Finite Group Databasebeta

← #20287754

Subgroups — #20287754

Order 6784.

418 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811355C211yesmaximal, characteristic, complemented
2#16811293C211yesmaximal, characteristic, complemented
3#16811009C211yesmaximal, characteristic, complemented
4#16810998C211yesmaximal, characteristic, complemented
5#16810840C211yesmaximal, characteristic, complemented
6#16810835C211yesmaximal, characteristic, complemented
7#16810821C211yesmaximal, characteristic, complemented
8#16810816C211yesmaximal, characteristic, complemented
9#16810816C211yesmaximal, characteristic, complemented
10#16810815C211yesmaximal, characteristic, complemented
11#16810751C211yesmaximal, characteristic, complemented
12#16810736C211yesmaximal, characteristic, complemented
13#16810735C211yesmaximal, characteristic, complemented
14SD16×Dic53C211yesmaximal, characteristic, complemented
15C53×#496C211yesmaximal, characteristic, complemented
16C2×#11719947—21no—
17C2×#11719924C2211yescharacteristic
18C2×D424C2211yescharacteristic
19C2×#11719922C2211yescharacteristic
20C2×#11719922C2211yescharacteristic
21C2×#11719922C2211yescharacteristic
22#13133213C2211yescharacteristic
23#13133213C2211yescharacteristic
24#13133213C2211yescharacteristic
25#13133206C2211yescharacteristic
26#13133203—21no—
27#13133201—21no—
28#13133199—21no—
29#13133189—21no—
30#13133189—21no—
31#13133188—21no—
32Q8×Dic53C2211yescharacteristic
33C4×Dic106C2211yescharacteristic
34C4×D212C2211yescharacteristic
35C4×D212C2211yescharacteristic
36D4×Dic53C2211yescharacteristic
37D4×Dic53C2211yescharacteristic
38D4×Dic53C2211yescharacteristic
39#13133177—21no—
40#13133149C2211yescharacteristic
41C4⋊#88322C2211yescharacteristic
42#13133146C2211yescharacteristic
43#13133146C2211yescharacteristic
44#13133141C2211yescharacteristic
45#13133139C2211yescharacteristic
46#13133139C2211yescharacteristic
47#13133138C2211yescharacteristic
48#13133138C2211yescharacteristic
49#13133138C2211yescharacteristic
50#13133127C2211yescharacteristic
51C8×Dic53C2211yescharacteristic
52C2×SD16×C53C2211yescharacteristic
53C2×SD16×C53C2211yescharacteristic
54C2×D8×C53C2211yescharacteristic
55C2×D8×C53C2211yescharacteristic
56#127×C53C2211yescharacteristic
57#124×C53C2211yescharacteristic
58#97×C53C2211yescharacteristic
59C2×Q8×C53C2311yescharacteristic
60C2×D4×C53—21no—
61C2×D4×C53—21no—
62C2×D4×C53C2311yescharacteristic
63C2×D4×C53C2311yescharacteristic
64C2×D4×C53C2311yescharacteristic
65C2×C22⋊D53—21no—
66C2×C22⋊D53—21no—
67C2×C22⋊D53—21no—
68C2×C22⋊D53—21no—
69C2×C22⋊D53—21no—
70C2×C22⋊D53—21no—
71C22×Dic53—21no—
72C22×Dic53—21no—
73C22×Dic53—21no—
74#11719947—44no—
75C2×D212C2311yescharacteristic
76C2×D212C2311yescharacteristic
77C2×C4×D53—21no—
78C2×C4×D53—21no—
79C2×Dic106D411yescharacteristic
80SD16×C53—22no—
81SD16×C53—41no—
82D8×C53—22no—
83D8×C53—41no—
84C2×C8×C53C2311yescharacteristic
85C2×C8×C53C2311yescharacteristic
86C22⋊C4×C53—21no—
87C22⋊C4×C53—21no—
88C42×C53C2311yescharacteristic
89C22⋊Dic53—21no—
90C22⋊Dic53—21no—
91C22⋊Dic53—21no—
92#11719924—41no—
93#11719922—41no—
94#11719922—41no—
95#11719922—41no—
96#11719921—21no—
97#11719921—21no—
98#11719921—21no—
99#11719921—21no—
100C4⋊Dic53—21no—
101C4⋊Dic53D411yescharacteristic
102C4⋊Dic53C2311yescharacteristic
103C4⋊Dic53C2311yescharacteristic
104#11719919D411yescharacteristic
105#11719919D411yescharacteristic
106C4×Dic53—21no—
107C4×Dic53C2311yescharacteristic
108C4×Dic53C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110C2×#88322C2311yescharacteristic
111D424—22no—
112C23×C53—21no—
113C23×C53—21no—
114C23×C53—21no—
115C22×D53—21no—
116C22×D53—21no—
117Q8×C53—21no—
118Q8×C53—21no—
119D4×C53—21no—
120D4×C53—21no—
121D4×C53—21no—
122D4×C53—21no—
123D4×C53—21no—
124D4×C53—21no—
125D4×C53—42no—
126D4×C53—81no—
127C2×C4×C53—21no—
128C2×C4×C53C2×D411yescharacteristic
129C2×C4×C53C2×D411yescharacteristic
130C2×C4×C53C2411yescharacteristic
131C22⋊D53—42no—
132C22⋊D53—42no—
133C22⋊D53—42no—
134C22⋊D53—81no—
135C22⋊D53—81no—
136C22⋊D53—81no—
137C2×Dic53—21no—
138C2×Dic53—21no—
139C2×Dic53—42no—
140C2×Dic53—42no—
141C2×Dic53—81no—
142C2×Dic53C2×D411yescharacteristic
143C2×Dic53C2×D411yescharacteristic
144C2×Dic53C2×D411yescharacteristic
145C2×Dic53C2×D411yescharacteristic
146D212—21no—
147D212—21no—
148D212—21no—
149D212—21no—
150C4×D53—42no—
151C4×D53—81no—
152Dic106—22no—
153C8×C53—21no—
154C8×C53#4112yescomplemented
155#88322—21no—
156#88322—21no—
157C22×C53—41no—
158C22×C53—41no—
159C22×C53—41no—
160C22×C53—41no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53—41no—
164C22×C53—41no—
165C22×C53—41no—
166C22×C53C22×D411yescharacteristic
167C2×D53—41no—
168C2×D53—41no—
169C2×D53—41no—
170C2×D53—41no—
171C2×D53—41no—
172C2×D53—41no—
173C4×C53—21no—
174C4×C53—21no—
175C4×C53—41no—
176C4×C53C2×#4111yescharacteristic
177C4×C53H(5,𝔽2)11yescharacteristic
178Dic53—21no—
179Dic53—21no—
180Dic53—21no—
181Dic53—41no—
182Dic53—41no—
183Dic53AGL(1,ℤ/8)12yes—
184#3399—531nomaximal, Sylow
185C2×C53—81no—
186C2×C53—81no—
187C2×C53—81no—
188C2×C53#55011yescharacteristic
189C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
190C2×C53#57611yescharacteristic
191D53—81no—
192D53—81no—
193#559—531no—
194#550—531no—
195#496—531no—
196#496D5311yescharacteristic, complemented
197#493—531no—
198#463—531no—
199#463—531no—
200#460—531no—
201#459—531no—
202#459—531no—
203#459—531no—
204#446—531no—
205#442—531no—
206#442—531no—
207C4×SD16—531no—
208C53#339911yescharacteristic, complemented, Sylow
209C2×#41—1061no—
210C2×SD16—531no—
211C2×SD16C2×D5311yescharacteristic
212C2×SD16C2×D5311yescharacteristic
213C2×D8—531no—
214C2×D8—531no—
215C2×D8—531no—
216C2×D8—531no—
217C2×D8C2×D5311yescharacteristic
218C2×D8C2×D5311yescharacteristic
219#127—531no—
220#127—531no—
221#127—531no—
222#127C2×D5311yescharacteristic
223#124—531no—
224#124—1061no—
225#124C2×D5311yescharacteristic
226#123—1061no—
227#123—1061no—
228#121—1061no—
229#121—1061no—
230#121—1061no—
231C4×Q8—531no—
232C4×Q8—531no—
233C4×D4—531no—
234C4×D4—531no—
235C4×D4—531no—
236C4×D4—531no—
237C4×D4—531no—
238C4×D4—1061no—
239#106—531no—
240C4⋊C8—531no—
241C4⋊C8—531no—
242C4⋊C8—531no—
243#103—531no—
244#102—531no—
245#102—531no—
246#102—531no—
247#102—531no—
248#102—531no—
249#97—531no—
250#97C2×D5311yescharacteristic
251C4×C8—531no—
252#41—2124no—
253C2×Q8—531no—
254C2×Q8C22×D5311yescharacteristic
255C2×D4—21no—
256C2×D4—21no—
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×D4C22×D5311yescharacteristic
266C2×D4C22×D5311yescharacteristic
267C2×D4C22×D5311yescharacteristic
268C22×C4—1061no—
269C22×C4—1061no—
270C22×C4—1061no—
271C22×C4—1061no—
272C22×C4—1061no—
273SD16—22no—
274SD16—41no—
275SD16—2121no—
276D8—22no—
277D8—41no—
278D8—1062no—
279D8—2121no—
280D8—2121no—
281D8—2121no—
282C2×C8—531no—
283C2×C8—531no—
284C2×C8C22×D5311yescharacteristic
285C2×C8C22×D5311yescharacteristic
286C4⋊C4—531no—
287C4⋊C4—531no—
288C4⋊C4—531no—
289C4⋊C4—531no—
290C4⋊C4—531no—
291C4⋊C4—1061no—
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—1061no—
295C22⋊C4—1061no—
296C22⋊C4—1061no—
297C22⋊C4—1061no—
298C22⋊C4—1061no—
299C22⋊C4—1061no—
300C22⋊C4—1061no—
301C42—531no—
302C42—531no—
303C42—1061no—
304C42C22×D5311yescharacteristic
305C23—21no—
306C23—21no—
307C23—21no—
308C23—1061no—
309C23—1061no—
310Q8—21no—
311Q8—21no—
312Q8—1062no—
313D4—21no—
314D4—21no—
315D4—21no—
316D4—21no—
317D4—21no—
318D4—21no—
319D4—42no—
320D4—81no—
321D4—1061no—
322D4—1061no—
323D4—1061no—
324D4—1061no—
325D4—4241no—
326D4—4241no—
327D4—4241no—
328D4—2122no—
329D4—2122no—
330D4—2122no—
331C2×C4—21no—
332C2×C4—531no—
333C2×C4—531no—
334C2×C4—531no—
335C2×C4—531no—
336C2×C4—1061no—
337C2×C4—1061no—
338C2×C4—4241no—
339C2×C4—4241no—
340C2×C4—2122no—
341C2×C4—2122no—
342C2×C4—2122no—
343C2×C4D4×D5311yescharacteristic
344C2×C4D4×D5311yescharacteristic
345C2×C4C23×D5311yescharacteristic
346C8—21no—
347C8—1061no—
348C8—1061no—
349C8#1171994712yescomplemented
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—2121no—
360C22—2121no—
361C22—2121no—
362C22—2121no—
363C22—2121no—
364C22—2121no—
365C22C2×D4×D5311yescharacteristic
366C4—21no—
367C4—21no—
368C4—41no—
369C4—532no—
370C4—1061no—
371C4—1061no—
372C4—1061no—
373C4—2121no—
374C4—2121no—
375C4C2×#1171994711yescharacteristic
376C4#1313326211yescharacteristic
377C2—81no—
378C2—81no—
379C2—81no—
380C2—4241no—
381C2—4241no—
382C2#1681129311yescharacteristic
383C2AGL(1,ℤ/8)×D5311yescharacteristic
384C2#1681142711yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#496C2
socleC22×C53C22×D4
hypercenter#496D53
hypocenterC53#3399
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #16811355C2semidirect
2. #16811293C2semidirect
3. #16811009C2semidirect
4. #16810998C2semidirect
5. #16810840C2semidirect
6. #16810835C2semidirect
7. #16810821C2semidirect
8. #16810816C2semidirect
9. #16810816C2semidirect
10. #16810815C2semidirect
11. #16810751C2semidirect
12. #16810736C2semidirect
13. #16810735C2semidirect
14. SD16×Dic53C2semidirect
15. C53×#496C2semidirect
154. C8×C53#41semidirect
196. #496D53semidirect
208. C53#3399semidirect
349. C8#11719947semidirect