Finite Group Databasebeta

← #27483747

Subgroups — #27483747

Order 13568.

754 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283074C211yesmaximal, characteristic, complemented
2#124×Dic53C211yesmaximal, characteristic
3#20276284C211yesmaximal, characteristic, complemented
4#20276283C211yesmaximal, characteristic, complemented
5#20275951C211yesmaximal, characteristic
6#20275400C211yesmaximal, characteristic
7#20275395C211yesmaximal, characteristic
8#20275393C211yesmaximal, characteristic, complemented
9#20275388C211yesmaximal, characteristic, complemented
10#20274718C211yesmaximal, characteristic, complemented
11#20274718C211yesmaximal, characteristic, complemented
12#20274713C211yesmaximal, characteristic
13#20274713C211yesmaximal, characteristic
14#20274563C211yesmaximal, characteristic, complemented
15C53×#2042C211yesmaximal, characteristic
16C2×#13133207—21no—
17C2×Q8×Dic53C2211yescharacteristic
18C2×D4×Dic53C2211yescharacteristic
19C2×#13133146C2211yescharacteristic
20C2×#13133146C2211yescharacteristic
21C2×#13133141C2211yescharacteristic
22C2×#13133141C2211yescharacteristic
23C2×#13133138C2211yescharacteristic
24C2×#13133138C2211yescharacteristic
25C2×C8×Dic53C2211yescharacteristic
26C2×C8×Dic53C2211yescharacteristic
27#16810530C2211yescharacteristic
28#16810526C2211yescharacteristic
29#16810522C2211yescharacteristic
30#16810521C2211yescharacteristic
31#16810520—21no—
32#16810519C2211yescharacteristic
33#16810518—21no—
34#16810516—21no—
35#16810516—21no—
36#16810492C2211yescharacteristic
37#16810492C2211yescharacteristic
38#16810489C2211yescharacteristic
39#16810487—21no—
40C22⋊C4×Dic53—21no—
41C22⋊C4×Dic53—21no—
42C42×Dic53C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810378C2211yescharacteristic
52C2×#124×C53C2211yescharacteristic
53C2×#103×C53C2211yescharacteristic
54C2×#103×C53C2211yescharacteristic
55C2×#102×C53C2211yescharacteristic
56C2×#102×C53C2211yescharacteristic
57C53×#389C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C23×Dic53—21no—
60C22×#88322C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62#13133207—44no—
63Q8×Dic53—22no—
64Q8×Dic53—22no—
65D4×Dic53—22no—
66D4×Dic53—22no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×C22⋊Dic53—21no—
80C2×C22⋊Dic53—21no—
81C2×C22⋊Dic53—21no—
82C2×C4×Dic53—21no—
83C2×C4×Dic53D411yescharacteristic
84C2×C4×Dic53D411yescharacteristic
85C2×C4×Dic53D411yescharacteristic
86C2×C4×Dic53D411yescharacteristic
87C2×C4×Dic53C2311yescharacteristic
88C2×C4×Dic53C2311yescharacteristic
89#13133146—22no—
90#13133146—22no—
91#13133141—22no—
92#13133141—41no—
93#13133138—22no—
94#13133138—41no—
95C8×Dic53—41no—
96C8×Dic53—41no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104C22×Q8×C53C2311yescharacteristic
105C22×D4×C53C2311yescharacteristic
106C22×C8×C53C2311yescharacteristic
107C22×C8×C53C2311yescharacteristic
108#124×C53—22no—
109#124×C53—22no—
110C2×C4⋊C4×C53—21no—
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C22⋊C4×C53—21no—
114C2×C22⋊C4×C53—21no—
115C2×C42×C53C2311yescharacteristic
116#103×C53—22no—
117#103×C53—41no—
118#102×C53—22no—
119#102×C53—41no—
120#95×C53—21no—
121C24×C53—21no—
122C2×Q8×C53—21no—
123C2×Q8×C53—21no—
124C2×Q8×C53—21no—
125C2×Q8×C53—21no—
126C2×Q8×C53—21no—
127C2×Q8×C53#4112yes—
128C2×D4×C53—21no—
129C2×D4×C53—21no—
130C2×D4×C53—21no—
131C2×D4×C53—21no—
132C2×D4×C53—21no—
133C2×D4×C53#4112yes—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2411yescharacteristic
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—42no—
145C22×Dic53—42no—
146C22×Dic53—42no—
147C22×Dic53C2×D411yescharacteristic
148C22×Dic53C2×D411yescharacteristic
149C22×Dic53C2×D411yescharacteristic
150C22×Dic53C2×D411yescharacteristic
151C2×C8×C53—21no—
152C2×C8×C53—21no—
153C2×C8×C53—21no—
154C2×C8×C53—21no—
155C2×C8×C53—21no—
156C2×C8×C53—21no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—42no—
162C22⋊C4×C53—24no—
163C22⋊C4×C53—24no—
164C42×C53—21no—
165C42×C53—21no—
166C22⋊Dic53—42no—
167C22⋊Dic53—42no—
168C22⋊Dic53—42no—
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—42no—
178C4⋊Dic53—42no—
179#11719919—42no—
180#11719919—42no—
181#11719919—42no—
182#11719919—42no—
183#11719919—42no—
184#11719919—42no—
185C4×Dic53—21no—
186C4×Dic53—21no—
187C4×Dic53—22no—
188C4×Dic53—22no—
189C4×Dic53—22no—
190C4×Dic53—22no—
191C4×Dic53—42no—
192C4×Dic53#4112yes—
193C4×Dic53#4112yes—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C23×C53—22no—
201C23×C53—41no—
202C23×C53—41no—
203C23×C53—41no—
204C23×C53—41no—
205C23×C53—41no—
206C23×C53—41no—
207C23×C53C22×D411yescharacteristic
208Q8×C53—22no—
209Q8×C53—22no—
210Q8×C53—41no—
211Q8×C53—41no—
212D4×C53—22no—
213D4×C53—22no—
214D4×C53—41no—
215D4×C53—41no—
216C2×C4×C53—22no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53C2×#4111yescharacteristic
232C2×C4×C53C2×#4111yescharacteristic
233C2×C4×C53C2×#4111yescharacteristic
234C2×C4×C53C2×#4111yescharacteristic
235C2×C4×C53H(5,𝔽2)11yescharacteristic
236C2×C4×C532−1+411yescharacteristic
237C2×Dic53—21no—
238C2×Dic53—21no—
239C2×Dic53—21no—
240C2×Dic53—21no—
241C2×Dic53—21no—
242C2×Dic53—21no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—84no—
256C2×Dic53#12412yes—
257C2×Dic53#12412yes—
258C2×Dic53#13512yes—
259C2×Dic53#13512yes—
260C2×Dic53AGL(1,ℤ/8)12yes—
261C2×Dic53#13712yes—
262C8×C53—41no—
263C8×C53—41no—
264#88322—41no—
265#88322—41no—
266#29122—531nomaximal, Sylow
267C22×C53—42no—
268C22×C53—42no—
269C22×C53—42no—
270C22×C53—81no—
271C22×C53—81no—
272C22×C53—81no—
273C22×C53—81no—
274C22×C53#52511yescharacteristic
275C22×C53C2×#12411yescharacteristic
276C22×C53#53711yescharacteristic
277C22×C53C2×#13511yescharacteristic
278C22×C53C2×#13511yescharacteristic
279C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
280C22×C53C2×#13711yescharacteristic
281C4×C53—42no—
282C4×C53—41no—
283C4×C53—41no—
284C4×C53—81no—
285C4×C53—81no—
286C4×C53#55211yescharacteristic
287C4×C53#55211yescharacteristic
288C4×C53#55311yescharacteristic
289C4×C53#55311yescharacteristic
290Dic53—22no—
291Dic53—22no—
292Dic53—22no—
293Dic53—81no—
294Dic53—81no—
295Dic53—81no—
296Dic53—81no—
297Dic53#48814yes—
298#2859—531no—
299C4×#124—531no—
300#2101—531no—
301#2101—531no—
302#2042—531no—
303#2042D5311yescharacteristic
304#1934—531no—
305#1934—531no—
306#1933—531no—
307#1933—531no—
308#1803—531no—
309#1803—531no—
310#1802—531no—
311#1802—531no—
312C4×C4⋊C8—531no—
313C2×C53—82no—
314C2×C53#256211yescharacteristic
315C2×C53#310611yescharacteristic
316C2×C53#310711yescharacteristic
317C2×C53C2×#48811yescharacteristic
318C2×C53#314111yescharacteristic
319C2×C53#324711yescharacteristic
320C2×C53#324811yescharacteristic
321C2×#125—1061no—
322C2×#124C2×D5311yescharacteristic
323C2×C4×Q8—531no—
324C2×C4×D4—531no—
325C2×C4⋊C8—531no—
326C2×C4⋊C8—531no—
327C2×#103—531no—
328C2×#103—531no—
329C2×#103C2×D5311yescharacteristic
330C2×#103C2×D5311yescharacteristic
331C2×#102—531no—
332C2×#102—531no—
333C2×#102C2×D5311yescharacteristic
334C2×#102C2×D5311yescharacteristic
335C2×C4×C8—531no—
336C2×C4×C8—531no—
337#391—531no—
338#390—531no—
339#389—531no—
340#389—531no—
341#389—531no—
342#389—1061no—
343#389—1061no—
344#389C2×D5311yescharacteristic
345#388—1061no—
346#388—1061no—
347#384—531no—
348#384—531no—
349#382—531no—
350#382—1061no—
351C4×C22⋊C4—1061no—
352C4×C22⋊C4—1061no—
353C43—531no—
354#340—531no—
355#340—531no—
356#340—531no—
357#340—531no—
358#340—531no—
359#340—531no—
360#340—531no—
361#340—531no—
362#336—531no—
363#336C2×D5311yescharacteristic
364C53#2912211yescharacteristic, complemented, Sylow
365C22×Q8C22×D5311yescharacteristic
366C22×D4C22×D5311yescharacteristic
367C23×C4—1061no—
368C22×C8—531no—
369C22×C8—531no—
370C22×C8C22×D5311yescharacteristic
371C22×C8C22×D5311yescharacteristic
372#125—2124no—
373#124—22no—
374#124—22no—
375C4×Q8—1062no—
376C4×Q8—1062no—
377C4×D4—1062no—
378C4×D4—1062no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—531no—
381C2×C4⋊C4—531no—
382C2×C4⋊C4—531no—
383C2×C4⋊C4—531no—
384C2×C4⋊C4—1061no—
385C2×C4⋊C4—1061no—
386C2×C4⋊C4—1061no—
387C2×C4⋊C4—1061no—
388C2×C4⋊C4—1061no—
389C2×C4⋊C4—1061no—
390C2×C4⋊C4—1061no—
391C2×C4⋊C4—1061no—
392C2×C4⋊C4C22×D5311yescharacteristic
393C2×C4⋊C4C22×D5311yescharacteristic
394C2×C22⋊C4—21no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—1061no—
397C2×C22⋊C4—1061no—
398C2×C22⋊C4—1061no—
399C2×C42—531no—
400C2×C42—531no—
401C2×C42—531no—
402C2×C42—531no—
403C2×C42—531no—
404C2×C42—531no—
405C2×C42—1061no—
406C2×C42C22×D5311yescharacteristic
407C4⋊C8—1062no—
408C4⋊C8—1062no—
409#103—22no—
410#103—41no—
411#103—1062no—
412#103—2121no—
413#102—22no—
414#102—41no—
415#102—1062no—
416#102—2121no—
417C4×C8—2121no—
418C4×C8—2121no—
419#95—21no—
420#95—1061no—
421#95—1061no—
422#95—1061no—
423#95—1061no—
424#95—1061no—
425#95—1061no—
426#95—1061no—
427C24—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8#1171994712yes—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4#1171994712yes—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—531no—
444C22×C4—531no—
445C22×C4—531no—
446C22×C4—531no—
447C22×C4—1061no—
448C22×C4—1061no—
449C22×C4—1061no—
450C22×C4—1061no—
451C22×C4—2122no—
452C22×C4—2122no—
453C22×C4—2122no—
454C22×C4D4×D5311yescharacteristic
455C22×C4D4×D5311yescharacteristic
456C22×C4C23×D5311yescharacteristic
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—1061no—
464C2×C8—1061no—
465C2×C8—1061no—
466C2×C8—1061no—
467C2×C8—1061no—
468C2×C8—1061no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—42no—
474C4⋊C4—1061no—
475C4⋊C4—1061no—
476C4⋊C4—1061no—
477C4⋊C4—1061no—
478C4⋊C4—1061no—
479C4⋊C4—1061no—
480C4⋊C4—1061no—
481C4⋊C4—1061no—
482C4⋊C4—2122no—
483C4⋊C4—2122no—
484C4⋊C4—2122no—
485C4⋊C4—2122no—
486C4⋊C4—2122no—
487C4⋊C4—2122no—
488C4⋊C4—2122no—
489C4⋊C4—2122no—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C22⋊C4—2122no—
493C22⋊C4—2122no—
494C22⋊C4—2122no—
495C42—21no—
496C42—21no—
497C42—532no—
498C42—532no—
499C42—1062no—
500C42—1062no—
501C42—1062no—
502C42—1062no—
503C42—1061no—
504C42—1061no—
505C42—2122no—
506C23—22no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23C2×D4×D5311yescharacteristic
514Q8—22no—
515Q8—22no—
516Q8—41no—
517Q8—41no—
518D4—22no—
519D4—22no—
520D4—41no—
521D4—41no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—532no—
538C2×C4—532no—
539C2×C4—532no—
540C2×C4—532no—
541C2×C4—532no—
542C2×C4—532no—
543C2×C4—1061no—
544C2×C4—1061no—
545C2×C4—1061no—
546C2×C4—1061no—
547C2×C4—1061no—
548C2×C4—1061no—
549C2×C4—4244no—
550C2×C4—2121no—
551C2×C4—2121no—
552C2×C4—2121no—
553C2×C4—2121no—
554C2×C4—2121no—
555C2×C4—2121no—
556C2×C4—2121no—
557C2×C4—2121no—
558C2×C4—2121no—
559C2×C4—2121no—
560C2×C4—2121no—
561C2×C4—2121no—
562C2×C4#41×D5311yescharacteristic
563C2×C4#41×D5311yescharacteristic
564C2×C4C2×#1171994711yescharacteristic
565C2×C4C2×#1171994711yescharacteristic
566C2×C4#1313326211yescharacteristic
567C2×C4#1313326411yescharacteristic
568C8—41no—
569C8—41no—
570C8—2121no—
571C8—2121no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—81no—
576C22—81no—
577C22—81no—
578C22—81no—
579C22#1681112911yescharacteristic
580C22#124×D5311yescharacteristic
581C22#1681120211yescharacteristic
582C22#135×D5311yescharacteristic
583C22#135×D5311yescharacteristic
584C22AGL(1,ℤ/8)×D5311yescharacteristic
585C22#137×D5311yescharacteristic
586C4—42no—
587C4—41no—
588C4—41no—
589C4—81no—
590C4—81no—
591C4—534no—
592C4—1062no—
593C4—1062no—
594C4—1062no—
595C4—4241no—
596C4—4241no—
597C4—4241no—
598C4—4241no—
599C4#1681131011yescharacteristic
600C4#1681131011yescharacteristic
601C4#1681131611yescharacteristic
602C4#1681131611yescharacteristic
603C2—82no—
604C2#2027987511yescharacteristic
605C2#2028476811yescharacteristic
606C2#2028477911yescharacteristic
607C2#488×D5311yescharacteristic
608C2#2028510711yescharacteristic
609C2#2028603811yescharacteristic
610C2#2028604911yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2042C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
hypercenter#2042D53
hypocenterC53#29122
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2042

Complements

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

NH ≅ G/Ntype
1. #20283074C2semidirect
3. #20276284C2semidirect
4. #20276283C2semidirect
8. #20275393C2semidirect
9. #20275388C2semidirect
10. #20274718C2semidirect
11. #20274718C2semidirect
14. #20274563C2semidirect
364. C53#29122semidirect

Maximal subgroups

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

subgroupindex
#202830742
#124×Dic532
#202762842
#202762832
#202759512
#202754002
#202753952
#202753932
#202753882
#202747182
#202747132
#202745632
C53×#20422
#2912253