Finite Group Databasebeta

← #20281161

Subgroups — #20281161

Order 6784.

456 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133212C211yesmaximal, characteristic
2C2×#13133193C211yesmaximal, characteristic
3#16810565C211yesmaximal, characteristic, complemented
4#16810561C211yesmaximal, characteristic, complemented
5#16810555C211yesmaximal, characteristic, complemented
6#16810552C211yesmaximal, characteristic
7#16810550C211yesmaximal, characteristic
8#16810542C211yesmaximal, characteristic, complemented
9#16810521C211yesmaximal, characteristic, complemented
10#16810520C211yesmaximal, characteristic, complemented
11#16810514C211yesmaximal, characteristic
12#16810510C211yesmaximal, characteristic, complemented
13#16810497C211yesmaximal, characteristic
14#16810485C211yesmaximal, characteristic
15C53×#391C211yesmaximal, characteristic, complemented
16C22×C4×D53C2211yescharacteristic
17#13133212—24no—
18#13133193—24no—
19C2×C4⋊Dic53C2211yescharacteristic
20C2×C4⋊Dic53C2211yescharacteristic
21C2×C4⋊Dic53C2211yescharacteristic
22C2×C4⋊Dic53C2211yescharacteristic
23C2×#11719919C2211yescharacteristic
24C2×#11719919C2211yescharacteristic
25C2×#11719919C2211yescharacteristic
26C2×#11719919C2211yescharacteristic
27C2×#11719919C2211yescharacteristic
28C2×#11719919C2211yescharacteristic
29C2×#11719921C2211yescharacteristic
30C2×#11719921C2211yescharacteristic
31C2×#11719921C2211yescharacteristic
32C2×#11719921C2211yescharacteristic
33C2×#11719921C2211yescharacteristic
34C2×#11719921C2211yescharacteristic
35C2×C4×Dic53C2211yescharacteristic
36#13133124C2211yescharacteristic
37#13133124C2211yescharacteristic
38#13133124C2211yescharacteristic
39#13133124C2211yescharacteristic
40#13133124C2211yescharacteristic
41#13133124C2211yescharacteristic
42#13133124C2211yescharacteristic
43#13133124C2211yescharacteristic
44#13133124C2211yescharacteristic
45#13133124C2211yescharacteristic
46C22×Q8×C53C2211yescharacteristic
47C2×C4⋊C4×C53C2211yescharacteristic
48C2×C42×C53C2211yescharacteristic
49#95×C53C2211yescharacteristic
50#95×C53C2211yescharacteristic
51#95×C53C2211yescharacteristic
52#95×C53C2211yescharacteristic
53C23×D53C2311yescharacteristic
54C2×Q8×C53—22no—
55C2×Q8×C53—22no—
56C2×Q8×C53—41no—
57C22×C4×C53C2311yescharacteristic
58C22×C4×C53C2311yescharacteristic
59C22×C4×C53C2311yescharacteristic
60C22×C4×C53C2311yescharacteristic
61C22×C4×C53C2311yescharacteristic
62C22×C4×C53C2311yescharacteristic
63C22×C4×C53C2311yescharacteristic
64C22×Dic53C2311yescharacteristic
65C22×Dic53C2311yescharacteristic
66C22×Dic53C2311yescharacteristic
67C22×Dic53C2311yescharacteristic
68C22×Dic53C2311yescharacteristic
69C22×Dic53C2311yescharacteristic
70C22×Dic53C2311yescharacteristic
71C2×C4×D53—22no—
72C2×C4×D53—22no—
73C2×C4×D53—41no—
74C4⋊C4×C53—22no—
75C42×C53—41no—
76#11719921—22no—
77#11719921—22no—
78#11719921—22no—
79#11719921—22no—
80#11719921—41no—
81#11719921D414yes—
82C4⋊Dic53—22no—
83C4⋊Dic53—22no—
84C4⋊Dic53—22no—
85C4⋊Dic53—41no—
86#11719919—22no—
87#11719919—22no—
88#11719919—22no—
89#11719919—22no—
90#11719919—22no—
91#11719919—41no—
92C4×Dic53—22no—
93C23×C53C2411yescharacteristic
94C22×D53—21no—
95C22×D53—21no—
96C22×D53—21no—
97C22×D53—21no—
98C22×D53—21no—
99C22×D53—21no—
100C22×D53C2×D412yes—
101Q8×C53—44no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53C2×D412yes—
121C2×C4×C53#4112yes—
122C2×C4×C53#4112yes—
123C2×Dic53—21no—
124C2×Dic53—21no—
125C2×Dic53—21no—
126C2×Dic53—21no—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53—21no—
136C2×Dic53—21no—
137C2×Dic53—21no—
138C2×Dic53—21no—
139C2×Dic53—21no—
140C2×Dic53—21no—
141C2×Dic53—21no—
142C2×Dic53—21no—
143C2×Dic53C2×D412yes—
144C4×D53—44no—
145C22×C53C22×D411yescharacteristic
146C22×C53C2×#4111yescharacteristic
147C22×C53C2×#4111yescharacteristic
148C22×C53H(5,𝔽2)11yescharacteristic
149C22×C532−1+411yescharacteristic
150C22×C532−1+411yescharacteristic
151C22×C532−1+411yescharacteristic
152C2×D53—22no—
153C2×D53—22no—
154C2×D53—22no—
155C2×D53—41no—
156C2×D53—41no—
157C2×D53—41no—
158C2×D53—41no—
159C4×C53—22no—
160C4×C53—22no—
161C4×C53—22no—
162C4×C53—41no—
163C4×C53—41no—
164C4×C53—41no—
165C4×C53—41no—
166Dic53—22no—
167Dic53—41no—
168Dic53—41no—
169Dic53—41no—
170Dic53—41no—
171Dic53—41no—
172Dic53—41no—
173#2689—531nomaximal, Sylow
174C2×C53#53611yescharacteristic
175C2×C53#53911yescharacteristic
176C2×C53#54111yescharacteristic
177C2×C53#54611yescharacteristic
178C2×C53#54711yescharacteristic
179C2×C53#55611yescharacteristic
180C2×C53#56311yescharacteristic
181D53—42no—
182C2×#126—531no—
183C2×C22⋊Q8—531no—
184#400—531no—
185#400—531no—
186#398—531no—
187#397—531no—
188#397—531no—
189#395—531no—
190#391D5311yescharacteristic, complemented
191#389—531no—
192#389—531no—
193#388—531no—
194#387—531no—
195#385—531no—
196#381—531no—
197C53#268911yescharacteristic, complemented, Sylow
198C22×Q8C2×D5311yescharacteristic
199C23×C4—531no—
200#126—1064no—
201C22⋊Q8—1064no—
202C2×C4⋊C4—531no—
203C2×C4⋊C4—531no—
204C2×C4⋊C4—531no—
205C2×C4⋊C4—531no—
206C2×C4⋊C4—531no—
207C2×C4⋊C4—531no—
208C2×C4⋊C4—531no—
209C2×C4⋊C4—531no—
210C2×C4⋊C4—531no—
211C2×C4⋊C4—531no—
212C2×C4⋊C4C2×D5311yescharacteristic
213C2×C22⋊C4—531no—
214C2×C22⋊C4—531no—
215C2×C22⋊C4—531no—
216C2×C22⋊C4—531no—
217C2×C22⋊C4—531no—
218C2×C22⋊C4—531no—
219C2×C42—531no—
220C2×C42C2×D5311yescharacteristic
221#95—531no—
222#95—531no—
223#95—531no—
224#95—531no—
225#95—531no—
226#95—531no—
227#95—531no—
228#95—531no—
229#95—531no—
230#95—531no—
231#95C2×D5311yescharacteristic
232#95C2×D5311yescharacteristic
233#95C2×D5311yescharacteristic
234#95C2×D5311yescharacteristic
235C24—531no—
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—41no—
239C22×C4—531no—
240C22×C4—531no—
241C22×C4—531no—
242C22×C4—531no—
243C22×C4—531no—
244C22×C4—531no—
245C22×C4—531no—
246C22×C4—1062no—
247C22×C4—1062no—
248C22×C4—2121no—
249C22×C4C22×D5311yescharacteristic
250C22×C4C22×D5311yescharacteristic
251C22×C4C22×D5311yescharacteristic
252C22×C4C22×D5311yescharacteristic
253C22×C4C22×D5311yescharacteristic
254C22×C4C22×D5311yescharacteristic
255C22×C4C22×D5311yescharacteristic
256C4⋊C4—22no—
257C4⋊C4—1062no—
258C4⋊C4—1062no—
259C4⋊C4—1062no—
260C4⋊C4—1062no—
261C4⋊C4—1062no—
262C4⋊C4—1062no—
263C4⋊C4—1062no—
264C4⋊C4—1062no—
265C4⋊C4—2121no—
266C4⋊C4—2121no—
267C22⋊C4—534no—
268C22⋊C4—1062no—
269C22⋊C4—1062no—
270C22⋊C4—1062no—
271C22⋊C4—1062no—
272C22⋊C4—2121no—
273C42—41no—
274C42—1062no—
275C23—532no—
276C23—1061no—
277C23—1061no—
278C23—1061no—
279C23—1061no—
280C23—1061no—
281C23—1061no—
282C23C23×D5311yescharacteristic
283Q8—44no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—532no—
303C2×C4—1061no—
304C2×C4—1061no—
305C2×C4—1061no—
306C2×C4—1061no—
307C2×C4—1061no—
308C2×C4—1061no—
309C2×C4—1061no—
310C2×C4—1061no—
311C2×C4—1061no—
312C2×C4—1061no—
313C2×C4—1061no—
314C2×C4—1061no—
315C2×C4—1061no—
316C2×C4—1061no—
317C2×C4—1061no—
318C2×C4—1061no—
319C2×C4—1061no—
320C2×C4—1061no—
321C2×C4—1061no—
322C2×C4—1061no—
323C2×C4—2124no—
324C2×C4#1171994512yes—
325C2×C4D4×D5312yes—
326C2×C4#1171994912yes—
327C22—1062no—
328C22—1062no—
329C22—1062no—
330C22—2121no—
331C22—2121no—
332C22—2121no—
333C22—2121no—
334C22C2×D4×D5311yescharacteristic
335C22C2×#1171994511yescharacteristic
336C22C2×#1171994911yescharacteristic
337C22#1313326211yescharacteristic
338C22#1313326411yescharacteristic
339C22#1313326411yescharacteristic
340C22#1313326511yescharacteristic
341C4—22no—
342C4—22no—
343C4—22no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—1062no—
349C4—2121no—
350C4—2121no—
351C4—2121no—
352C4—2121no—
353C4—2121no—
354C4—2121no—
355C2—2122no—
356C2#1681119311yescharacteristic
357C2#1681122211yescharacteristic
358C2#1681123111yescharacteristic
359C2#1681126311yescharacteristic
360C2#1681127611yescharacteristic
361C2#1681134611yescharacteristic
362C2#1681137411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#391C2
hypercenter#391D53
hypocenterC53#2689
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
3. #16810565C2semidirect
4. #16810561C2semidirect
5. #16810555C2semidirect
8. #16810542C2semidirect
9. #16810521C2semidirect
10. #16810520C2semidirect
12. #16810510C2semidirect
15. C53×#391C2semidirect
190. #391D53semidirect
197. C53#2689semidirect