Finite Group Databasebeta

← #20286726

Subgroups — #20286726

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#16811323C211yesmaximal, characteristic, complemented
2#16811295C211yesmaximal, characteristic, complemented
3#16810988C211yesmaximal, characteristic, complemented
4#16810950C211yesmaximal, characteristic, complemented
5#16810938C211yesmaximal, characteristic, complemented
6#16810930C211yesmaximal, characteristic, complemented
7#16810875C211yesmaximal, characteristic, complemented
8#16810858C211yesmaximal, characteristic, complemented
9#16810818C211yesmaximal, characteristic, complemented
10#16810767C211yesmaximal, characteristic, complemented
11#16810767C211yesmaximal, characteristic, complemented
12#16810696C211yesmaximal, characteristic, complemented
13#16810623C211yesmaximal, characteristic, complemented
14#16810618C211yesmaximal, characteristic, complemented
15C53×#440C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×#11719949C2211yescharacteristic
18C2×#11719922C2211yescharacteristic
19C2×#11719922C2211yescharacteristic
20C2×#11719912C2211yescharacteristic
21C2×C8×D53C2211yescharacteristic
22#13133214—21no—
23#13133214C2211yescharacteristic
24#13133210—21no—
25#13133209C2211yescharacteristic
26#13133200—21no—
27#13133192—21no—
28#13133192—21no—
29#13133192C2211yescharacteristic
30#13133192C2211yescharacteristic
31#13133190—21no—
32#13133188C2211yescharacteristic
33#13133188C2211yescharacteristic
34C4×D212—21no—
35C4×D212C2211yescharacteristic
36C4×C22⋊D53—21no—
37#13133175C2211yescharacteristic
38C4⋊C4×D53C2211yescharacteristic
39#13133150C2211yescharacteristic
40#13133148C2211yescharacteristic
41C4⋊#88322C2211yescharacteristic
42#13133141C2211yescharacteristic
43#13133141C2211yescharacteristic
44#13133139C2211yescharacteristic
45#13133139C2211yescharacteristic
46#13133139C2211yescharacteristic
47#13133139C2211yescharacteristic
48#13133137C2211yescharacteristic
49#13133137C2211yescharacteristic
50#13133129C2211yescharacteristic
51#13133129C2211yescharacteristic
52C2×SD16×C53C2211yescharacteristic
53C4×Q8×C53C2211yescharacteristic
54C4×D4×C53C2211yescharacteristic
55#107×C53C2211yescharacteristic
56#103×C53C2211yescharacteristic
57#102×C53C2211yescharacteristic
58#97×C53C2211yescharacteristic
59C2×Q8×C53C2311yescharacteristic
60C2×D4×C53C2311yescharacteristic
61C22×C4×C53—21no—
62C2×C22⋊D53—21no—
63C2×C22⋊D53—21no—
64#11719949—22no—
65#11719949—22no—
66#11719949—41no—
67#11719949—41no—
68C2×D212—21no—
69C2×D212—21no—
70C2×D212—21no—
71C2×D212—21no—
72C2×D212—21no—
73C2×D212—21no—
74C2×D212D411yescharacteristic
75C2×D212C2311yescharacteristic
76C2×D212C2311yescharacteristic
77C2×C4×D53—21no—
78C2×C4×D53—21no—
79C2×C4×D53—21no—
80C2×C4×D53C2311yescharacteristic
81C2×C4×D53C2311yescharacteristic
82SD16×C53—22no—
83C2×C8×C53C2311yescharacteristic
84C2×C8×C53C2311yescharacteristic
85C4⋊C4×C53—21no—
86C4⋊C4×C53C2311yescharacteristic
87C4⋊C4×C53C2311yescharacteristic
88C22⋊C4×C53—21no—
89C42×C53—21no—
90C42×C53C2311yescharacteristic
91C22⋊Dic53—21no—
92C22⋊Dic53—21no—
93#11719922—41no—
94#11719922—41no—
95#11719921—21no—
96#11719921—21no—
97#11719921—21no—
98#11719921D411yescharacteristic
99#11719921D411yescharacteristic
100C4⋊Dic53D411yescharacteristic
101C4⋊Dic53C2311yescharacteristic
102C4⋊Dic53C2311yescharacteristic
103#11719919—21no—
104#11719919—21no—
105#11719919—21no—
106C4×Dic53—21no—
107C2×#88322C2311yescharacteristic
108C2×#88322C2311yescharacteristic
109#11719912—41no—
110C8×D53—41no—
111C23×C53—21no—
112C22×D53—21no—
113C22×D53—21no—
114C22×D53C2×D411yescharacteristic
115C22×D53C2×D411yescharacteristic
116Q8×C53—21no—
117Q8×C53#4112yes—
118D4×C53—21no—
119D4×C53—21no—
120C2×C4×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—81no—
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53C2×D411yescharacteristic
132C2×Dic53C2×D411yescharacteristic
133D212—21no—
134D212—21no—
135D212—21no—
136D212—21no—
137D212—41no—
138D212—42no—
139D212—42no—
140D212—42no—
141D212—81no—
142D212—81no—
143D212—81no—
144C4×D53—21no—
145C4×D53—21no—
146C4×D53—21no—
147C4×D53—21no—
148C4×D53—42no—
149C4×D53—42no—
150C4×D53—81no—
151C8×C53—21no—
152C8×C53—21no—
153#88322—21no—
154#88322—21no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53C22×D411yescharacteristic
159C2×D53—21no—
160C2×D53—21no—
161C2×D53—21no—
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—21no—
174C4×C53—41no—
175C4×C53—41no—
176C4×C53C2×#4111yescharacteristic
177C4×C53H(5,𝔽2)11yescharacteristic
178Dic53—21no—
179Dic53—21no—
180Dic53—41no—
181Dic53—41no—
182#3309—531nomaximal, Sylow
183C2×C53—81no—
184C2×C53#54611yescharacteristic
185C2×C53#57511yescharacteristic
186C2×C53#57611yescharacteristic
187D53—41no—
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#440D5311yescharacteristic, complemented
203#435—531no—
204#417—531no—
205#416—531no—
206C53#330911yescharacteristic, complemented, Sylow
207C2×#41—531no—
208C2×#41—531no—
209C2×SD16C2×D5311yescharacteristic
210C2×D8—531no—
211C2×D8—531no—
212C2×M16—531no—
213C22×C8—531no—
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×Q8C2×D5311yescharacteristic
227C4×D4—531no—
228C4×D4—1061no—
229C4×D4—1061no—
230C4×D4C2×D5311yescharacteristic
231#117—531no—
232C2×C4⋊C4—531no—
233#107—531no—
234#107C2×D5311yescharacteristic
235#106—531no—
236C4⋊C8—531no—
237#103—531no—
238#103—531no—
239#103C2×D5311yescharacteristic
240#102—531no—
241#102—531no—
242#102—531no—
243#102—531no—
244#102—531no—
245#102—531no—
246#102C2×D5311yescharacteristic
247C22⋊C8—531no—
248C22⋊C8—531no—
249#97C2×D5311yescharacteristic
250#41—1062no—
251#41—1062no—
252#41—2121no—
253#41—2121no—
254C2×Q8C22×D5311yescharacteristic
255C2×D4—531no—
256C2×D4—531no—
257C2×D4—531no—
258C2×D4—1061no—
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—531no—
270C22×C4—1061no—
271C22×C4—1061no—
272C22×C4—1061no—
273SD16—22no—
274D8—2121no—
275D8—2121no—
276M16—2121no—
277C2×C8—531no—
278C2×C8—531no—
279C2×C8—2121no—
280C2×C8C22×D5311yescharacteristic
281C2×C8C22×D5311yescharacteristic
282C4⋊C4—21no—
283C4⋊C4—531no—
284C4⋊C4—531no—
285C4⋊C4—531no—
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—1061no—
301C42C22×D5311yescharacteristic
302C23—21no—
303C23—531no—
304C23—531no—
305C23—1061no—
306C23—1061no—
307Q8—21no—
308Q8#1171994512yes—
309D4—21no—
310D4—21no—
311D4—1061no—
312D4—1061no—
313D4—1061no—
314D4—1061no—
315D4—4241no—
316D4—4241no—
317D4—4241no—
318D4—4241no—
319D4—2121no—
320D4—2122no—
321D4—2122no—
322D4—2122no—
323D4—2122no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—81no—
328C2×C4—531no—
329C2×C4—531no—
330C2×C4—1061no—
331C2×C4—1061no—
332C2×C4—1061no—
333C2×C4—1061no—
334C2×C4—1061no—
335C2×C4—1061no—
336C2×C4—4241no—
337C2×C4—2122no—
338C2×C4—2122no—
339C2×C4D4×D5311yescharacteristic
340C2×C4D4×D5311yescharacteristic
341C2×C4C23×D5311yescharacteristic
342C8—21no—
343C8—21no—
344C8—1061no—
345C8—1061no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—1061no—
350C22—1061no—
351C22—1061no—
352C22—1061no—
353C22—1061no—
354C22—1061no—
355C22—2121no—
356C22—2121no—
357C22—2121no—
358C22—2121no—
359C22—2121no—
360C22—2121no—
361C22C2×D4×D5311yescharacteristic
362C4—22no—
363C4—21no—
364C4—21no—
365C4—41no—
366C4—41no—
367C4—1061no—
368C4—1061no—
369C4—2121no—
370C4—2121no—
371C4C2×#1171994511yescharacteristic
372C4#1313326311yescharacteristic
373C2—81no—
374C2—4241no—
375C2—4241no—
376C2—2121no—
377C2—2121no—
378C2#1681126911yescharacteristic
379C2#1681142211yescharacteristic
380C2#1681142711yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#440C2
socleC22×C53C22×D4
hypercenter#440D53
hypocenterC53#3309
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. #16811323C2semidirect
2. #16811295C2semidirect
3. #16810988C2semidirect
4. #16810950C2semidirect
5. #16810938C2semidirect
6. #16810930C2semidirect
7. #16810875C2semidirect
8. #16810858C2semidirect
9. #16810818C2semidirect
10. #16810767C2semidirect
11. #16810767C2semidirect
12. #16810696C2semidirect
13. #16810623C2semidirect
14. #16810618C2semidirect
15. C53×#440C2semidirect
202. #440D53semidirect
206. C53#3309semidirect