Finite Group Databasebeta

← #20279611

Subgroups — #20279611

Order 6784.

732 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 151 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×#13133189C212yesmaximal, complemented
2C2×D4×Dic53C211yesmaximal, characteristic, complemented
3C2×#13133177C211yesmaximal, characteristic, complemented
4#16810550C211yesmaximal, characteristic, complemented
5#16810548C211yesmaximal, characteristic, complemented
6#16810531C211yesmaximal, characteristic, complemented
7#16810503C212yesmaximal, complemented
8#16810499C212yesmaximal, complemented
9#16810487C211yesmaximal, characteristic, complemented
10#16810481C212yesmaximal, complemented
11C53×#392C211yesmaximal, characteristic, complemented
12C22×C22⋊D53C2212yes—
13C22×C22⋊D53C2211yescharacteristic
14C23×Dic53C2212yes—
15C23×Dic53C2211yescharacteristic
16C22×C4×D53C2211yescharacteristic
17#13133189—28no—
18D4×Dic53—24no—
19#13133177—24no—
20C2×C4⋊Dic53C2211yescharacteristic
21C2×#11719919C2211yescharacteristic
22C2×#11719919C2212yes—
23C2×C22⋊Dic53C2212yes—
24C2×C22⋊Dic53C2212yes—
25C2×C22⋊Dic53C2212yes—
26C2×#11719921C2212yes—
27C2×#11719921C2211yescharacteristic
28C2×C4×Dic53C2211yescharacteristic
29#13133124C2212yes—
30#13133124C2212yes—
31#13133124C2212yes—
32#13133124C2211yescharacteristic
33C22×D4×C53C2211yescharacteristic
34C22×D4×C53C2212yes—
35C2×C22⋊C4×C53C2211yescharacteristic
36C2×C22⋊C4×C53C2212yes—
37#95×C53C2211yescharacteristic
38C24×C53C2311yescharacteristic
39C24×C53C2312yes—
40C23×D53C2311yescharacteristic
41C2×D4×C53—24no—
42C2×D4×C53—24no—
43C2×D4×C53—24no—
44C2×D4×C53—22no—
45C2×D4×C53—42no—
46C22×C4×C53C2312yes—
47C22×C4×C53C2311yescharacteristic
48C22×C4×C53C2311yescharacteristic
49C2×C22⋊D53—24no—
50C2×C22⋊D53—24no—
51C2×C22⋊D53—22no—
52C2×C22⋊D53—22no—
53C2×C22⋊D53—42no—
54C2×C22⋊D53D414yescomplemented
55C22×Dic53—22no—
56C22×Dic53—24no—
57C22×Dic53—24no—
58C22×Dic53—24no—
59C22×Dic53—42no—
60C22×Dic53C2312yes—
61C22×Dic53C2311yescharacteristic
62C22×Dic53C2312yes—
63C22×Dic53C2311yescharacteristic
64C22×Dic53C2311yescharacteristic
65C2×C4×D53—22no—
66C2×C4×D53—42no—
67C22⋊C4×C53—22no—
68C22⋊C4×C53—42no—
69C22⋊Dic53—24no—
70C22⋊Dic53—24no—
71C22⋊Dic53—42no—
72#11719921—24no—
73#11719921—22no—
74C4⋊Dic53—22no—
75#11719919—22no—
76#11719919—24no—
77C4×Dic53D414yescomplemented
78C23×C53—22no—
79C23×C53—21no—
80C23×C53—22no—
81C23×C53—22no—
82C23×C53—21no—
83C23×C53—22no—
84C23×C53—22no—
85C23×C53—22no—
86C23×C53—22no—
87C23×C53—22no—
88C23×C53C2×D412yes—
89C23×C53#4114yes—
90C23×C53C2411yescharacteristic
91C22×D53—21no—
92C22×D53—21no—
93C22×D53—22no—
94C22×D53—22no—
95C22×D53C2×D412yes—
96D4×C53—28no—
97D4×C53—48no—
98C2×C4×C53—22no—
99C2×C4×C53—21no—
100C2×C4×C53—22no—
101C2×C4×C53—22no—
102C2×C4×C53—22no—
103C2×C4×C53—22no—
104C2×C4×C53C2×D412yes—
105C22⋊D53—28no—
106C22⋊D53—48no—
107C2×Dic53—22no—
108C2×Dic53—21no—
109C2×Dic53—22no—
110C2×Dic53—22no—
111C2×Dic53—22no—
112C2×Dic53—22no—
113C2×Dic53—22no—
114C2×Dic53—22no—
115C2×Dic53—22no—
116C2×Dic53—44no—
117C2×Dic53—48no—
118C2×Dic53C2×D412yes—
119C2×Dic53C2×D412yes—
120C2×Dic53C2×D412yes—
121C2×Dic53#4112yes—
122C4×D53—44no—
123C22×C53—22no—
124C22×C53—24no—
125C22×C53—24no—
126C22×C53—24no—
127C22×C53—24no—
128C22×C53—42no—
129C22×C53—41no—
130C22×C53—41no—
131C22×C53—42no—
132C22×C53—42no—
133C22×C53—42no—
134C22×C53—42no—
135C22×C53C22×D411yescharacteristic
136C22×C53C22×D411yescharacteristic
137C22×C53C2×#4111yescharacteristic
138C22×C53C2×#4112yes—
139C22×C53H(5,𝔽2)12yes—
140C2×D53—22no—
141C2×D53—24no—
142C2×D53—41no—
143C2×D53—41no—
144C2×D53—42no—
145C4×C53—22no—
146C4×C53—41no—
147C4×C53—42no—
148Dic53—22no—
149Dic53—42no—
150Dic53—41no—
151Dic53—42no—
152Dic53#12114yes—
153#2535—531nomaximal, Sylow
154C2×C53—42no—
155C2×C53—44no—
156C2×C53C2×#12111yescharacteristic
157C2×C53#52511yescharacteristic
158C2×C53#53511yescharacteristic
159C2×C53#54612yes—
160C2×C53#55312yes—
161D53—42no—
162C2×#121—532no—
163C2×C4×D4—531no—
164C2×C4×D4—531no—
165#397—531no—
166#397—531no—
167#392—531no—
168#392D5311yescharacteristic, complemented
169#386—532no—
170#386—532no—
171#382—531no—
172#380—532no—
173C53#253511yescharacteristic, complemented, Sylow
174C22×D4—531no—
175C22×D4—532no—
176C22×D4C2×D5311yescharacteristic
177C22×D4C2×D5312yes—
178C23×C4—532no—
179C23×C4—531no—
180C23×C4—531no—
181#121—1068no—
182C4×D4—1064no—
183C4×D4—1064no—
184C2×C4⋊C4—531no—
185C2×C4⋊C4—531no—
186C2×C4⋊C4—532no—
187C2×C22⋊C4—532no—
188C2×C22⋊C4—532no—
189C2×C22⋊C4—531no—
190C2×C22⋊C4—532no—
191C2×C22⋊C4—532no—
192C2×C22⋊C4C2×D5311yescharacteristic
193C2×C22⋊C4C2×D5312yes—
194C2×C42—531no—
195#95—532no—
196#95—532no—
197#95—532no—
198#95—531no—
199#95C2×D5311yescharacteristic
200C24—531no—
201C24C22×D5311yescharacteristic
202C24C22×D5312yes—
203C2×D4—24no—
204C2×D4—24no—
205C2×D4—24no—
206C2×D4—22no—
207C2×D4—42no—
208C2×D4—534no—
209C2×D4—1062no—
210C2×D4—1062no—
211C2×D4—1064no—
212C2×D4—1064no—
213C2×D4—2122no—
214C22×C4—531no—
215C22×C4—531no—
216C22×C4—532no—
217C22×C4—532no—
218C22×C4—531no—
219C22×C4—1062no—
220C22×C4—1064no—
221C22×C4—1062no—
222C22×C4—1064no—
223C22×C4—1064no—
224C22×C4—2122no—
225C22×C4—2122no—
226C22×C4C22×D5312yes—
227C22×C4C22×D5311yescharacteristic
228C22×C4C22×D5311yescharacteristic
229C4⋊C4—1062no—
230C4⋊C4—1064no—
231C4⋊C4—1062no—
232C22⋊C4—22no—
233C22⋊C4—42no—
234C22⋊C4—1064no—
235C22⋊C4—1064no—
236C22⋊C4—1064no—
237C22⋊C4—1062no—
238C22⋊C4—2122no—
239C42—534no—
240C23—21no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—21no—
245C23—22no—
246C23—22no—
247C23—22no—
248C23—22no—
249C23—22no—
250C23—532no—
251C23—1061no—
252C23—1061no—
253C23—1062no—
254C23—1062no—
255C23D4×D5312yes—
256C23#1171994714yes—
257C23C23×D5311yescharacteristic
258D4—28no—
259D4—48no—
260D4—1068no—
261D4—2128no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—532no—
269C2×C4—532no—
270C2×C4—532no—
271C2×C4—532no—
272C2×C4—1062no—
273C2×C4—1062no—
274C2×C4—1062no—
275C2×C4—1061no—
276C2×C4—1062no—
277C2×C4—1062no—
278C2×C4—1062no—
279C2×C4—1062no—
280C2×C4—1062no—
281C2×C4—2124no—
282C2×C4—2128no—
283C2×C4—2124no—
284C2×C4D4×D5312yes—
285C22—24no—
286C22—22no—
287C22—24no—
288C22—24no—
289C22—24no—
290C22—42no—
291C22—42no—
292C22—42no—
293C22—41no—
294C22—41no—
295C22—42no—
296C22—42no—
297C22—1062no—
298C22—1064no—
299C22—2122no—
300C22—2121no—
301C22—2121no—
302C22C2×D4×D5311yescharacteristic
303C22C2×D4×D5311yescharacteristic
304C22#41×D5311yescharacteristic
305C22C2×#1171994712yes—
306C22#1313326212yes—
307C4—22no—
308C4—41no—
309C4—42no—
310C4—534no—
311C4—1062no—
312C4—2122no—
313C4—2121no—
314C4—2122no—
315C2—44no—
316C2—42no—
317C2—2122no—
318C2#121×D5311yescharacteristic
319C2#1681112911yescharacteristic
320C2#1681118611yescharacteristic
321C2#1681126012yes—
322C2#1681131612yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#392C2
hypercenter#392D53
hypocenterC53#2535
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#13133189C2semidirect
2. C2×D4×Dic53C2semidirect
3. C2×#13133177C2semidirect
4. #16810550C2semidirect
5. #16810548C2semidirect
6. #16810531C2semidirect
7. #16810503C2semidirect
8. #16810499C2semidirect
9. #16810487C2semidirect
10. #16810481C2semidirect
11. C53×#392C2semidirect
54. C2×C22⋊D53D4semidirect
77. C4×Dic53D4semidirect
168. #392D53semidirect
173. C53#2535semidirect