Finite Group Databasebeta

← #20286717

Subgroups — #20286717

Order 6784.

404 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811317C211yesmaximal, characteristic, complemented
2#16811292C211yesmaximal, characteristic, complemented
3#16810986C211yesmaximal, characteristic, complemented
4#16810947C211yesmaximal, characteristic, complemented
5#16810935C211yesmaximal, characteristic, complemented
6#16810932C211yesmaximal, characteristic, complemented
7#16810875C211yesmaximal, characteristic, complemented
8#16810859C211yesmaximal, characteristic, complemented
9#16810817C211yesmaximal, characteristic, complemented
10#16810771C211yesmaximal, characteristic, complemented
11#16810768C211yesmaximal, characteristic, complemented
12#16810726C211yesmaximal, characteristic, complemented
13#16810694C211yesmaximal, characteristic, complemented
14#16810623C211yesmaximal, characteristic, complemented
15C53×#416C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×#11719924C2211yescharacteristic
18C2×D424C2211yescharacteristic
19C2×#11719922C2211yescharacteristic
20C2×#11719912C2211yescharacteristic
21#13133213—21no—
22#13133213C2211yescharacteristic
23#13133211—21no—
24#13133208C2211yescharacteristic
25#13133199—21no—
26#13133192C2211yescharacteristic
27#13133191—21no—
28#13133191C2211yescharacteristic
29#13133191C2211yescharacteristic
30#13133190—21no—
31#13133188—21no—
32#13133188C2211yescharacteristic
33Q8×Dic53C2211yescharacteristic
34C4×D212—21no—
35#13133179C2211yescharacteristic
36C4×C22⋊D53—21no—
37C4×C22⋊D53C2211yescharacteristic
38C4⋊C4×D53C2211yescharacteristic
39#13133151C2211yescharacteristic
40#13133150C2211yescharacteristic
41#13133148C2211yescharacteristic
42#13133146C2211yescharacteristic
43#13133141C2211yescharacteristic
44#13133139C2211yescharacteristic
45#13133139C2211yescharacteristic
46#13133138C2211yescharacteristic
47#13133137C2211yescharacteristic
48#13133137C2211yescharacteristic
49C22⋊#88322C2211yescharacteristic
50#13133129C2211yescharacteristic
51#13133127C2211yescharacteristic
52C2×#41×C53C2211yescharacteristic
53C22×C8×C53C2211yescharacteristic
54#117×C53C2211yescharacteristic
55#103×C53C2211yescharacteristic
56#103×C53C2211yescharacteristic
57#102×C53C2211yescharacteristic
58#102×C53C2211yescharacteristic
59#41×C53—22no—
60#41×C53—41no—
61C2×Q8×C53C2311yescharacteristic
62C2×D4×C53—21no—
63C2×D4×C53C2311yescharacteristic
64C22×C4×C53—21no—
65C22×C4×C53C2311yescharacteristic
66C2×C22⋊D53—21no—
67C2×C22⋊D53—21no—
68C2×C22⋊D53—21no—
69C2×C22⋊D53—21no—
70C2×C22⋊D53—21no—
71C2×C22⋊D53D411yescharacteristic
72#11719949—22no—
73#11719949—41no—
74C2×D212—21no—
75C2×D212—21no—
76C2×D212C2311yescharacteristic
77C2×D212C2311yescharacteristic
78C2×C4×D53—21no—
79C2×C4×D53—21no—
80C2×C4×D53—21no—
81C2×C4×D53C2311yescharacteristic
82C2×C8×C53—41no—
83C2×C8×C53C2311yescharacteristic
84C2×C8×C53C2311yescharacteristic
85C4⋊C4×C53C2311yescharacteristic
86C4⋊C4×C53C2311yescharacteristic
87C22⋊C4×C53—21no—
88C42×C53—21no—
89C22⋊Dic53—21no—
90C22⋊Dic53D411yescharacteristic
91#11719924—22no—
92#11719922—41no—
93#11719921—21no—
94#11719921—21no—
95#11719921—21no—
96#11719921—21no—
97#11719921D411yescharacteristic
98C4⋊Dic53—21no—
99C4⋊Dic53—21no—
100C4⋊Dic53C2311yescharacteristic
101C4⋊Dic53C2311yescharacteristic
102#11719919—21no—
103#11719919—21no—
104#11719919D411yescharacteristic
105C4×Dic53—21no—
106C4×Dic53C2311yescharacteristic
107C2×#88322C2311yescharacteristic
108C2×#88322C2311yescharacteristic
109D424—41no—
110#11719912—41no—
111C23×C53—21no—
112C23×C53C2×D411yescharacteristic
113C22×D53—21no—
114C22×D53—21no—
115C22×D53C2×D411yescharacteristic
116Q8×C53—21no—
117Q8×C53#4112yes—
118D4×C53—21no—
119D4×C53—21no—
120D4×C53—42no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—42no—
127C2×C4×C53C2×D411yescharacteristic
128C2×C4×C53C2411yescharacteristic
129C22⋊D53—41no—
130C22⋊D53—42no—
131C22⋊D53—81no—
132C22⋊D53—81no—
133C22⋊D53—81no—
134C22⋊D53—81no—
135C2×Dic53—21no—
136C2×Dic53—21no—
137C2×Dic53C2×D411yescharacteristic
138C2×Dic53C2×D411yescharacteristic
139C2×Dic53C2×D411yescharacteristic
140D212—21no—
141D212—21no—
142D212—21no—
143D212—21no—
144D212—42no—
145D212—42no—
146C4×D53—21no—
147C4×D53—21no—
148C4×D53—42no—
149C4×D53—81no—
150C4×D53—81no—
151C8×C53—21no—
152C8×C53—21no—
153#88322—21no—
154#88322—21no—
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—41no—
159C22×C53—41no—
160C22×C53—41no—
161C22×C53C22×D411yescharacteristic
162C2×D53—21no—
163C2×D53—21no—
164C2×D53—21no—
165C2×D53—41no—
166C2×D53—41no—
167C2×D53—41no—
168C2×D53—41no—
169C2×D53—41no—
170C2×D53—41no—
171C4×C53—22no—
172C4×C53—21no—
173C4×C53—41no—
174C4×C53—41no—
175C4×C53C2×#4111yescharacteristic
176C4×C53H(5,𝔽2)11yescharacteristic
177Dic53—21no—
178Dic53—21no—
179Dic53—21no—
180Dic53—41no—
181Dic53—41no—
182#3309—531nomaximal, Sylow
183C2×C53—41no—
184C2×C53—81no—
185C2×C53#54611yescharacteristic
186C2×C53#57511yescharacteristic
187C2×C53#57611yescharacteristic
188D53—41no—
189D53—81no—
190D53—81no—
191#553—531no—
192#550—531no—
193#490—531no—
194#482—531no—
195#480—531no—
196#479—531no—
197#469—531no—
198#466—531no—
199#459—531no—
200#449—531no—
201#449—531no—
202#440—531no—
203#435—531no—
204#417—531no—
205#416D5311yescharacteristic, complemented
206C53#330911yescharacteristic, complemented, Sylow
207C2×#41—531no—
208C2×#41C2×D5311yescharacteristic
209C2×SD16—531no—
210C2×D8—531no—
211C2×D8—531no—
212C2×M16—531no—
213C22×C8C2×D5311yescharacteristic
214#127—531no—
215#127—1061no—
216#126—1061no—
217#125—531no—
218#123—1061no—
219#121—531no—
220#121—531no—
221#121—531no—
222#121—531no—
223#121—1061no—
224#121—1061no—
225#121—1061no—
226C4×Q8—531no—
227C4×D4—531no—
228C4×D4—531no—
229C4×D4—1061no—
230C4×D4—1061no—
231#117C2×D5311yescharacteristic
232C2×C4⋊C4—531no—
233#107—531no—
234#107—531no—
235#106—531no—
236C4⋊C8—531no—
237#103—531no—
238#103C2×D5311yescharacteristic
239#103C2×D5311yescharacteristic
240#102—531no—
241#102—531no—
242#102—531no—
243#102—531no—
244#102—531no—
245#102C2×D5311yescharacteristic
246#102C2×D5311yescharacteristic
247C22⋊C8—531no—
248C22⋊C8—531no—
249#97—531no—
250#41—22no—
251#41—41no—
252#41—1062no—
253#41—2121no—
254C2×Q8C22×D5311yescharacteristic
255C2×D4—21no—
256C2×D4—531no—
257C2×D4—531no—
258C2×D4—531no—
259C2×D4—1061no—
260C2×D4—1061no—
261C2×D4—1061no—
262C2×D4—1061no—
263C2×D4—1061no—
264C2×D4—1061no—
265C2×D4—1061no—
266C2×D4C22×D5311yescharacteristic
267C22×C4—21no—
268C22×C4—531no—
269C22×C4—1061no—
270C22×C4—1061no—
271C22×C4—1061no—
272C22×C4C22×D5311yescharacteristic
273SD16—1062no—
274D8—2121no—
275D8—2121no—
276M16—2121no—
277C2×C8—41no—
278C2×C8—531no—
279C2×C8—531no—
280C2×C8C22×D5311yescharacteristic
281C2×C8C22×D5311yescharacteristic
282C4⋊C4—531no—
283C4⋊C4—531no—
284C4⋊C4—531no—
285C4⋊C4—1061no—
286C4⋊C4—1061no—
287C4⋊C4—1061no—
288C4⋊C4—1061no—
289C4⋊C4C22×D5311yescharacteristic
290C4⋊C4C22×D5311yescharacteristic
291C22⋊C4—21no—
292C22⋊C4—531no—
293C22⋊C4—531no—
294C22⋊C4—1061no—
295C22⋊C4—1061no—
296C22⋊C4—1061no—
297C22⋊C4—1061no—
298C22⋊C4—1061no—
299C42—21no—
300C42—531no—
301C42—1061no—
302C23—21no—
303C23—531no—
304C23—1061no—
305C23—1061no—
306C23D4×D5311yescharacteristic
307Q8—21no—
308Q8#1171994512yes—
309D4—21no—
310D4—21no—
311D4—42no—
312D4—1061no—
313D4—1061no—
314D4—1061no—
315D4—1061no—
316D4—4241no—
317D4—4241no—
318D4—4241no—
319D4—4241no—
320D4—2122no—
321D4—2122no—
322D4—2121no—
323D4—2122no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—42no—
330C2×C4—531no—
331C2×C4—531no—
332C2×C4—531no—
333C2×C4—1061no—
334C2×C4—1061no—
335C2×C4—1061no—
336C2×C4—1061no—
337C2×C4—4241no—
338C2×C4—4241no—
339C2×C4—2122no—
340C2×C4D4×D5311yescharacteristic
341C2×C4C23×D5311yescharacteristic
342C8—21no—
343C8—21no—
344C8—1061no—
345C8—1061no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—1061no—
353C22—1061no—
354C22—1061no—
355C22—2121no—
356C22—2121no—
357C22—2121no—
358C22—2121no—
359C22—2121no—
360C22—2121no—
361C22C2×D4×D5311yescharacteristic
362C4—21no—
363C4—22no—
364C4—41no—
365C4—41no—
366C4—1061no—
367C4—1061no—
368C4—1061no—
369C4—2121no—
370C4—2121no—
371C4C2×#1171994511yescharacteristic
372C4#1313326211yescharacteristic
373C2—41no—
374C2—81no—
375C2—4241no—
376C2—4241no—
377C2—2121no—
378C2#1681126311yescharacteristic
379C2#1681142011yescharacteristic
380C2#1681142511yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#416C2
socleC22×C53C22×D4
hypercenter#416D53
hypocenterC53#3309
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. #16811317C2semidirect
2. #16811292C2semidirect
3. #16810986C2semidirect
4. #16810947C2semidirect
5. #16810935C2semidirect
6. #16810932C2semidirect
7. #16810875C2semidirect
8. #16810859C2semidirect
9. #16810817C2semidirect
10. #16810771C2semidirect
11. #16810768C2semidirect
12. #16810726C2semidirect
13. #16810694C2semidirect
14. #16810623C2semidirect
15. C53×#416C2semidirect
205. #416D53semidirect
206. C53#3309semidirect