Finite Group Databasebeta

← #27395299

Subgroups — #27395299

Order 13568.

1296 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810759C211yesmaximal, characteristic
2C4×#13133191C211yesmaximal, characteristic, complemented
3C2×#16810525C211yesmaximal, characteristic, complemented
4#20275591C212yesmaximal, complemented
5#20275412C212yesmaximal, complemented
6#20275343C212yesmaximal, complemented
7#20275062C212yesmaximal, complemented
8#20275030C211yesmaximal, characteristic, complemented
9#20274910C211yesmaximal, characteristic, complemented
10#20274647C211yesmaximal, characteristic, complemented
11C53×#1749C211yesmaximal, characteristic, complemented
12C23×D212C2211yescharacteristic
13C22×D424C2212yes—
14C2×#13133191C2211yescharacteristic
15C2×C4×D212C2211yescharacteristic, complemented
16C2×C4×C22⋊D53—21no—
17C22×C4⋊Dic53C2211yescharacteristic
18C22×#11719921—21no—
19C22×#11719921—21no—
20#16810759C418yescomplemented
21C2×#13133149C2212yescomplemented
22C2×#13133139C2212yes—
23C2×#13133139C2212yes—
24C2×#13133139C2212yes—
25C2×#13133139C2212yescomplemented
26#16810525—21no—
27#16810525—21no—
28#16810525C2212yes—
29#16810525C2212yes—
30#16810525C2211yescharacteristic, complemented
31#16810521C2211yescharacteristic, complemented
32#16810514—21no—
33#16810502—21no—
34#16810483C2211yescharacteristic
35C4×C4⋊Dic53C2211yescharacteristic, complemented
36C4×#11719921—21no—
37#16810386C2212yescomplemented
38#16810386C2212yescomplemented
39C22×C42×C53C2211yescharacteristic
40C2×C22⋊C8×C53C2211yescharacteristic
41C2×C22⋊C8×C53C2211yescharacteristic
42C2×#97×C53C2212yescomplemented
43C53×#336C2212yescomplemented
44C24×D53—21no—
45C22×D212—21no—
46C22×D212—22no—
47C22×D212—42no—
48C22×D212—42no—
49C22×D212C2×C414yescomplemented
50C22×D212C2312yes—
51C22×D212C2311yescharacteristic
52C22×C22⋊D53—21no—
53C23×Dic53—21no—
54C22×C4×D53—21no—
55C2×D424—42no—
56C2×D424—42no—
57C2×D424C2×C418yescomplemented
58#13133191—22no—
59#13133191C2×C414yes—
60C4×D212—42no—
61C4×C22⋊D53—44no—
62C2×C4⋊Dic53—21no—
63C2×C4⋊Dic53—42no—
64C2×C4⋊Dic53C2311yescharacteristic
65C2×C4⋊Dic53C2312yes—
66C2×C4⋊Dic53C2311yescharacteristic
67C2×C4⋊Dic53C2311yescharacteristic
68C2×#11719919—21no—
69C2×C22⋊Dic53—21no—
70C2×#11719921—21no—
71C2×#11719921—21no—
72C2×#11719921—21no—
73C2×#11719921—21no—
74C2×#11719921—22no—
75C2×#11719921—22no—
76C2×#11719921—22no—
77C2×#11719921—22no—
78C2×#11719921—44no—
79C2×#11719921—44no—
80C2×C4×Dic53—21no—
81#13133149—42no—
82#13133139—42no—
83#13133139—42no—
84#13133139—42no—
85#13133139C2×C418yescomplemented
86#13133124—21no—
87#13133124—21no—
88#13133124—21no—
89C23×C4×C53D411yescharacteristic
90C23×C4×C53D411yescharacteristic
91C23×C4×C53C2311yescharacteristic
92C22×C8×C53C2312yes—
93C22×C8×C53C2312yes—
94C2×C42×C53—42no—
95C2×C42×C53D411yescharacteristic
96C2×C42×C53D411yescharacteristic
97C2×C42×C53C2311yescharacteristic
98C2×C42×C53C2311yescharacteristic
99C22⋊C8×C53—22no—
100C22⋊C8×C53C2×C414yescomplemented
101#97×C53—42no—
102C24×C53C2×D411yescharacteristic
103C23×D53—24no—
104C23×D53—21no—
105C23×D53—22no—
106C23×D53—42no—
107C23×D53—42no—
108C23×D53—42no—
109C23×D53—42no—
110C23×D53—42no—
111C23×D53—42no—
112C22×C4×C53—22no—
113C22×C4×C53—22no—
114C22×C4×C53—22no—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—41no—
118C22×C4×C53—41no—
119C22×C4×C53—41no—
120C22×C4×C53C22×C412yes—
121C22×C4×C53C2×D411yescharacteristic
122C22×C4×C53C2×D411yescharacteristic
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53C2×D411yescharacteristic
126C22×C4×C53#4112yes—
127C22×C4×C53C2411yescharacteristic
128C2×C22⋊D53—24no—
129C2×C22⋊D53—42no—
130C2×C22⋊D53—42no—
131C22×Dic53—21no—
132C22×Dic53—21no—
133C22×Dic53—21no—
134C22×Dic53—22no—
135C22×Dic53—42no—
136C22×Dic53—42no—
137C22×Dic53—42no—
138C2×D212—22no—
139C2×D212—22no—
140C2×D212—22no—
141C2×D212—22no—
142C2×D212—22no—
143C2×D212—22no—
144C2×D212—24no—
145C2×D212—21no—
146C2×D212—22no—
147C2×D212—48no—
148C2×D212—42no—
149C2×D212—42no—
150C2×D212—48no—
151C2×D212C22×C414yes—
152C2×D212C22×C412yes—
153C2×C4×D53—42no—
154C2×C4×D53—42no—
155C2×C4×D53—42no—
156C2×C8×C53—22no—
157C2×C8×C53—22no—
158C2×C8×C53—22no—
159C2×C8×C53—22no—
160C2×C8×C53—22no—
161C2×C8×C53C22×C414yes—
162C42×C53—22no—
163C42×C53—22no—
164C42×C53—41no—
165C42×C53—41no—
166C22⋊Dic53—42no—
167#11719921—24no—
168#11719921—42no—
169#11719921—44no—
170#11719921—44no—
171#11719921—44no—
172#11719921—42no—
173#11719921—42no—
174#11719921—44no—
175C4⋊Dic53—21no—
176C4⋊Dic53—22no—
177C4⋊Dic53—22no—
178C4⋊Dic53—22no—
179C4⋊Dic53—22no—
180C4⋊Dic53—42no—
181C4⋊Dic53C22×C412yes—
182#11719919—42no—
183C4×Dic53—42no—
184D424—48no—
185C23×C53—21no—
186C23×C53—21no—
187C23×C53—21no—
188C23×C53—21no—
189C23×C53C4×D412yes—
190C23×C53AGL(1,ℤ/8)12yes—
191C23×C53AGL(1,ℤ/8)12yes—
192C23×C53C22×D411yescharacteristic
193C22×D53—24no—
194C22×D53—22no—
195C22×D53—48no—
196C22×D53—48no—
197C22×D53—41no—
198C22×D53—42no—
199C22×D53—42no—
200C22×D53—41no—
201C22×D53—41no—
202C22×D53—41no—
203C22×D53—41no—
204C22×D53—42no—
205C22×D53—42no—
206C22×D53—42no—
207C22×D53—42no—
208C22×D53—41no—
209C22×D53—48no—
210C22×D53—84no—
211C22×D53—84no—
212C22×D53—84no—
213C22×D53—84no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—24no—
219C2×C4×C53—24no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—44no—
230C2×C4×C53—44no—
231C2×C4×C53C4×D412yes—
232C2×C4×C53AGL(1,ℤ/8)12yes—
233C2×C4×C53C23×C411yescharacteristic
234C2×C4×C53C2×#4111yescharacteristic
235C2×C4×C53H(5,𝔽2)12yes—
236C2×C4×C53H(5,𝔽2)12yes—
237C22⋊D53—48no—
238C2×Dic53—22no—
239C2×Dic53—41no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C2×Dic53—42no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—42no—
247C2×Dic53—42no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—84no—
251D212—24no—
252D212—24no—
253D212—24no—
254D212—48no—
255D212—42no—
256D212—42no—
257D212—42no—
258C4×D53—84no—
259C8×C53—24no—
260C8×C53—42no—
261#22046—531nomaximal, Sylow
262C22×C53—22no—
263C22×C53—22no—
264C22×C53—22no—
265C22×C53—22no—
266C22×C53—22no—
267C22×C53—22no—
268C22×C53#44214yes—
269C22×C53C2×C4×D411yescharacteristic
270C22×C53#53411yescharacteristic
271C22×C53#53511yescharacteristic
272C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
273C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
274C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
275C22×C53#57511yescharacteristic
276C2×D53—44no—
277C2×D53—42no—
278C2×D53—42no—
279C2×D53—44no—
280C2×D53—42no—
281C2×D53—44no—
282C2×D53—816no—
283C2×D53—81no—
284C2×D53—82no—
285C2×D53—81no—
286C2×D53—82no—
287C2×D53—81no—
288C2×D53—82no—
289C2×D53—81no—
290C2×D53—82no—
291C4×C53—22no—
292C4×C53—22no—
293C4×C53—41no—
294C4×C53—41no—
295C4×C53—41no—
296C4×C53#51812yes—
297C4×C53#53812yes—
298Dic53—42no—
299Dic53—81no—
300Dic53—81no—
301Dic53—82no—
302C2×#447—531no—
303C4×#121—531no—
304C2×#390—531no—
305#1972—532no—
306#1936—532no—
307#1926—532no—
308#1879—532no—
309#1874—531no—
310#1853—531no—
311#1786—531no—
312#1749D5311yescharacteristic, complemented
313C2×C53—24no—
314C2×C53#232111yescharacteristic
315C2×C53C2×#44211yescharacteristic
316C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
317C2×C53#319211yescharacteristic
318C2×C53#319411yescharacteristic
319C2×C53#321311yescharacteristic
320C2×C53#321711yescharacteristic
321D53—82no—
322D53—84no—
323C23×D4—531no—
324C22×D8—532no—
325C2×#121—531no—
326C2×C4×D4—531no—
327C2×C4×D4—1061no—
328C22×C4⋊C4—531no—
329C22×C22⋊C4—1061no—
330C22×C22⋊C4—1061no—
331C22×C42C2×D5311yescharacteristic
332#447—538no—
333C2×#106—532no—
334C2×#102—532no—
335C2×#102—532no—
336C2×#102—532no—
337C2×#102—532no—
338C2×C22⋊C8C2×D5311yescharacteristic
339C2×C22⋊C8C2×D5311yescharacteristic
340C2×#97C2×D5312yescomplemented
341#390—532no—
342#390—532no—
343#390—531no—
344#390—1061no—
345#390—1061no—
346#389—531no—
347#388—1061no—
348#386—1061no—
349#380—531no—
350C4×C4⋊C4—531no—
351C4×C22⋊C4—1061no—
352#340—532no—
353#340—532no—
354#336C2×D5312yescomplemented
355C53#2204611yescharacteristic, complemented, Sylow
356C25—1061no—
357C22×D4—534no—
358C22×D4—531no—
359C22×D4—532no—
360C22×D4—1061no—
361C22×D4—1061no—
362C22×D4—1062no—
363C22×D4—2122no—
364C22×D4—2122no—
365C23×C4—1061no—
366C23×C4—1061no—
367C23×C4D21211yescharacteristic
368C23×C4D21211yescharacteristic
369C23×C4C22×D5311yescharacteristic
370C2×D8—538no—
371C2×D8—2122no—
372C2×D8—2122no—
373C22×C8C22×D5312yes—
374C22×C8C22×D5312yes—
375#121—534no—
376#121—1062no—
377C4×D4—2122no—
378C4×D4—2124no—
379C2×C4⋊C4—531no—
380C2×C4⋊C4—532no—
381C2×C4⋊C4—531no—
382C2×C4⋊C4—531no—
383C2×C4⋊C4—1061no—
384C2×C4⋊C4—1061no—
385C2×C4⋊C4—2122no—
386C2×C22⋊C4—1061no—
387C2×C22⋊C4—1061no—
388C2×C22⋊C4—1062no—
389C2×C22⋊C4—1062no—
390C2×C22⋊C4—1062no—
391C2×C22⋊C4—1062no—
392C2×C22⋊C4—1061no—
393C2×C22⋊C4—1061no—
394C2×C22⋊C4—1061no—
395C2×C22⋊C4—2124no—
396C2×C22⋊C4—2124no—
397C2×C42—42no—
398C2×C42—1061no—
399C2×C42D21211yescharacteristic
400C2×C42D21211yescharacteristic
401C2×C42C22×D5311yescharacteristic
402C2×C42C22×D5311yescharacteristic
403#106—2122no—
404#102—538no—
405#102—2122no—
406#102—2122no—
407#102—2122no—
408C22⋊C8—22no—
409C22⋊C8C4×D5314yescomplemented
410#97—42no—
411#95—1061no—
412#95—1061no—
413#95—1061no—
414C24—1064no—
415C24—1061no—
416C24—1062no—
417C24—2122no—
418C24—2122no—
419C24—2122no—
420C24—2122no—
421C24—2122no—
422C24—2122no—
423C24C2×D21211yescharacteristic
424C2×D4—534no—
425C2×D4—532no—
426C2×D4—1064no—
427C2×D4—1064no—
428C2×D4—1061no—
429C2×D4—1062no—
430C2×D4—1062no—
431C2×D4—1062no—
432C2×D4—1062no—
433C2×D4—1062no—
434C2×D4—1062no—
435C2×D4—1062no—
436C2×D4—2128no—
437C2×D4—2122no—
438C2×D4—2122no—
439C2×D4—2128no—
440C2×D4—2122no—
441C2×D4—2122no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—1061no—
451C22×C4—1061no—
452C22×C4—1061no—
453C22×C4—1062no—
454C22×C4—2122no—
455C22×C4—2122no—
456C22×C4—2122no—
457C22×C4—2122no—
458C22×C4—2122no—
459C22×C4—2122no—
460C22×C4C2×C4×D5312yes—
461C22×C4C2×D21211yescharacteristic
462C22×C4C2×D21211yescharacteristic
463C22×C4C2×D21211yescharacteristic
464C22×C4C2×D21211yescharacteristic
465C22×C4C2×D21211yescharacteristic
466C22×C4#1171994512yes—
467C22×C4C23×D5311yescharacteristic
468D8—2128no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8C2×C4×D5314yes—
475C4⋊C4—532no—
476C4⋊C4—1061no—
477C4⋊C4—1062no—
478C4⋊C4—1062no—
479C4⋊C4—1062no—
480C4⋊C4—1062no—
481C4⋊C4—2122no—
482C4⋊C4—2122no—
483C22⋊C4—1064no—
484C22⋊C4—2124no—
485C22⋊C4—2124no—
486C22⋊C4—2124no—
487C22⋊C4—2122no—
488C22⋊C4—2122no—
489C22⋊C4—2122no—
490C22⋊C4—2122no—
491C22⋊C4—2124no—
492C42—22no—
493C42—22no—
494C42—41no—
495C42—41no—
496C42—2122no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—1062no—
502C23—1064no—
503C23—4244no—
504C23—4244no—
505C23—4244no—
506C23—4244no—
507C23—2128no—
508C23—2128no—
509C23—2122no—
510C23—2122no—
511C23—2121no—
512C23—2121no—
513C23—2121no—
514C23—2122no—
515C23—2121no—
516C23—2122no—
517C23—2121no—
518C23—2122no—
519C23—2122no—
520C23—2121no—
521C23—2128no—
522C23C4×D21212yes—
523C23#1313324512yes—
524C23#1313324512yes—
525C23C22×D21211yescharacteristic
526D4—1064no—
527D4—1064no—
528D4—1064no—
529D4—2128no—
530D4—2128no—
531D4—2122no—
532D4—2122no—
533D4—2122no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—1062no—
552C2×C4—4244no—
553C2×C4—4244no—
554C2×C4—2121no—
555C2×C4—2122no—
556C2×C4—2122no—
557C2×C4—2121no—
558C2×C4—2121no—
559C2×C4—2121no—
560C2×C4—2121no—
561C2×C4—2122no—
562C2×C4—2121no—
563C2×C4—2121no—
564C2×C4—2121no—
565C2×C4C4×D21212yes—
566C2×C4#1313324512yes—
567C2×C4C22×C4×D5311yescharacteristic
568C2×C4C2×#1171994511yescharacteristic
569C2×C4#1313326212yes—
570C2×C4#1313326312yes—
571C8—24no—
572C8—42no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—42416no—
580C22—4242no—
581C22—4241no—
582C22—4242no—
583C22—4241no—
584C22—4241no—
585C22—4242no—
586C22—4242no—
587C22—4241no—
588C22—2122no—
589C22—2124no—
590C22—2124no—
591C22—2122no—
592C22—2124no—
593C22—2122no—
594C22#1681073614yes—
595C22C2×C4×D21211yescharacteristic
596C22#1681118511yescharacteristic
597C22#1681119211yescharacteristic
598C22C2×#1313324511yescharacteristic
599C22C2×#1313324511yescharacteristic
600C22C2×#1313324511yescharacteristic
601C22#1681142311yescharacteristic
602C4—22no—
603C4—22no—
604C4—41no—
605C4—41no—
606C4—41no—
607C4—4241no—
608C4—4242no—
609C4—4241no—
610C4—2122no—
611C4#1681109512yes—
612C4#1681120612yes—
613C2—24no—
614C2—4244no—
615C2—4242no—
616C2#2027737511yescharacteristic
617C2C2×#1681073611yescharacteristic
618C2C4×#1313324511yescharacteristic
619C2#2028551511yescharacteristic
620C2#2028553111yescharacteristic
621C2#2028570411yescharacteristic
622C2#2028573511yescharacteristic
subgroupG/H
centerC23C22×D212
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1749C2
socleC23×C53C22×D4
prime radicalC2×#16810759C2
cosocleC22×C4×C53C24
hypercenter#1749D53
hypocenterC53#22046
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
2. C4×#13133191C2semidirect
3. C2×#16810525C2semidirect
4. #20275591C2semidirect
5. #20275412C2semidirect
6. #20275343C2semidirect
7. #20275062C2semidirect
8. #20275030C2semidirect
9. #20274910C2semidirect
10. #20274647C2semidirect
11. C53×#1749C2semidirect
15. C2×C4×D212C22semidirect
20. #16810759C4semidirect
21. C2×#13133149C22semidirect
25. C2×#13133139C22semidirect
30. #16810525C22semidirect
31. #16810521C22semidirect
35. C4×C4⋊Dic53C22semidirect
37. #16810386C22semidirect
38. #16810386C22semidirect
42. C2×#97×C53C22semidirect
43. C53×#336C22semidirect
49. C22×D212C2×C4semidirect
57. C2×D424C2×C4semidirect
85. #13133139C2×C4semidirect
100. C22⋊C8×C53C2×C4semidirect
312. #1749D53semidirect
340. C2×#97C2×D53semidirect
354. #336C2×D53semidirect
355. C53#22046semidirect
409. C22⋊C8C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168107592
C4×#131331912
C2×#168105252
#202755912
#202754122
#202753432
#202750622
#202750302
#202749102
#202746472
C53×#17492
#2204653