Finite Group Databasebeta

← #27370036

Subgroups — #27370036

Order 13568.

744 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285635C211yesmaximal, characteristic, complemented
2#20284806C211yesmaximal, characteristic, complemented
3C2×#16810818C211yesmaximal, characteristic, complemented
4#20274248C212yesmaximal, complemented
5#20274194C212yesmaximal, complemented
6#20274124C212yesmaximal, complemented
7#20274070C212yesmaximal, complemented
8#20273500C211yesmaximal, characteristic, complemented
9#20273428C211yesmaximal, characteristic, complemented
10#20273411C211yesmaximal, characteristic, complemented
11C53×#2974C211yesmaximal, characteristic, complemented
12C22×#11719922—21no—
13#16811173C2211yescharacteristic
14C2×#13133214C2211yescharacteristic
15#16811000C2212yescomplemented
16#16810970C2212yescomplemented
17#16810941—21no—
18#16810913C2211yescharacteristic, complemented
19#16810913C2211yescharacteristic, complemented
20#16810819—22no—
21#16810818—21no—
22#16810818—21no—
23#16810818C2212yes—
24#16810818C2212yes—
25#16810818C2211yescharacteristic, complemented
26#16810818C2211yescharacteristic, complemented
27#16810759—21no—
28#16810759—21no—
29#16810757—21no—
30#16810733—21no—
31#16810733—21no—
32#16810680C2211yescharacteristic
33C2×C4⋊#88322C2211yescharacteristic
34C2×#13133137—21no—
35#16810376C2212yescomplemented
36#16810374C2212yescomplemented
37#16810371C2212yescomplemented
38#16810366C2212yescomplemented
39#16810366C2212yescomplemented
40#16810366C2212yescomplemented
41C2×C4×D4×C53C2211yescharacteristic
42C53×#436C2212yescomplemented
43C53×#435—22no—
44C8×D4×C53C2212yescomplemented
45C53×#432C2211yescharacteristic
46C53×#431C2211yescharacteristic
47C53×#407—21no—
48C2×C22⋊C8×C53—21no—
49C2×#11719945—21no—
50C22×D212—21no—
51C22×D212—21no—
52C22×D212—21no—
53C2×D424—22no—
54C2×D424—22no—
55C2×#11719922—21no—
56C2×#11719922—21no—
57C2×#11719922—22no—
58C2×#11719922—22no—
59C2×#11719922—44no—
60C22×#88322—21no—
61#13133217C2311yescharacteristic
62#13133214—21no—
63#13133214—21no—
64#13133214—21no—
65#13133214C2312yes—
66#13133214C2311yescharacteristic
67#13133205—21no—
68#13133191—21no—
69#13133191—21no—
70C4×D212—21no—
71C4×D212—21no—
72#13133148—21no—
73#13133148—21no—
74C4⋊#88322—21no—
75C4⋊#88322C2312yes—
76C4⋊#88322C2311yescharacteristic
77C4⋊#88322C2311yescharacteristic
78#13133139—22no—
79#13133139—22no—
80#13133138—21no—
81#13133138—21no—
82#13133137—22no—
83#13133137—22no—
84#13133137—21no—
85#13133137—21no—
86C22⋊#88322—21no—
87#13133128—21no—
88C22×D4×C53D411yescharacteristic
89C23×C4×C53—21no—
90C2×M16×C53—22no—
91C22×C8×C53—22no—
92C4×D4×C53—21no—
93C4×D4×C53—21no—
94C4×D4×C53D411yescharacteristic
95C4×D4×C53D411yescharacteristic
96C4×D4×C53C2311yescharacteristic
97C4×D4×C53C2311yescharacteristic
98C2×C4⋊C4×C53D411yescharacteristic
99C2×C22⋊C4×C53—21no—
100C2×C42×C53C2311yescharacteristic
101C4⋊C8×C53D412yes—
102C4⋊C8×C53C2312yes—
103C22⋊C8×C53—22no—
104C22⋊C8×C53—22no—
105C22⋊C8×C53—21no—
106C22⋊C8×C53D411yescharacteristic
107C22⋊C8×C53D411yescharacteristic
108#97×C53—21no—
109C4×C8×C53C2312yes—
110C24×C53—21no—
111C23×D53—41no—
112C2×D4×C53—22no—
113C2×D4×C53—21no—
114C2×D4×C53—41no—
115C2×D4×C53C2×D411yescharacteristic
116C2×D4×C53C2×D411yescharacteristic
117C22×C4×C53—21no—
118C22×C4×C53—21no—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—82no—
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53#4111yescharacteristic
126C2×C22⋊D53—41no—
127#11719945—82no—
128C2×D212—21no—
129C2×D212—22no—
130C2×D212—22no—
131C2×D212—21no—
132C2×D212—21no—
133C2×D212—22no—
134C2×D212—42no—
135C2×D212—42no—
136C2×D212—41no—
137C2×D212—41no—
138C2×D212—41no—
139C2×D212—82no—
140C2×D212—82no—
141C2×C4×D53—41no—
142C2×Dic106—21no—
143M16×C53—82no—
144C2×C8×C53—22no—
145C2×C8×C53—82no—
146C2×C8×C53C2×D412yes—
147C2×C8×C53C2×D412yes—
148C4⋊C4×C53—21no—
149C4⋊C4×C53C2×D411yescharacteristic
150C4⋊C4×C53C2×D411yescharacteristic
151C22⋊C4×C53—21no—
152C22⋊C4×C53—21no—
153C22⋊C4×C53—21no—
154C22⋊C4×C53—21no—
155C42×C53C2×D411yescharacteristic
156C42×C53C2×D411yescharacteristic
157C42×C53#4111yescharacteristic
158C42×C53C2411yescharacteristic
159#11719922—42no—
160#11719922—44no—
161#11719922—42no—
162#11719922—44no—
163#11719921—41no—
164#11719921—41no—
165C4⋊Dic53—21no—
166C4⋊Dic53—21no—
167C2×#88322—21no—
168C2×#88322—21no—
169C2×#88322—22no—
170C2×#88322—42no—
171D424—82no—
172D424—82no—
173C23×C53—21no—
174C23×C53—21no—
175C23×C53—42no—
176C23×C53—42no—
177C23×C53—81no—
178C23×C53#12111yescharacteristic
179C22×D53—41no—
180C22×D53—42no—
181C22×D53—82no—
182C22×D53—82no—
183C22×D53—82no—
184D4×C53—21no—
185D4×C53—21no—
186D4×C53—21no—
187D4×C53—42no—
188D4×C53AGL(1,ℤ/8)12yes—
189C2×C4×C53—22no—
190C2×C4×C53—21no—
191C2×C4×C53—21no—
192C2×C4×C53—22no—
193C2×C4×C53—21no—
194C2×C4×C53—21no—
195C2×C4×C53—42no—
196C2×C4×C53—42no—
197C2×C4×C53—81no—
198C2×C4×C53—81no—
199C2×C4×C53—81no—
200C2×C4×C53#12111yescharacteristic
201C2×C4×C53#12111yescharacteristic
202C2×C4×C53#12111yescharacteristic
203C2×C4×C53C22×D411yescharacteristic
204C2×C4×C53C22×D411yescharacteristic
205C2×C4×C53C2×#4111yescharacteristic
206C22⋊D53—82no—
207C2×Dic53—41no—
208D212—42no—
209D212—41no—
210D212—42no—
211D212—41no—
212D212—42no—
213D212—42no—
214D212—42no—
215D212—82no—
216D212—81no—
217D212—82no—
218D212—82no—
219D212—82no—
220C4×D53—82no—
221Dic106—81no—
222C8×C53—22no—
223C8×C53—22no—
224C8×C53—42no—
225#88322—42no—
226#88322—41no—
227#88322—41no—
228#19458—531nomaximal, Sylow
229C22×C53—21no—
230C22×C53—21no—
231C22×C53—22no—
232C22×C53—41no—
233C22×C53—41no—
234C22×C53—41no—
235C22×C53—41no—
236C22×C53—41no—
237C22×C53—84no—
238C22×C53#47212yes—
239C22×C53C2×#12111yescharacteristic
240C2×D53—82no—
241C2×D53—81no—
242C2×D53—82no—
243C2×D53—81no—
244C2×D53—81no—
245C2×D53—82no—
246C2×D53—164no—
247C4×C53—21no—
248C4×C53—21no—
249C4×C53—21no—
250C4×C53—21no—
251C4×C53—41no—
252C4×C53—41no—
253C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
254C4×C53#57511yescharacteristic
255C4×C53#57612yes—
256Dic53—81no—
257#3205—531no—
258#3109—531no—
259C2×#459—531no—
260#2974D5311yescharacteristic, complemented
261#1697—532no—
262#1687—532no—
263#1677—532no—
264#1667—532no—
265#1573—531no—
266#1555—531no—
267#1551—531no—
268C2×C53—22no—
269C2×C53—42no—
270C2×C53—81no—
271C2×C53#304111yescharacteristic
272C2×C53#305511yescharacteristic
273C2×C53C2×#47211yescharacteristic
274D53—161no—
275D53—162no—
276C22×D8—1061no—
277#532—531no—
278C2×#127—531no—
279C2×C4×D4C2×D5311yescharacteristic
280#493—532no—
281#486—532no—
282#481—1061no—
283#476—531no—
284#476—531no—
285#459—531no—
286#459—531no—
287#459—532no—
288#459—532no—
289#459—1061no—
290#459—1061no—
291#459—1062no—
292#447—1061no—
293#447—1061no—
294#447—1061no—
295#442—1061no—
296#442—1061no—
297#436C2×D5312yescomplemented
298#435—22no—
299C8×D4C2×D5312yescomplemented
300#432—531no—
301#432C2×D5311yescharacteristic
302#431C2×D5311yescharacteristic
303C2×C4⋊C8—531no—
304C2×#102—1061no—
305#407—21no—
306C2×C22⋊C8—21no—
307#335—532no—
308#334—532no—
309#333—532no—
310#331—532no—
311#331—532no—
312#331—532no—
313C53#1945811yescharacteristic, complemented, Sylow
314C2×#41—1061no—
315C22×D4—1061no—
316C22×D4—1061no—
317C22×D4—1061no—
318C22×D4D21211yescharacteristic
319C23×C4—21no—
320C2×D8—1062no—
321C2×D8—1062no—
322C2×D8—1061no—
323C2×D8—1061no—
324C2×D8—1062no—
325C2×D8—1062no—
326C2×D8—2124no—
327C2×M16—22no—
328C22×C8—22no—
329C22×C8—1061no—
330C4⋊Q8—531no—
331#127—532no—
332#127—531no—
333#127—1061no—
334#127—1061no—
335#127—1061no—
336#124—1061no—
337#121—1061no—
338#121—1061no—
339C4×D4—21no—
340C4×D4—21no—
341C4×D4—1061no—
342C4×D4—1061no—
343C4×D4D21211yescharacteristic
344C4×D4D21211yescharacteristic
345C4×D4C22×D5311yescharacteristic
346C4×D4C22×D5311yescharacteristic
347C2×C4⋊C4D21211yescharacteristic
348C2×C22⋊C4—21no—
349C2×C42C22×D5311yescharacteristic
350#106—1061no—
351#106—1061no—
352C4⋊C8—531no—
353C4⋊C8—531no—
354C4⋊C8—532no—
355C4⋊C8—1061no—
356C4⋊C8C22⋊D5312yes—
357C4⋊C8C22×D5312yes—
358#102—1061no—
359#102—1061no—
360#102—1062no—
361#102—1062no—
362#102—1062no—
363#102—1062no—
364#102—1061no—
365#102—1061no—
366C22⋊C8—21no—
367C22⋊C8—22no—
368C22⋊C8—22no—
369C22⋊C8—1061no—
370C22⋊C8C22⋊D5311yescharacteristic
371C22⋊C8C22⋊D5311yescharacteristic
372#97—21no—
373#97—1061no—
374C4×C8C22×D5312yes—
375C24—21no—
376C24—2121no—
377#41—4242no—
378C2×Q8—1061no—
379C2×D4—22no—
380C2×D4—21no—
381C2×D4—41no—
382C2×D4—1062no—
383C2×D4—1061no—
384C2×D4—1061no—
385C2×D4—1062no—
386C2×D4—1061no—
387C2×D4—1062no—
388C2×D4—4242no—
389C2×D4—4242no—
390C2×D4—2122no—
391C2×D4—2122no—
392C2×D4—2121no—
393C2×D4—2121no—
394C2×D4—2121no—
395C2×D4—2121no—
396C2×D4C2×D21211yescharacteristic
397C2×D4C2×D21211yescharacteristic
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—82no—
404C22×C4—2121no—
405C22×C4C2×D21211yescharacteristic
406C22×C4#1171994511yescharacteristic
407C22×C4C2×C22⋊D5311yescharacteristic
408D8—4242no—
409D8—4242no—
410D8—2124no—
411D8—2122no—
412D8—2124no—
413D8—2122no—
414M16—82no—
415C2×C8—22no—
416C2×C8—82no—
417C2×C8—1061no—
418C2×C8—1061no—
419C2×C8—1062no—
420C2×C8—2122no—
421C2×C8C2×C22⋊D5312yes—
422C2×C8C2×C22⋊D5312yes—
423C4⋊C4—21no—
424C4⋊C4—1061no—
425C4⋊C4—1061no—
426C4⋊C4C2×D21211yescharacteristic
427C4⋊C4C2×D21211yescharacteristic
428C22⋊C4—21no—
429C22⋊C4—21no—
430C22⋊C4—21no—
431C22⋊C4—21no—
432C22⋊C4—2121no—
433C22⋊C4—2121no—
434C42C2×D21211yescharacteristic
435C42#1171994511yescharacteristic
436C42C2×C22⋊D5311yescharacteristic
437C42C23×D5311yescharacteristic
438C23—21no—
439C23—21no—
440C23—42no—
441C23—42no—
442C23—81no—
443C23—4242no—
444C23—4242no—
445C23—4242no—
446C23—2121no—
447C23—2122no—
448C23#1313319111yescharacteristic
449Q8—4241no—
450D4—21no—
451D4—21no—
452D4—21no—
453D4—42no—
454D4—4242no—
455D4—4241no—
456D4—4242no—
457D4—4242no—
458D4—4242no—
459D4—4242no—
460D4—2122no—
461D4—2121no—
462D4—2121no—
463D4—2122no—
464D4—2122no—
465D4—2122no—
466D4—2122no—
467D4#1313324512yes—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—4242no—
480C2×C4—2121no—
481C2×C4#1313319111yescharacteristic
482C2×C4#1313319111yescharacteristic
483C2×C4#1313319111yescharacteristic
484C2×C4C22×C22⋊D5311yescharacteristic
485C2×C4C22×D21211yescharacteristic
486C2×C4C2×#1171994511yescharacteristic
487C8—22no—
488C8—22no—
489C8—42no—
490C8—2122no—
491C8—2121no—
492C8—2121no—
493C22—21no—
494C22—21no—
495C22—22no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—84no—
502C22—8484no—
503C22—4241no—
504C22—4242no—
505C22—4242no—
506C22—4241no—
507C22—4242no—
508C22—4241no—
509C22#1681089312yes—
510C22C2×#1313319111yescharacteristic
511C4—21no—
512C4—21no—
513C4—21no—
514C4—21no—
515C4—41no—
516C4—41no—
517C4—4241no—
518C4C2×#1313324511yescharacteristic
519C4#1681142311yescharacteristic
520C4#1681142612yes—
521C2—22no—
522C2—42no—
523C2—81no—
524C2—8482no—
525C2—8481no—
526C2#2028421811yescharacteristic
527C2#2028433411yescharacteristic
528C2C2×#1681089311yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#2974C2
socleC22×C53C2×#121
hypercenter#2974D53
hypocenterC53#19458
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2974

Complements

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

NH ≅ G/Ntype
1. #20285635C2semidirect
2. #20284806C2semidirect
3. C2×#16810818C2semidirect
4. #20274248C2semidirect
5. #20274194C2semidirect
6. #20274124C2semidirect
7. #20274070C2semidirect
8. #20273500C2semidirect
9. #20273428C2semidirect
10. #20273411C2semidirect
11. C53×#2974C2semidirect
15. #16811000C22semidirect
16. #16810970C22semidirect
18. #16810913C22semidirect
19. #16810913C22semidirect
25. #16810818C22semidirect
26. #16810818C22semidirect
35. #16810376C22semidirect
36. #16810374C22semidirect
37. #16810371C22semidirect
38. #16810366C22semidirect
39. #16810366C22semidirect
40. #16810366C22semidirect
42. C53×#436C22semidirect
44. C8×D4×C53C22semidirect
260. #2974D53semidirect
297. #436C2×D53semidirect
299. C8×D4C2×D53semidirect
313. C53#19458semidirect

Maximal subgroups

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

subgroupindex
#202856352
#202848062
C2×#168108182
#202742482
#202741942
#202741242
#202740702
#202735002
#202734282
#202734112
C53×#29742
#1945853