Finite Group Databasebeta

← #20280281

Subgroups — #20280281

Order 6784.

604 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 145 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#16810540C211yesmaximal, characteristic, complemented
5#16810532C211yesmaximal, characteristic
6#16810514C212yesmaximal, complemented
7#16810502C212yesmaximal, complemented
8#16810499C212yesmaximal, complemented
9#16810494C212yesmaximal, complemented
10#16810491C211yesmaximal, characteristic, complemented
11C2×#117×C53C211yesmaximal, characteristic, complemented
12C22×D212C2211yescharacteristic
13C22×C22⋊D53C2212yes—
14C22×C4×D53C2212yes—
15C4×D212—24no—
16C2×C4⋊Dic53C2211yescharacteristic
17C2×#11719919C2212yes—
18C2×C22⋊Dic53C2212yes—
19C2×#11719921C2212yes—
20C2×#11719921C2212yes—
21C2×#11719921C2212yes—
22C2×#11719921C2212yes—
23#13133124C2212yes—
24#13133124C2212yes—
25#13133124C2212yes—
26#13133124C2212yes—
27#13133124C2211yescharacteristic
28#13133124C2211yescharacteristic
29C23×C4×C53C2211yescharacteristic
30#117×C53—24no—
31C2×C4⋊C4×C53C2211yescharacteristic
32C2×C4⋊C4×C53C2211yescharacteristic
33C2×C22⋊C4×C53C2211yescharacteristic
34C2×C22⋊C4×C53C2211yescharacteristic
35C2×C42×C53C2211yescharacteristic
36C2×C42×C53C2211yescharacteristic
37C24×C53C2311yescharacteristic
38C23×D53C2312yes—
39C22×C4×C53—22no—
40C22×C4×C53—22no—
41C22×C4×C53—22no—
42C22×C4×C53C2311yescharacteristic
43C22×C4×C53C2311yescharacteristic
44C22×C4×C53C2311yescharacteristic
45C22×C4×C53C2311yescharacteristic
46C22×C4×C53C2311yescharacteristic
47C22×C4×C53C2311yescharacteristic
48C2×C22⋊D53—24no—
49C2×C22⋊D53—24no—
50C2×C22⋊D53—42no—
51C22×Dic53C2312yes—
52C22×Dic53C2312yes—
53C22×Dic53C2312yes—
54C2×D212—24no—
55C2×D212—22no—
56C2×C4×D53—24no—
57C2×C4×D53—42no—
58C2×C4×D53—42no—
59C4⋊C4×C53—22no—
60C4⋊C4×C53—22no—
61C22⋊C4×C53—22no—
62C22⋊C4×C53—22no—
63C42×C53—22no—
64C42×C53D414yescomplemented
65C22⋊Dic53—42no—
66#11719921—24no—
67#11719921—24no—
68#11719921—24no—
69#11719921—42no—
70C4⋊Dic53—22no—
71#11719919—42no—
72C23×C53—21no—
73C23×C53—22no—
74C23×C53—21no—
75C23×C53—22no—
76C23×C53#4112yes—
77C23×C53C2411yescharacteristic
78C22×D53—22no—
79C22×D53—22no—
80C22×D53—22no—
81C22×D53—22no—
82C22×D53—22no—
83C22×D53—22no—
84C22×D53#4114yes—
85C2×C4×C53—28no—
86C2×C4×C53—21no—
87C2×C4×C53—22no—
88C2×C4×C53—21no—
89C2×C4×C53—21no—
90C2×C4×C53—21no—
91C2×C4×C53—22no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—22no—
95C2×C4×C53—22no—
96C2×C4×C53C2×D412yes—
97C2×C4×C53C2×D412yes—
98C2×C4×C53C2×D412yes—
99C2×C4×C53#4112yes—
100C22⋊D53—48no—
101C2×Dic53—22no—
102C2×Dic53—22no—
103C2×Dic53—22no—
104C2×Dic53—22no—
105C2×Dic53—22no—
106C2×Dic53—22no—
107C2×Dic53—22no—
108C2×Dic53—22no—
109C2×Dic53—22no—
110D212—28no—
111C4×D53—48no—
112C22×C53—22no—
113C22×C53—24no—
114C22×C53—41no—
115C22×C53—41no—
116C22×C53—42no—
117C22×C53C22×D411yescharacteristic
118C22×C53C2×#4112yes—
119C22×C53C2×#4111yescharacteristic
120C22×C53C2×#4111yescharacteristic
121C22×C53H(5,𝔽2)12yes—
122C2×D53—24no—
123C2×D53—24no—
124C2×D53—24no—
125C2×D53—42no—
126C2×D53—42no—
127C2×D53—42no—
128C2×D53—42no—
129C4×C53—22no—
130C4×C53—22no—
131C4×C53—22no—
132C4×C53—22no—
133C4×C53—41no—
134C4×C53—41no—
135Dic53—42no—
136Dic53—42no—
137Dic53—42no—
138#2598—531nomaximal, Sylow
139C2×C53—42no—
140C2×C53#52511yescharacteristic
141C2×C53#52511yescharacteristic
142C2×C53#53511yescharacteristic
143C2×C53#55112yes—
144C2×C53#55312yes—
145D53—44no—
146C2×C4×D4—531no—
147C2×#117D5311yescharacteristic, complemented
148#400—531no—
149#399—531no—
150#394—531no—
151#392—531no—
152#388—532no—
153#386—532no—
154#386—532no—
155#385—532no—
156#383—531no—
157C53#259811yescharacteristic, complemented, Sylow
158C22×D4—532no—
159C22×D4—531no—
160C23×C4—532no—
161C23×C4C2×D5311yescharacteristic
162C4×D4—1064no—
163#117—24no—
164C2×C4⋊C4—531no—
165C2×C4⋊C4—532no—
166C2×C4⋊C4C2×D5311yescharacteristic
167C2×C4⋊C4C2×D5311yescharacteristic
168C2×C22⋊C4—532no—
169C2×C22⋊C4—532no—
170C2×C22⋊C4—532no—
171C2×C22⋊C4—532no—
172C2×C22⋊C4—532no—
173C2×C22⋊C4C2×D5311yescharacteristic
174C2×C22⋊C4C2×D5311yescharacteristic
175C2×C42C2×D5311yescharacteristic
176C2×C42C2×D5311yescharacteristic
177#95—532no—
178#95—532no—
179#95—531no—
180#95—531no—
181#95—532no—
182#95—532no—
183C24—532no—
184C24C22×D5311yescharacteristic
185C2×D4—1064no—
186C2×D4—1064no—
187C2×D4—1064no—
188C2×D4—1062no—
189C2×D4—2122no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—532no—
194C22×C4—532no—
195C22×C4—532no—
196C22×C4—1064no—
197C22×C4—2122no—
198C22×C4—2122no—
199C22×C4C22×D5311yescharacteristic
200C22×C4C22×D5311yescharacteristic
201C22×C4C22×D5311yescharacteristic
202C22×C4C22×D5311yescharacteristic
203C22×C4C22×D5311yescharacteristic
204C22×C4C22×D5311yescharacteristic
205C4⋊C4—22no—
206C4⋊C4—22no—
207C4⋊C4—1062no—
208C4⋊C4—2122no—
209C22⋊C4—22no—
210C22⋊C4—22no—
211C22⋊C4—1064no—
212C22⋊C4—1064no—
213C22⋊C4—1064no—
214C22⋊C4—2122no—
215C22⋊C4—2122no—
216C42—22no—
217C42C22⋊D5314yescomplemented
218C23—21no—
219C23—22no—
220C23—21no—
221C23—22no—
222C23—534no—
223C23—1062no—
224C23—1062no—
225C23—1062no—
226C23—1062no—
227C23—1062no—
228C23—1062no—
229C23#1171994512yes—
230C23C23×D5311yescharacteristic
231D4—1068no—
232D4—2128no—
233C2×C4—28no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—1062no—
245C2×C4—1062no—
246C2×C4—1062no—
247C2×C4—1062no—
248C2×C4—1062no—
249C2×C4—1062no—
250C2×C4—1062no—
251C2×C4—1062no—
252C2×C4—1062no—
253C2×C4—2128no—
254C2×C4#1171994512yes—
255C2×C4C2×C22⋊D5312yes—
256C2×C4C2×C22⋊D5312yes—
257C2×C4C2×C22⋊D5312yes—
258C22—24no—
259C22—22no—
260C22—42no—
261C22—41no—
262C22—41no—
263C22—1064no—
264C22—1064no—
265C22—1064no—
266C22—2122no—
267C22—2122no—
268C22—2122no—
269C22—2122no—
270C22C22×C22⋊D5311yescharacteristic
271C22#41×D5312yes—
272C22C2×#1171994511yescharacteristic
273C22C2×#1171994511yescharacteristic
274C22#1313326312yes—
275C4—22no—
276C4—22no—
277C4—22no—
278C4—22no—
279C4—41no—
280C4—41no—
281C4—2122no—
282C4—2122no—
283C4—2122no—
284C2—42no—
285C2—2124no—
286C2#1681113011yescharacteristic
287C2#1681113411yescharacteristic
288C2#1681118711yescharacteristic
289C2#1681129712yes—
290C2#1681131712yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×#117×C53C2
hypercenterC2×#117D53
hypocenterC53#2598
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. C2×C4×D212C2semidirect
2. #16810562C2semidirect
3. #16810559C2semidirect
4. #16810540C2semidirect
6. #16810514C2semidirect
7. #16810502C2semidirect
8. #16810499C2semidirect
9. #16810494C2semidirect
10. #16810491C2semidirect
11. C2×#117×C53C2semidirect
64. C42×C53D4semidirect
147. C2×#117D53semidirect
157. C53#2598semidirect
217. C42C22⋊D53semidirect