Finite Group Databasebeta

← #20287722

Subgroups — #20287722

Order 6784.

384 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811356C211yesmaximal, characteristic, complemented
2#16811355C211yesmaximal, characteristic, complemented
3#16811005C211yesmaximal, characteristic, complemented
4#16811004C211yesmaximal, characteristic, complemented
5#16810998C211yesmaximal, characteristic
6#16810841C211yesmaximal, characteristic, complemented
7#16810839C211yesmaximal, characteristic, complemented
8#16810836C211yesmaximal, characteristic, complemented
9#16810822C211yesmaximal, characteristic, complemented
10#16810821C211yesmaximal, characteristic, complemented
11#16810747C211yesmaximal, characteristic, complemented
12#16810715C211yesmaximal, characteristic, complemented
13C4×#11719922C211yesmaximal, characteristic, complemented
14#16810703C211yesmaximal, characteristic, complemented
15C53×#463C211yesmaximal, characteristic, complemented
16C2×#11719925C2211yescharacteristic
17C2×#11719913C2211yescharacteristic
18C2×#11719923C2211yescharacteristic
19C2×#11719924C2211yescharacteristic
20C2×D424C2211yescharacteristic
21C2×#11719922C2211yescharacteristic
22C2×#11719922C2211yescharacteristic
23#13133213C2211yescharacteristic
24#13133213C2211yescharacteristic
25#13133206C2211yescharacteristic
26#13133205C2211yescharacteristic
27#13133204C2211yescharacteristic
28#13133203—21no—
29#13133203—21no—
30#13133201—21no—
31#13133200—21no—
32#13133199—21no—
33#13133199—21no—
34Q8×Dic53C2211yescharacteristic
35C4×Dic106C2211yescharacteristic
36#13133182C2211yescharacteristic
37C4×D212C2211yescharacteristic
38D4×Dic53C2211yescharacteristic
39#13133179C2211yescharacteristic
40C4×C22⋊D53—21no—
41#13133177—21no—
42#13133150C2211yescharacteristic
43#13133146C2211yescharacteristic
44#13133146C2211yescharacteristic
45#13133142C2211yescharacteristic
46#13133139C2211yescharacteristic
47#13133138C2211yescharacteristic
48#13133137C2211yescharacteristic
49C4×#88322C2211yescharacteristic
50C8×Dic53C2211yescharacteristic
51C8×Dic53C2211yescharacteristic
52C2×SD16×C53C2211yescharacteristic
53C2×D8×C53C2211yescharacteristic
54#124×C53C2211yescharacteristic
55C4×D4×C53C2211yescharacteristic
56C4⋊C8×C53C2211yescharacteristic
57#103×C53C2211yescharacteristic
58#102×C53C2211yescharacteristic
59C2×Q8×C53C2311yescharacteristic
60C2×D4×C53C2311yescharacteristic
61C2×D4×C53C2311yescharacteristic
62C22×C4×C53—21no—
63C2×C22⋊D53—21no—
64C2×C22⋊D53—21no—
65C2×C22⋊D53—21no—
66C2×C22⋊D53—21no—
67C22×Dic53—21no—
68C2×D212C2311yescharacteristic
69C2×D212C2311yescharacteristic
70C2×C4×D53—21no—
71C2×C4×D53—21no—
72C2×Dic106D411yescharacteristic
73C2×Dic106C2311yescharacteristic
74SD16×C53—41no—
75D8×C53—41no—
76C2×C8×C53C2311yescharacteristic
77C2×C8×C53C2311yescharacteristic
78C4⋊C4×C53C2311yescharacteristic
79C22⋊C4×C53—21no—
80C22⋊C4×C53—21no—
81C22⋊C4×C53—21no—
82C42×C53C2311yescharacteristic
83C22⋊Dic53—21no—
84C22⋊Dic53—21no—
85C22⋊Dic53—21no—
86#11719925—22no—
87#11719924—41no—
88#11719923—41no—
89#11719922—22no—
90#11719922—22no—
91#11719921—21no—
92#11719921—21no—
93#11719921—21no—
94#11719921—21no—
95#11719921—21no—
96#11719921—21no—
97C4⋊Dic53—21no—
98C4⋊Dic53D411yescharacteristic
99C4⋊Dic53C2311yescharacteristic
100#11719919—21no—
101#11719919D411yescharacteristic
102#11719919D411yescharacteristic
103C4×Dic53—21no—
104C4×Dic53—21no—
105C4×Dic53C2311yescharacteristic
106C4×Dic53C2311yescharacteristic
107C2×#88322C2311yescharacteristic
108C2×#88322C2311yescharacteristic
109D424—41no—
110#11719913—41no—
111C23×C53—21no—
112C23×C53—21no—
113C22×D53—21no—
114C22×D53—21no—
115Q8×C53—21no—
116Q8×C53—21no—
117D4×C53—21no—
118D4×C53—21no—
119D4×C53—21no—
120D4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—81no—
124C2×C4×C53C2×D411yescharacteristic
125C2×C4×C53C2×D411yescharacteristic
126C2×C4×C53C2411yescharacteristic
127C22⋊D53—42no—
128C22⋊D53—42no—
129C22⋊D53—81no—
130C22⋊D53—81no—
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—81no—
134C2×Dic53C2×D411yescharacteristic
135C2×Dic53C2×D411yescharacteristic
136C2×Dic53C2×D411yescharacteristic
137C2×Dic53C2×D411yescharacteristic
138D212—21no—
139D212—21no—
140D212—21no—
141D212—21no—
142C4×D53—81no—
143C4×D53—81no—
144Dic106—22no—
145Dic106—21no—
146Dic106—21no—
147C8×C53—21no—
148C8×C53—21no—
149#88322—21no—
150#88322#4112yes—
151C22×C53—41no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53—41no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53C22×D411yescharacteristic
158C2×D53—41no—
159C2×D53—41no—
160C2×D53—41no—
161C2×D53—41no—
162C2×D53—41no—
163C2×D53—41no—
164C4×C53—21no—
165C4×C53—21no—
166C4×C53—41no—
167C4×C53—41no—
168C4×C53C2×#4111yescharacteristic
169C4×C53H(5,𝔽2)11yescharacteristic
170Dic53—21no—
171Dic53—21no—
172Dic53—21no—
173Dic53—41no—
174Dic53—41no—
175Dic53#13512yes—
176#3396—531nomaximal, Sylow
177C2×C53—81no—
178C2×C53—81no—
179C2×C53#55011yescharacteristic
180C2×C53C2×#13511yescharacteristic
181C2×C53#57611yescharacteristic
182D53—81no—
183D53—81no—
184#559—531no—
185#559—531no—
186#495—531no—
187#495—531no—
188#493—531no—
189#463—531no—
190#463—531no—
191#463—531no—
192#463D5311yescharacteristic, complemented
193#460—531no—
194#460—531no—
195C8×Q8—531no—
196C4×Q16—531no—
197C4×D8—531no—
198C4×D8—531no—
199C53#339611yescharacteristic, complemented, Sylow
200C2×Q16—531no—
201C2×SD16—531no—
202C2×SD16—531no—
203C2×SD16—531no—
204C2×SD16C2×D5311yescharacteristic
205C2×D8—531no—
206C2×D8—531no—
207C2×D8—531no—
208C2×D8C2×D5311yescharacteristic
209#127—531no—
210#127—531no—
211#124—531no—
212#124—531no—
213#124—531no—
214#124—1061no—
215#124—1061no—
216#124C2×D5311yescharacteristic
217#123—1061no—
218#123—1061no—
219#123—1061no—
220#123—1061no—
221C4×Q8—531no—
222C4×Q8—531no—
223C4×Q8—531no—
224C4×D4—531no—
225C4×D4—531no—
226C4×D4—531no—
227C4×D4—1061no—
228C4×D4—1061no—
229C4×D4C2×D5311yescharacteristic
230#107—531no—
231C4⋊C8—531no—
232C4⋊C8—531no—
233C4⋊C8C2×D5311yescharacteristic
234#103—531no—
235#103C2×D5311yescharacteristic
236#102—531no—
237#102—531no—
238#102—531no—
239#102C2×D5311yescharacteristic
240C4×C8—531no—
241C4×C8—531no—
242C4×C8—531no—
243C2×Q8—531no—
244C2×Q8—531no—
245C2×Q8C22×D5311yescharacteristic
246C2×D4—531no—
247C2×D4—531no—
248C2×D4—1061no—
249C2×D4—1061no—
250C2×D4—1061no—
251C2×D4—1061no—
252C2×D4C22×D5311yescharacteristic
253C2×D4C22×D5311yescharacteristic
254C22×C4—21no—
255C22×C4—1061no—
256C22×C4—1061no—
257C22×C4—1061no—
258Q16—1062no—
259SD16—41no—
260SD16—2121no—
261SD16—2121no—
262SD16—2121no—
263D8—41no—
264D8—1062no—
265D8—1062no—
266D8—2121no—
267C2×C8—531no—
268C2×C8—531no—
269C2×C8C22×D5311yescharacteristic
270C2×C8C22×D5311yescharacteristic
271C4⋊C4—531no—
272C4⋊C4—531no—
273C4⋊C4—531no—
274C4⋊C4—531no—
275C4⋊C4—1061no—
276C4⋊C4—1061no—
277C4⋊C4C22×D5311yescharacteristic
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—21no—
281C22⋊C4—1061no—
282C22⋊C4—1061no—
283C22⋊C4—1061no—
284C22⋊C4—1061no—
285C22⋊C4—1061no—
286C22⋊C4—1061no—
287C22⋊C4—1061no—
288C22⋊C4—1061no—
289C22⋊C4—1061no—
290C42—531no—
291C42—531no—
292C42—1061no—
293C42—1061no—
294C42C22×D5311yescharacteristic
295C23—21no—
296C23—21no—
297C23—1061no—
298C23—1061no—
299Q8—21no—
300Q8—21no—
301Q8—1062no—
302Q8—1061no—
303Q8—1061no—
304D4—21no—
305D4—21no—
306D4—21no—
307D4—21no—
308D4—1061no—
309D4—1061no—
310D4—1061no—
311D4—1061no—
312D4—4241no—
313D4—4241no—
314D4—2122no—
315D4—2122no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—81no—
319C2×C4—531no—
320C2×C4—531no—
321C2×C4—531no—
322C2×C4—531no—
323C2×C4—1061no—
324C2×C4—1061no—
325C2×C4—4241no—
326C2×C4—4241no—
327C2×C4—4241no—
328C2×C4D4×D5311yescharacteristic
329C2×C4D4×D5311yescharacteristic
330C2×C4C23×D5311yescharacteristic
331C8—21no—
332C8—21no—
333C8—532no—
334C8—1061no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—2121no—
342C22—2121no—
343C22—2121no—
344C22—2121no—
345C22—2121no—
346C22—2121no—
347C22C2×D4×D5311yescharacteristic
348C4—21no—
349C4—21no—
350C4—41no—
351C4—41no—
352C4—532no—
353C4—1061no—
354C4—1061no—
355C4—1061no—
356C4—2121no—
357C4—2121no—
358C4#41×D5311yescharacteristic
359C4#1313326211yescharacteristic
360C2—81no—
361C2—81no—
362C2—4241no—
363C2—4241no—
364C2#1681129111yescharacteristic
365C2#135×D5311yescharacteristic
366C2#1681142711yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#463C2
socleC22×C53C22×D4
hypercenter#463D53
hypocenterC53#3396
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. #16811356C2semidirect
2. #16811355C2semidirect
3. #16811005C2semidirect
4. #16811004C2semidirect
6. #16810841C2semidirect
7. #16810839C2semidirect
8. #16810836C2semidirect
9. #16810822C2semidirect
10. #16810821C2semidirect
11. #16810747C2semidirect
12. #16810715C2semidirect
13. C4×#11719922C2semidirect
14. #16810703C2semidirect
15. C53×#463C2semidirect
192. #463D53semidirect
199. C53#3396semidirect