Finite Group Databasebeta

← #20286152

Subgroups — #20286152

Order 6784.

370 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811227C211yesmaximal, characteristic, complemented
2#16811224C211yesmaximal, characteristic, complemented
3#16810925C211yesmaximal, characteristic, complemented
4#16810924C211yesmaximal, characteristic, complemented
5#16810923C211yesmaximal, characteristic, complemented
6#16810921C211yesmaximal, characteristic, complemented
7#16810875C211yesmaximal, characteristic, complemented
8#16810873C211yesmaximal, characteristic, complemented
9#16810858C211yesmaximal, characteristic, complemented
10#16810856C211yesmaximal, characteristic, complemented
11#16810816C211yesmaximal, characteristic, complemented
12#16810816C211yesmaximal, characteristic, complemented
13#16810815C211yesmaximal, characteristic, complemented
14#16810648C211yesmaximal, characteristic, complemented
15C53×#459C211yesmaximal, characteristic, complemented
16C2×#11719922C2211yescharacteristic
17C2×#11719922C2211yescharacteristic
18C2×#11719912C2211yescharacteristic
19C2×C8×D53C2211yescharacteristic
20#13133213C2211yescharacteristic
21#13133209C2211yescharacteristic
22#13133208C2211yescharacteristic
23#13133200—21no—
24#13133199—21no—
25#13133198—21no—
26#13133198—21no—
27#13133192C2211yescharacteristic
28#13133191—21no—
29#13133190—21no—
30#13133189—21no—
31#13133189—21no—
32#13133188C2211yescharacteristic
33#13133188C2211yescharacteristic
34#13133188C2211yescharacteristic
35C4×D212C2211yescharacteristic
36D4×Dic53C2211yescharacteristic
37D4×Dic53C2211yescharacteristic
38#13133174C2211yescharacteristic
39#13133151C2211yescharacteristic
40#13133150C2211yescharacteristic
41#13133149C2211yescharacteristic
42#13133148C2211yescharacteristic
43C4⋊#88322C2211yescharacteristic
44#13133146C2211yescharacteristic
45#13133146C2211yescharacteristic
46#13133139C2211yescharacteristic
47#13133139C2211yescharacteristic
48#13133138C2211yescharacteristic
49#13133138C2211yescharacteristic
50#13133138C2211yescharacteristic
51#13133138C2211yescharacteristic
52C2×D8×C53C2211yescharacteristic
53C2×D8×C53C2211yescharacteristic
54#127×C53C2211yescharacteristic
55C4×D4×C53C2211yescharacteristic
56C4⋊C8×C53C2211yescharacteristic
57#102×C53C2211yescharacteristic
58#102×C53C2211yescharacteristic
59C2×D4×C53—21no—
60C2×D4×C53—21no—
61C2×D4×C53C2311yescharacteristic
62C2×D4×C53C2311yescharacteristic
63C2×D4×C53C2311yescharacteristic
64C22×C4×C53—21no—
65C2×C22⋊D53—21no—
66C2×C22⋊D53—21no—
67C2×C22⋊D53—21no—
68C2×C22⋊D53—21no—
69C2×C22⋊D53—21no—
70C22×Dic53—21no—
71C22×Dic53—21no—
72C2×D212—21no—
73C2×D212C2311yescharacteristic
74C2×C4×D53—21no—
75C2×C4×D53C2311yescharacteristic
76D8×C53—41no—
77D8×C53—41no—
78C2×C8×C53C2311yescharacteristic
79C2×C8×C53C2311yescharacteristic
80C4⋊C4×C53C2311yescharacteristic
81C22⋊C4×C53—21no—
82C42×C53C2311yescharacteristic
83C22⋊Dic53—21no—
84C22⋊Dic53—21no—
85C22⋊Dic53—21no—
86C22⋊Dic53—21no—
87C22⋊Dic53—21no—
88#11719922—41no—
89#11719922—41no—
90#11719921—21no—
91#11719921—21no—
92#11719921D411yescharacteristic
93#11719921D411yescharacteristic
94C4⋊Dic53—21no—
95C4⋊Dic53C2311yescharacteristic
96C4⋊Dic53C2311yescharacteristic
97C4⋊Dic53C2311yescharacteristic
98#11719919—21no—
99#11719919—21no—
100#11719919—21no—
101#11719919D411yescharacteristic
102#11719919D411yescharacteristic
103C4×Dic53C2311yescharacteristic
104C2×#88322C2311yescharacteristic
105C2×#88322C2311yescharacteristic
106#11719912—41no—
107C8×D53—41no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—21no—
111C22×D53—21no—
112C22×D53C2×D411yescharacteristic
113D4×C53—21no—
114D4×C53—21no—
115D4×C53—21no—
116D4×C53—21no—
117D4×C53—21no—
118D4×C53—21no—
119D4×C53—81no—
120D4×C53—81no—
121C2×C4×C53—21no—
122C2×C4×C53—81no—
123C2×C4×C53C2×D411yescharacteristic
124C2×C4×C53C2×D411yescharacteristic
125C2×C4×C53C2411yescharacteristic
126C22⋊D53—81no—
127C22⋊D53—81no—
128C22⋊D53—81no—
129C22⋊D53—81no—
130C22⋊D53—81no—
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—81no—
135C2×Dic53—81no—
136C2×Dic53C2×D411yescharacteristic
137C2×Dic53C2×D411yescharacteristic
138C2×Dic53C2×D411yescharacteristic
139D212—21no—
140D212—21no—
141D212—81no—
142C4×D53—21no—
143C4×D53—21no—
144C4×D53—81no—
145C8×C53—21no—
146C8×C53—21no—
147#88322—21no—
148#88322—21no—
149C22×C53—41no—
150C22×C53—41no—
151C22×C53—41no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53—41no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53C22×D411yescharacteristic
159C2×D53—21no—
160C2×D53—21no—
161C2×D53—21no—
162C2×D53—41no—
163C2×D53—41no—
164C2×D53—41no—
165C4×C53—21no—
166C4×C53—21no—
167C4×C53—41no—
168C4×C53H(5,𝔽2)11yescharacteristic
169C4×C532−1+411yescharacteristic
170Dic53—21no—
171Dic53—21no—
172Dic53—21no—
173Dic53—41no—
174Dic53—41no—
175Dic53—41no—
176#3259—531nomaximal, Sylow
177C2×C53—81no—
178C2×C53—81no—
179C2×C53—81no—
180C2×C53#53711yescharacteristic
181C2×C53#57611yescharacteristic
182C2×C53#57611yescharacteristic
183D53—41no—
184D53—81no—
185#540—531no—
186#540—531no—
187#478—531no—
188#478—531no—
189#478—531no—
190#478—531no—
191#469—531no—
192#469—531no—
193#466—531no—
194#466—531no—
195#459—531no—
196#459—531no—
197#459—531no—
198#459D5311yescharacteristic, complemented
199#424—531no—
200C53#325911yescharacteristic, complemented, Sylow
201C2×D8—531no—
202C2×D8—531no—
203C2×D8C2×D5311yescharacteristic
204C2×D8C2×D5311yescharacteristic
205C2×M16—531no—
206C22×C8—531no—
207#127—531no—
208#127C2×D5311yescharacteristic
209#125—531no—
210#125—531no—
211#123—1061no—
212#123—1061no—
213#123—1061no—
214#123—1061no—
215#121—531no—
216#121—531no—
217#121—531no—
218#121—531no—
219#121—1061no—
220#121—1061no—
221#121—1061no—
222#121—1061no—
223C4×D4—531no—
224C4×D4—531no—
225C4×D4—531no—
226C4×D4C2×D5311yescharacteristic
227#117—531no—
228#107—531no—
229#107—531no—
230#106—531no—
231#106—531no—
232C4⋊C8—531no—
233C4⋊C8—531no—
234C4⋊C8—531no—
235C4⋊C8C2×D5311yescharacteristic
236#102—531no—
237#102—531no—
238#102—531no—
239#102—531no—
240#102—531no—
241#102—531no—
242#102C2×D5311yescharacteristic
243#102C2×D5311yescharacteristic
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—531no—
247C2×D4—1061no—
248C2×D4—1061no—
249C2×D4—1061no—
250C2×D4—1061no—
251C2×D4—1061no—
252C2×D4—1061no—
253C2×D4C22×D5311yescharacteristic
254C2×D4C22×D5311yescharacteristic
255C2×D4C22×D5311yescharacteristic
256C22×C4—21no—
257C22×C4—531no—
258C22×C4—1061no—
259C22×C4—1061no—
260C22×C4—1061no—
261D8—41no—
262D8—41no—
263D8—2121no—
264D8—2121no—
265M16—2121no—
266C2×C8—531no—
267C2×C8—531no—
268C2×C8—2121no—
269C2×C8C22×D5311yescharacteristic
270C2×C8C22×D5311yescharacteristic
271C4⋊C4—531no—
272C4⋊C4—531no—
273C4⋊C4—531no—
274C4⋊C4—531no—
275C4⋊C4—531no—
276C4⋊C4—1061no—
277C4⋊C4—1061no—
278C4⋊C4—1061no—
279C4⋊C4—1061no—
280C4⋊C4C22×D5311yescharacteristic
281C22⋊C4—21no—
282C22⋊C4—531no—
283C22⋊C4—531no—
284C22⋊C4—1061no—
285C22⋊C4—1061no—
286C22⋊C4—1061no—
287C22⋊C4—1061no—
288C22⋊C4—1061no—
289C22⋊C4—1061no—
290C22⋊C4—1061no—
291C42—531no—
292C42C22×D5311yescharacteristic
293C23—21no—
294C23—21no—
295C23—21no—
296C23—531no—
297C23—1061no—
298D4—21no—
299D4—21no—
300D4—21no—
301D4—21no—
302D4—21no—
303D4—21no—
304D4—81no—
305D4—81no—
306D4—1061no—
307D4—1061no—
308D4—4241no—
309D4—4241no—
310D4—4241no—
311D4—4241no—
312D4—4241no—
313D4—4241no—
314C2×C4—21no—
315C2×C4—81no—
316C2×C4—531no—
317C2×C4—531no—
318C2×C4—531no—
319C2×C4—1061no—
320C2×C4—1061no—
321C2×C4—1061no—
322C2×C4—1061no—
323C2×C4—1061no—
324C2×C4—4241no—
325C2×C4—4241no—
326C2×C4—4241no—
327C2×C4D4×D5311yescharacteristic
328C2×C4D4×D5311yescharacteristic
329C2×C4C23×D5311yescharacteristic
330C8—21no—
331C8—21no—
332C8—1061no—
333C8—1061no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—1061no—
344C22—1061no—
345C22—1061no—
346C22—2121no—
347C22—2121no—
348C22—2121no—
349C22C2×D4×D5311yescharacteristic
350C4—21no—
351C4—21no—
352C4—41no—
353C4—1061no—
354C4—1061no—
355C4—1061no—
356C4—2121no—
357C4—2121no—
358C4—2121no—
359C4#1313326211yescharacteristic
360C4#1313326411yescharacteristic
361C2—81no—
362C2—81no—
363C2—81no—
364C2—4241no—
365C2—2121no—
366C2#1681120211yescharacteristic
367C2#1681142411yescharacteristic
368C2#1681142711yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#459C2
socleC22×C53C22×D4
hypercenter#459D53
hypocenterC53#3259
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #16811227C2semidirect
2. #16811224C2semidirect
3. #16810925C2semidirect
4. #16810924C2semidirect
5. #16810923C2semidirect
6. #16810921C2semidirect
7. #16810875C2semidirect
8. #16810873C2semidirect
9. #16810858C2semidirect
10. #16810856C2semidirect
11. #16810816C2semidirect
12. #16810816C2semidirect
13. #16810815C2semidirect
14. #16810648C2semidirect
15. C53×#459C2semidirect
198. #459D53semidirect
200. C53#3259semidirect