Finite Group Databasebeta

← #20286153

Subgroups — #20286153

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#16811229C211yesmaximal, characteristic, complemented
2#16811224C211yesmaximal, characteristic, complemented
3#16810926C211yesmaximal, characteristic, complemented
4#16810922C211yesmaximal, characteristic, complemented
5#16810920C211yesmaximal, characteristic, complemented
6#16810877C211yesmaximal, characteristic, complemented
7#16810875C211yesmaximal, characteristic, complemented
8#16810860C211yesmaximal, characteristic, complemented
9#16810858C211yesmaximal, characteristic, complemented
10#16810819C211yesmaximal, characteristic, complemented
11#16810818C211yesmaximal, characteristic, complemented
12#16810817C211yesmaximal, characteristic, complemented
13#16810817C211yesmaximal, characteristic, complemented
14#16810648C211yesmaximal, characteristic, complemented
15C53×#478C211yesmaximal, characteristic, complemented
16C2×D424C2211yescharacteristic
17C2×D424C2211yescharacteristic
18C2×#11719922C2211yescharacteristic
19C2×#11719922C2211yescharacteristic
20C2×#11719912C2211yescharacteristic
21C2×C8×D53C2211yescharacteristic
22#13133214C2211yescharacteristic
23#13133213C2211yescharacteristic
24#13133208C2211yescharacteristic
25#13133202—21no—
26#13133202—21no—
27#13133200—21no—
28#13133199—21no—
29#13133192—21no—
30#13133192—21no—
31#13133192C2211yescharacteristic
32#13133192C2211yescharacteristic
33#13133192C2211yescharacteristic
34#13133191—21no—
35#13133190—21no—
36#13133188C2211yescharacteristic
37C4×D212C2211yescharacteristic
38#13133179C2211yescharacteristic
39#13133179C2211yescharacteristic
40#13133174C2211yescharacteristic
41#13133150C2211yescharacteristic
42#13133148C2211yescharacteristic
43C4⋊#88322C2211yescharacteristic
44#13133146C2211yescharacteristic
45#13133146C2211yescharacteristic
46#13133139C2211yescharacteristic
47#13133139C2211yescharacteristic
48#13133137C2211yescharacteristic
49#13133137C2211yescharacteristic
50#13133137C2211yescharacteristic
51#13133137C2211yescharacteristic
52#125×C53C2211yescharacteristic
53C4×D4×C53C2211yescharacteristic
54#107×C53C2211yescharacteristic
55#106×C53C2211yescharacteristic
56C4⋊C8×C53C2211yescharacteristic
57#102×C53C2211yescharacteristic
58#102×C53C2211yescharacteristic
59C2×D4×C53C2311yescharacteristic
60C22×C4×C53—21no—
61C2×C22⋊D53—21no—
62C2×C22⋊D53—21no—
63C2×C22⋊D53—21no—
64C2×D212—21no—
65C2×D212—21no—
66C2×D212—21no—
67C2×D212—21no—
68C2×D212—21no—
69C2×D212C2311yescharacteristic
70C2×D212C2311yescharacteristic
71C2×D212C2311yescharacteristic
72C2×C4×D53—21no—
73C2×C4×D53—21no—
74C2×C4×D53—21no—
75C2×C4×D53C2311yescharacteristic
76C2×C8×C53C2311yescharacteristic
77C2×C8×C53C2311yescharacteristic
78C4⋊C4×C53—21no—
79C4⋊C4×C53—21no—
80C4⋊C4×C53C2311yescharacteristic
81C4⋊C4×C53C2311yescharacteristic
82C4⋊C4×C53C2311yescharacteristic
83C22⋊C4×C53—21no—
84C42×C53C2311yescharacteristic
85C22⋊Dic53—21no—
86#11719922—41no—
87#11719922—41no—
88#11719921—21no—
89#11719921—21no—
90#11719921—21no—
91#11719921—21no—
92#11719921—21no—
93#11719921—21no—
94#11719921D411yescharacteristic
95#11719921D411yescharacteristic
96C4⋊Dic53—21no—
97C4⋊Dic53C2311yescharacteristic
98#11719919—21no—
99#11719919D411yescharacteristic
100#11719919D411yescharacteristic
101C4×Dic53C2311yescharacteristic
102C2×#88322C2311yescharacteristic
103C2×#88322C2311yescharacteristic
104D424—41no—
105D424—41no—
106#11719912—41no—
107C8×D53—41no—
108C23×C53—21no—
109C22×D53—21no—
110C22×D53—21no—
111C22×D53—21no—
112C22×D53C2×D411yescharacteristic
113D4×C53—21no—
114D4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—81no—
119C2×C4×C53C2×D411yescharacteristic
120C2×C4×C53C2×D411yescharacteristic
121C2×C4×C53C2411yescharacteristic
122C22⋊D53—81no—
123C22⋊D53—81no—
124C22⋊D53—81no—
125C2×Dic53—21no—
126C2×Dic53C2×D411yescharacteristic
127C2×Dic53C2×D411yescharacteristic
128C2×Dic53C2×D411yescharacteristic
129D212—21no—
130D212—21no—
131D212—21no—
132D212—21no—
133D212—21no—
134D212—21no—
135D212—81no—
136D212—81no—
137D212—81no—
138D212—81no—
139D212—81no—
140C4×D53—21no—
141C4×D53—21no—
142C4×D53—81no—
143C4×D53—81no—
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×C53C22×D411yescharacteristic
153C2×D53—21no—
154C2×D53—21no—
155C2×D53—21no—
156C2×D53—41no—
157C2×D53—41no—
158C2×D53—41no—
159C2×D53—41no—
160C2×D53—41no—
161C2×D53—41no—
162C2×D53—41no—
163C2×D53—41no—
164C2×D53—41no—
165C4×C53—21no—
166C4×C53—21no—
167C4×C53—41no—
168C4×C53—41no—
169C4×C53—41no—
170C4×C53H(5,𝔽2)11yescharacteristic
171C4×C532−1+411yescharacteristic
172Dic53—21no—
173Dic53—21no—
174Dic53—21no—
175Dic53—41no—
176#3259—531nomaximal, Sylow
177C2×C53—81no—
178C2×C53#53711yescharacteristic
179C2×C53#57611yescharacteristic
180C2×C53#57611yescharacteristic
181D53—41no—
182D53—81no—
183D53—81no—
184D53—81no—
185#540—531no—
186#540—531no—
187#478—531no—
188#478—531no—
189#478—531no—
190#478D5311yescharacteristic, complemented
191#469—531no—
192#469—531no—
193#466—531no—
194#466—531no—
195#459—531no—
196#459—531no—
197#459—531no—
198#459—531no—
199#424—531no—
200C53#325911yescharacteristic, complemented, Sylow
201C2×D8—531no—
202C2×D8—531no—
203C2×D8—531no—
204C2×D8—531no—
205C2×M16—531no—
206C22×C8—531no—
207#127—531no—
208#127—531no—
209#125—531no—
210#125C2×D5311yescharacteristic
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#107C2×D5311yescharacteristic
230#106—531no—
231#106C2×D5311yescharacteristic
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—531no—
245C2×D4—531no—
246C2×D4—531no—
247C2×D4—1061no—
248C2×D4—1061no—
249C2×D4—1061no—
250C2×D4—1061no—
251C2×D4—1061no—
252C2×D4—1061no—
253C2×D4—1061no—
254C2×D4—1061no—
255C2×D4C22×D5311yescharacteristic
256C22×C4—21no—
257C22×C4—531no—
258C22×C4—1061no—
259C22×C4—1061no—
260C22×C4—1061no—
261D8—2121no—
262D8—2121no—
263D8—2121no—
264D8—2121no—
265M16—2121no—
266C2×C8—531no—
267C2×C8—531no—
268C2×C8—2121no—
269C2×C8C22×D5311yescharacteristic
270C2×C8C22×D5311yescharacteristic
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—531no—
274C4⋊C4—531no—
275C4⋊C4—531no—
276C4⋊C4—1061no—
277C4⋊C4—1061no—
278C4⋊C4C22×D5311yescharacteristic
279C4⋊C4C22×D5311yescharacteristic
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—531no—
295C23—1061no—
296C23—1061no—
297C23—1061no—
298D4—21no—
299D4—21no—
300D4—1061no—
301D4—1061no—
302D4—1061no—
303D4—1061no—
304D4—1061no—
305D4—1061no—
306D4—4241no—
307D4—4241no—
308D4—4241no—
309D4—4241no—
310D4—4241no—
311D4—4241no—
312D4—4241no—
313D4—4241no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—81no—
318C2×C4—531no—
319C2×C4—531no—
320C2×C4—531no—
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—1061no—
338C22—1061no—
339C22—1061no—
340C22—2121no—
341C22—2121no—
342C22—2121no—
343C22—2121no—
344C22—2121no—
345C22—2121no—
346C22—2121no—
347C22—2121no—
348C22—2121no—
349C22C2×D4×D5311yescharacteristic
350C4—21no—
351C4—21no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—1061no—
356C4—1061no—
357C4—1061no—
358C4—2121no—
359C4#1313326311yescharacteristic
360C4#1313326511yescharacteristic
361C2—81no—
362C2—4241no—
363C2—4241no—
364C2—4241no—
365C2—2121no—
366C2#1681120311yescharacteristic
367C2#1681142511yescharacteristic
368C2#1681142711yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#478C2
socleC22×C53C22×D4
hypercenter#478D53
hypocenterC53#3259
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #16811229C2semidirect
2. #16811224C2semidirect
3. #16810926C2semidirect
4. #16810922C2semidirect
5. #16810920C2semidirect
6. #16810877C2semidirect
7. #16810875C2semidirect
8. #16810860C2semidirect
9. #16810858C2semidirect
10. #16810819C2semidirect
11. #16810818C2semidirect
12. #16810817C2semidirect
13. #16810817C2semidirect
14. #16810648C2semidirect
15. C53×#478C2semidirect
190. #478D53semidirect
200. C53#3259semidirect