #20280612

Subgroups — #20280612

Order 6784.

562 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 562 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×C4×D212C211yesmaximal, characteristic, complemented
2#16810562C211yesmaximal, characteristic, complemented
3#16810559C211yesmaximal, characteristic, complemented
4#16810545C212yesmaximal, complemented
5#16810540C211yesmaximal, characteristic, complemented
6#16810539C212yesmaximal, complemented
7#16810532C211yesmaximal, characteristic
8#16810499C212yesmaximal, complemented
9#16810494C212yesmaximal, complemented
10#16810489C211yesmaximal, characteristic, complemented
11C2×#126×C53C211yesmaximal, characteristic, complemented
12C22×D212C2211yescharacteristic
13C22×C22⋊D53C2212yes—
14C22×C4×D53C2212yes—
15C4×D212—24no—
16C2×C4⋊Dic53C2211yescharacteristic
17C2×C4⋊Dic53C2212yes—
18C2×#11719919C2212yes—
19C2×C22⋊Dic53C2212yes—
20C2×#11719921C2212yes—
21C2×#11719921C2212yes—
22C2×#11719921C2212yes—
23C2×#11719921C2212yes—
24#13133124C2212yes—
25#13133124C2212yes—
26#13133124C2211yescharacteristic
27#13133124C2212yes—
28#13133124C2211yescharacteristic
29#126×C53—24no—
30C2×C4⋊C4×C53C2211yescharacteristic
31C2×C4⋊C4×C53C2211yescharacteristic
32C2×C4⋊C4×C53C2211yescharacteristic
33C2×C22⋊C4×C53C2211yescharacteristic
34C2×C22⋊C4×C53C2211yescharacteristic
35C2×C22⋊C4×C53C2211yescharacteristic
36C2×C42×C53C2211yescharacteristic
37C24×C53C2311yescharacteristic
38C23×D53C2312yes—
39C22×C4×C53C2311yescharacteristic
40C22×C4×C53C2311yescharacteristic
41C22×C4×C53C2311yescharacteristic
42C22×C4×C53C2311yescharacteristic
43C22×C4×C53C2311yescharacteristic
44C22×C4×C53C2311yescharacteristic
45C2×C22⋊D53—24no—
46C2×C22⋊D53—42no—
47C2×C22⋊D53—42no—
48C22×Dic53C2312yes—
49C22×Dic53C2312yes—
50C22×Dic53C2312yes—
51C2×D212—24no—
52C2×D212—22no—
53C2×C4×D53—24no—
54C2×C4×D53—42no—
55C2×C4×D53—42no—
56C4⋊C4×C53—22no—
57C4⋊C4×C53—22no—
58C4⋊C4×C53—22no—
59C22⋊C4×C53—22no—
60C22⋊C4×C53—22no—
61C22⋊C4×C53—22no—
62C42×C53D414yescomplemented
63C22⋊Dic53—42no—
64#11719921—24no—
65#11719921—24no—
66#11719921—42no—
67#11719921—42no—
68C4⋊Dic53—22no—
69C4⋊Dic53—42no—
70#11719919—42no—
71C23×C53—21no—
72C23×C53—22no—
73C23×C53—21no—
74C23×C53—22no—
75C23×C53—21no—
76C23×C53C2411yescharacteristic
77C22×D53—22no—
78C22×D53—22no—
79C22×D53—22no—
80C22×D53—22no—
81C22×D53—22no—
82C22×D53—22no—
83C22×D53#4114yes—
84C2×C4×C53—22no—
85C2×C4×C53—22no—
86C2×C4×C53—21no—
87C2×C4×C53—21no—
88C2×C4×C53—21no—
89C2×C4×C53—21no—
90C2×C4×C53—21no—
91C2×C4×C53—22no—
92C2×C4×C53—22no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53C2×D412yes—
96C2×C4×C53C2×D412yes—
97C2×C4×C53C2×D412yes—
98C22⋊D53—48no—
99C2×Dic53—22no—
100C2×Dic53—22no—
101C2×Dic53—22no—
102C2×Dic53—22no—
103C2×Dic53—22no—
104C2×Dic53—22no—
105C2×Dic53—22no—
106C2×Dic53—22no—
107C2×Dic53—22no—
108D212—28no—
109C4×D53—48no—
110C22×C53—41no—
111C22×C53—41no—
112C22×C53—41no—
113C22×C53—42no—
114C22×C53—42no—
115C22×C53C22×D411yescharacteristic
116C22×C53C2×#4112yes—
117C22×C53H(5,𝔽2)12yes—
118C22×C53H(5,𝔽2)11yescharacteristic
119C22×C532−1+411yescharacteristic
120C2×D53—24no—
121C2×D53—24no—
122C2×D53—24no—
123C2×D53—42no—
124C2×D53—42no—
125C2×D53—42no—
126C2×D53—42no—
127C4×C53—22no—
128C4×C53—22no—
129C4×C53—22no—
130C4×C53—41no—
131C4×C53—41no—
132C4×C53—41no—
133Dic53—42no—
134Dic53—42no—
135Dic53—42no—
136#2636—531nomaximal, Sylow
137C2×C53—81no—
138C2×C53#52511yescharacteristic
139C2×C53#53511yescharacteristic
140C2×C53#53711yescharacteristic
141C2×C53#53812yes—
142C2×C53#53912yes—
143D53—44no—
144C2×#126D5311yescharacteristic, complemented
145C2×C4×D4—531no—
146#400—531no—
147#399—531no—
148#396—532no—
149#394—531no—
150#394—532no—
151#392—531no—
152#386—532no—
153#385—532no—
154#382—531no—
155C53#263611yescharacteristic, complemented, Sylow
156C22×D4—532no—
157C22×D4—531no—
158C23×C4—532no—
159#126—24no—
160C4×D4—1064no—
161C2×C4⋊C4—531no—
162C2×C4⋊C4—532no—
163C2×C4⋊C4—532no—
164C2×C4⋊C4C2×D5311yescharacteristic
165C2×C4⋊C4C2×D5311yescharacteristic
166C2×C4⋊C4C2×D5311yescharacteristic
167C2×C22⋊C4—532no—
168C2×C22⋊C4—532no—
169C2×C22⋊C4—532no—
170C2×C22⋊C4—532no—
171C2×C22⋊C4—532no—
172C2×C22⋊C4C2×D5311yescharacteristic
173C2×C22⋊C4C2×D5311yescharacteristic
174C2×C22⋊C4C2×D5311yescharacteristic
175C2×C42C2×D5311yescharacteristic
176#95—532no—
177#95—531no—
178#95—531no—
179#95—532no—
180#95—532no—
181C24—532no—
182C24C22×D5311yescharacteristic
183C2×D4—1064no—
184C2×D4—1064no—
185C2×D4—1062no—
186C2×D4—2122no—
187C2×D4—2122no—
188C22×C4—532no—
189C22×C4—532no—
190C22×C4—532no—
191C22×C4—1064no—
192C22×C4—2122no—
193C22×C4—2122no—
194C22×C4C22×D5311yescharacteristic
195C22×C4C22×D5311yescharacteristic
196C22×C4C22×D5311yescharacteristic
197C22×C4C22×D5311yescharacteristic
198C22×C4C22×D5311yescharacteristic
199C22×C4C22×D5311yescharacteristic
200C4⋊C4—22no—
201C4⋊C4—22no—
202C4⋊C4—22no—
203C4⋊C4—1062no—
204C4⋊C4—2122no—
205C4⋊C4—2122no—
206C22⋊C4—22no—
207C22⋊C4—22no—
208C22⋊C4—22no—
209C22⋊C4—1064no—
210C22⋊C4—1064no—
211C22⋊C4—2122no—
212C22⋊C4—2122no—
213C22⋊C4—2122no—
214C42C22⋊D5314yescomplemented
215C23—21no—
216C23—22no—
217C23—22no—
218C23—21no—
219C23—21no—
220C23—534no—
221C23—1062no—
222C23—1062no—
223C23—1062no—
224C23—1062no—
225C23—1062no—
226C23—1062no—
227C23C23×D5311yescharacteristic
228D4—1068no—
229D4—2128no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—1062no—
242C2×C4—1062no—
243C2×C4—1062no—
244C2×C4—1062no—
245C2×C4—1062no—
246C2×C4—1062no—
247C2×C4—1062no—
248C2×C4—1062no—
249C2×C4—1062no—
250C2×C4—2128no—
251C2×C4C2×C22⋊D5312yes—
252C2×C4C2×C22⋊D5312yes—
253C2×C4C2×C22⋊D5312yes—
254C22—41no—
255C22—41no—
256C22—41no—
257C22—42no—
258C22—42no—
259C22—1064no—
260C22—1064no—
261C22—1064no—
262C22—2122no—
263C22—2122no—
264C22—2122no—
265C22—2122no—
266C22C22×C22⋊D5311yescharacteristic
267C22#41×D5312yes—
268C22#1313326312yes—
269C22#1313326311yescharacteristic
270C22#1313326411yescharacteristic
271C4—22no—
272C4—22no—
273C4—22no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—2122no—
278C4—2122no—
279C4—2122no—
280C2—81no—
281C2—2124no—
282C2#1681113011yescharacteristic
283C2#1681118711yescharacteristic
284C2#1681120111yescharacteristic
285C2#1681121212yes—
286C2#1681121412yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×#126×C53C2
hypercenterC2×#126D53
hypocenterC53#2636
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. C2×C4×D212C2semidirect
2. #16810562C2semidirect
3. #16810559C2semidirect
4. #16810545C2semidirect
5. #16810540C2semidirect
6. #16810539C2semidirect
8. #16810499C2semidirect
9. #16810494C2semidirect
10. #16810489C2semidirect
11. C2×#126×C53C2semidirect
62. C42×C53D4semidirect
144. C2×#126D53semidirect
155. C53#2636semidirect
214. C42C22⋊D53semidirect