Finite Group Databasebeta

← #27367620

Subgroups — #27367620

Order 13568.

704 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20286031C211yesmaximal, characteristic, complemented
2#20285630C211yesmaximal, characteristic
3#20285090C211yesmaximal, characteristic, complemented
4#20274276C212yesmaximal, complemented
5#20274248C212yesmaximal, complemented
6#20274152C212yesmaximal, complemented
7#20274124C212yesmaximal, complemented
8#20273467C211yesmaximal, characteristic, complemented
9#20273373C211yesmaximal, characteristic, complemented
10C2×#16810366C211yesmaximal, characteristic, complemented
11C53×#2935C211yesmaximal, characteristic, complemented
12#16811180C2211yescharacteristic
13C2×#13133214C2211yescharacteristic
14#16810999—21no—
15#16810988—22no—
16#16810971—21no—
17#16810970C2212yes—
18#16810970C2212yes—
19#16810913C2212yescomplemented
20#16810905—21no—
21#16810850—22no—
22#16810828C2212yescomplemented
23#16810820—21no—
24#16810818C2212yes—
25#16810818C2212yes—
26#16810756—22no—
27#16810676C2211yescharacteristic
28C2×C4⋊#88322C2211yescharacteristic
29#16810629—21no—
30#16810629—21no—
31#16810376C2212yescomplemented
32#16810374C2212yescomplemented
33#16810371C2211yescharacteristic, complemented
34#16810371C2211yescharacteristic, complemented
35#16810368C2211yescharacteristic, complemented
36#16810368C2211yescharacteristic, complemented
37#16810366C2212yes—
38#16810366C2212yes—
39C2×#131×C53—21no—
40C53×#532C2211yescharacteristic
41C53×#436C2212yescomplemented
42C53×#435—22no—
43C8×D4×C53C2212yescomplemented
44C2×C4⋊C8×C53C2211yescharacteristic
45C53×#408—21no—
46C4×M16×C53C2211yescharacteristic
47C22×D212—21no—
48C22×D212—21no—
49C22×D212—21no—
50C2×#11719923—22no—
51C2×#11719922—22no—
52C2×#11719922—22no—
53C22×#88322—21no—
54#13133217C2311yescharacteristic
55#13133217C2311yescharacteristic
56#13133214—21no—
57#13133214—21no—
58#13133214—21no—
59#13133214C2312yes—
60#13133209—21no—
61#13133196—21no—
62#13133196—21no—
63C4×Dic106—21no—
64C4×Dic106—21no—
65#13133176—21no—
66#13133148—22no—
67C4⋊#88322—21no—
68C4⋊#88322C2311yescharacteristic
69C4⋊#88322C2311yescharacteristic
70C4⋊#88322C2312yes—
71#13133140—22no—
72#13133139—22no—
73#13133139—22no—
74#13133139—22no—
75#13133139—22no—
76#13133138—22no—
77#13133137—22no—
78#13133137—22no—
79C22⋊#88322—21no—
80C4×#88322—21no—
81C2×#41×C53—21no—
82#131×C53—44no—
83C2×M16×C53—22no—
84C2×M16×C53D411yescharacteristic
85C2×M16×C53D411yescharacteristic
86C22×C8×C53—22no—
87C22×C8×C53—21no—
88C4⋊Q8×C53D411yescharacteristic
89#127×C53D411yescharacteristic
90#124×C53—21no—
91#121×C53—21no—
92#121×C53D412yes—
93C4×D4×C53—21no—
94C4×D4×C53C2312yes—
95C2×C42×C53C2311yescharacteristic
96C4⋊C8×C53—21no—
97C4⋊C8×C53C2311yescharacteristic
98C4⋊C8×C53C2311yescharacteristic
99C22⋊C8×C53—22no—
100C22⋊C8×C53—22no—
101#97×C53D412yes—
102C4×C8×C53C2312yes—
103C23×D53—41no—
104#41×C53—42no—
105#41×C53—42no—
106C2×Q8×C53—21no—
107C2×D4×C53—22no—
108C2×D4×C53—21no—
109C2×D4×C53—21no—
110C2×D4×C53C2×D412yes—
111C22×C4×C53—22no—
112C22×C4×C53C2×D411yescharacteristic
113C22×C4×C53C2×D411yescharacteristic
114C22×C4×C53#4111yescharacteristic
115C2×D212—22no—
116C2×D212—22no—
117C2×D212—22no—
118C2×D212—42no—
119C2×D212—42no—
120C2×D212—41no—
121C2×D212—41no—
122C2×D212—41no—
123C2×D212—81no—
124C2×D212—81no—
125C2×D212—82no—
126C2×Dic106—21no—
127C2×Dic106—21no—
128M16×C53—22no—
129M16×C53—22no—
130M16×C53—44no—
131C2×C8×C53—21no—
132C2×C8×C53—21no—
133C2×C8×C53—44no—
134C2×C8×C53—42no—
135C2×C8×C53C2×D412yes—
136C2×C8×C53C2×D412yes—
137C4⋊C4×C53—21no—
138C4⋊C4×C53C2×D412yes—
139C22⋊C4×C53—22no—
140C22⋊C4×C53—22no—
141C42×C53C2×D411yescharacteristic
142C42×C53C2×D411yescharacteristic
143C42×C53#4111yescharacteristic
144C42×C53C2411yescharacteristic
145C22⋊Dic53—41no—
146#11719923—82no—
147#11719922—82no—
148#11719922—82no—
149C4⋊Dic53—21no—
150C4⋊Dic53—21no—
151C4⋊Dic53—21no—
152C4⋊Dic53—21no—
153#11719919—41no—
154#11719919—41no—
155C4×Dic53—41no—
156C2×#88322—21no—
157C2×#88322—21no—
158C2×#88322—22no—
159C2×#88322—81no—
160C23×C53—22no—
161C23×C53#12111yescharacteristic
162C22×D53—42no—
163C22×D53—82no—
164C22×D53—82no—
165C22×D53—82no—
166Q8×C53—42no—
167D4×C53—22no—
168D4×C53—22no—
169D4×C53—44no—
170D4×C53—42no—
171D4×C53—81no—
172C2×C4×C53—21no—
173C2×C4×C53—21no—
174C2×C4×C53—22no—
175C2×C4×C53—41no—
176C2×C4×C53—42no—
177C2×C4×C53—42no—
178C2×C4×C53#12111yescharacteristic
179C2×C4×C53#12111yescharacteristic
180C2×C4×C53#12111yescharacteristic
181C2×C4×C53#13512yes—
182C2×C4×C53AGL(1,ℤ/8)12yes—
183C2×C4×C53C22×D411yescharacteristic
184C2×C4×C53C22×D411yescharacteristic
185C2×C4×C53C2×#4111yescharacteristic
186C2×Dic53—41no—
187C2×Dic53—41no—
188D212—42no—
189D212—42no—
190D212—42no—
191D212—42no—
192D212—82no—
193D212—82no—
194D212—82no—
195D212—82no—
196Dic106—41no—
197Dic106—41no—
198Dic106—81no—
199C8×C53—22no—
200C8×C53—22no—
201C8×C53—41no—
202C8×C53—41no—
203#88322—42no—
204#88322—41no—
205#88322—41no—
206#19234—531nomaximal, Sylow
207C22×C53—21no—
208C22×C53—21no—
209C22×C53—42no—
210C22×C53—42no—
211C22×C53—42no—
212C22×C53#46312yes—
213C22×C53C2×#12111yescharacteristic
214C2×D53—82no—
215C2×D53—82no—
216C2×D53—82no—
217C2×D53—164no—
218C4×C53—21no—
219C4×C53—21no—
220C4×C53—21no—
221C4×C53—42no—
222C4×C53#46312yes—
223C4×C53C2×#13511yescharacteristic
224C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
225C4×C53#57611yescharacteristic
226C4×C53#57711yescharacteristic
227Dic53—81no—
228Dic53—81no—
229#3246—531no—
230#3205—531no—
231#3138—531no—
232#2935D5311yescharacteristic, complemented
233#1701—532no—
234#1697—532no—
235#1681—532no—
236#1677—532no—
237#1565—531no—
238#1542—531no—
239C2×#331—531no—
240C2×C53—22no—
241C2×C53—82no—
242C2×C53C2×#46311yescharacteristic
243C2×C53#306211yescharacteristic
244C2×C53#306811yescharacteristic
245D53—162no—
246C2×#131—21no—
247#533—531no—
248#532C2×D5311yescharacteristic
249C2×#127—531no—
250#493—1061no—
251#490—1062no—
252#486—532no—
253#486—532no—
254#486—1061no—
255#476—532no—
256#475—1061no—
257#465—1062no—
258#461—532no—
259#460—1061no—
260#459—532no—
261#459—532no—
262#447—1062no—
263#436C2×D5312yescomplemented
264#435—22no—
265C8×D4C2×D5312yescomplemented
266#431—531no—
267C2×C4⋊C8—531no—
268C2×C4⋊C8C2×D5311yescharacteristic
269#418—1061no—
270#418—1061no—
271#408—21no—
272C4×M16C2×D5311yescharacteristic
273#335—532no—
274#334—532no—
275#333—531no—
276#333—531no—
277#332—531no—
278#332—531no—
279#331—532no—
280#331—532no—
281C53#1923411yescharacteristic, complemented, Sylow
282C2×#41—21no—
283C22×D4—1061no—
284C22×D4—1061no—
285C22×D4—1061no—
286C2×SD16—1062no—
287C2×D8—1062no—
288C2×D8—1062no—
289#131—44no—
290C2×M16—22no—
291C2×M16C22⋊D5311yescharacteristic
292C2×M16C22⋊D5311yescharacteristic
293C22×C8—21no—
294C22×C8—22no—
295C22×C8—1061no—
296C4⋊Q8—531no—
297C4⋊Q8—531no—
298C4⋊Q8D21211yescharacteristic
299#127—532no—
300#127—1061no—
301#127—1061no—
302#127—1061no—
303#127D21211yescharacteristic
304#125—1061no—
305#124—21no—
306C22⋊Q8—1061no—
307C22⋊Q8—1061no—
308#121—21no—
309#121D21212yes—
310C4×Q8—1061no—
311C4×Q8—1061no—
312C4×D4—21no—
313C4×D4C22×D5312yes—
314#117—1061no—
315C2×C42C22×D5311yescharacteristic
316#106—1062no—
317C4⋊C8—21no—
318C4⋊C8—532no—
319C4⋊C8—531no—
320C4⋊C8—531no—
321C4⋊C8—1061no—
322C4⋊C8C22×D5311yescharacteristic
323C4⋊C8C22×D5311yescharacteristic
324#103—1062no—
325#102—1062no—
326#102—1062no—
327#102—1062no—
328#102—1062no—
329#102—1062no—
330#102—1062no—
331#102—1062no—
332C22⋊C8—22no—
333C22⋊C8—22no—
334C22⋊C8—1061no—
335#97C22⋊D5312yes—
336C4×C8—1061no—
337C4×C8C22×D5312yes—
338C24—2121no—
339#41—42no—
340#41—42no—
341C2×Q8—21no—
342C2×Q8—1061no—
343C2×Q8—1061no—
344C2×D4—22no—
345C2×D4—21no—
346C2×D4—21no—
347C2×D4—1062no—
348C2×D4—1062no—
349C2×D4—1062no—
350C2×D4—4241no—
351C2×D4—4241no—
352C2×D4—4242no—
353C2×D4—2121no—
354C2×D4—2121no—
355C2×D4—2121no—
356C2×D4—2122no—
357C2×D4—2122no—
358C2×D4C2×D21212yes—
359C22×C4—22no—
360C22×C4C2×D21211yescharacteristic
361C22×C4#1171994511yescharacteristic
362C22×C4C2×C22⋊D5311yescharacteristic
363SD16—4242no—
364D8—4242no—
365D8—4242no—
366M16—22no—
367M16—22no—
368M16—44no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—44no—
372C2×C8—42no—
373C2×C8—1061no—
374C2×C8—1061no—
375C2×C8—1062no—
376C2×C8—4241no—
377C2×C8C2×C22⋊D5312yes—
378C2×C8C2×C22⋊D5312yes—
379C4⋊C4—21no—
380C4⋊C4—1061no—
381C4⋊C4—1061no—
382C4⋊C4—1061no—
383C4⋊C4—1061no—
384C4⋊C4—2121no—
385C4⋊C4—2121no—
386C4⋊C4C2×D21212yes—
387C22⋊C4—22no—
388C22⋊C4—22no—
389C22⋊C4—2121no—
390C42—2121no—
391C42C2×D21211yescharacteristic
392C42#1171994511yescharacteristic
393C42C2×C22⋊D5311yescharacteristic
394C42C23×D5311yescharacteristic
395C23—22no—
396C23—4242no—
397C23—4242no—
398C23—4242no—
399C23—2122no—
400C23#1313319111yescharacteristic
401Q8—42no—
402Q8—4241no—
403Q8—2121no—
404Q8—2121no—
405D4—22no—
406D4—22no—
407D4—44no—
408D4—42no—
409D4—81no—
410D4—4242no—
411D4—4242no—
412D4—4242no—
413D4—4242no—
414D4—2122no—
415D4—2122no—
416D4—2122no—
417D4—2122no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—2121no—
425C2×C4—2121no—
426C2×C4#1313319111yescharacteristic
427C2×C4#1313319111yescharacteristic
428C2×C4#1313319111yescharacteristic
429C2×C4#1313323712yes—
430C2×C4#1313324312yes—
431C2×C4C22×C22⋊D5311yescharacteristic
432C2×C4C22×D21211yescharacteristic
433C2×C4C2×#1171994511yescharacteristic
434C8—22no—
435C8—22no—
436C8—41no—
437C8—41no—
438C8—2122no—
439C8—2121no—
440C8—2121no—
441C22—21no—
442C22—21no—
443C22—42no—
444C22—42no—
445C22—42no—
446C22—8484no—
447C22—4242no—
448C22—4242no—
449C22—4242no—
450C22#1681083812yes—
451C22C2×#1313319111yescharacteristic
452C4—21no—
453C4—21no—
454C4—21no—
455C4—42no—
456C4—4241no—
457C4—4241no—
458C4#1681083812yes—
459C4C2×#1313323711yescharacteristic
460C4C2×#1313324311yescharacteristic
461C4#1681142811yescharacteristic
462C4#1681143511yescharacteristic
463C2—22no—
464C2—82no—
465C2—8482no—
466C2C2×#1681083811yescharacteristic
467C2#2028439511yescharacteristic
468C2#2028444311yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#2935C2
socleC22×C53C2×#121
prime radical#20285630C2
hypercenter#2935D53
hypocenterC53#19234
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2935

Complements

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

NH ≅ G/Ntype
1. #20286031C2semidirect
3. #20285090C2semidirect
4. #20274276C2semidirect
5. #20274248C2semidirect
6. #20274152C2semidirect
7. #20274124C2semidirect
8. #20273467C2semidirect
9. #20273373C2semidirect
10. C2×#16810366C2semidirect
11. C53×#2935C2semidirect
19. #16810913C22semidirect
22. #16810828C22semidirect
31. #16810376C22semidirect
32. #16810374C22semidirect
33. #16810371C22semidirect
34. #16810371C22semidirect
35. #16810368C22semidirect
36. #16810368C22semidirect
41. C53×#436C22semidirect
43. C8×D4×C53C22semidirect
232. #2935D53semidirect
263. #436C2×D53semidirect
265. C8×D4C2×D53semidirect
281. C53#19234semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202860312
#202856302
#202850902
#202742762
#202742482
#202741522
#202741242
#202734672
#202733732
C2×#168103662
C53×#29352
#1923453