Finite Group Databasebeta

← #27465607

Subgroups — #27465607

Order 13568.

730 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277408C211yesmaximal, characteristic
2#20277324C211yesmaximal, characteristic, complemented
3#20276148C211yesmaximal, characteristic, complemented
4#20276143C211yesmaximal, characteristic, complemented
5#20275898C211yesmaximal, characteristic
6#20275898C211yesmaximal, characteristic
7#20275485C211yesmaximal, characteristic, complemented
8#20275474C211yesmaximal, characteristic, complemented
9#20275343C211yesmaximal, characteristic, complemented
10#20275336C211yesmaximal, characteristic, complemented
11#20275308C211yesmaximal, characteristic, complemented
12#20275287C211yesmaximal, characteristic, complemented
13#20274726C211yesmaximal, characteristic, complemented
14#20274716C211yesmaximal, characteristic, complemented
15C53×#1798C211yesmaximal, characteristic, complemented
16C2×#13133191C2211yescharacteristic
17C2×C4×D212C2211yescharacteristic
18C2×#13133151C2211yescharacteristic
19C2×#13133151C2211yescharacteristic
20C2×#13133151C2211yescharacteristic
21C2×#13133149C2211yescharacteristic
22C2×#13133139C2211yescharacteristic
23C2×#13133139C2211yescharacteristic
24C2×#13133139C2211yescharacteristic
25C2×#13133139C2211yescharacteristic
26#16810525C2211yescharacteristic
27#16810523—21no—
28#16810521C2211yescharacteristic
29#16810516—21no—
30#16810516—21no—
31#16810513—21no—
32#16810509—21no—
33#16810509—21no—
34#16810499—21no—
35#16810494—21no—
36#16810493C2211yescharacteristic
37#16810489C2211yescharacteristic
38#16810483C2211yescharacteristic
39#16810483C2211yescharacteristic
40#16810483C2211yescharacteristic
41C4×C4⋊Dic53C2211yescharacteristic
42C4×C4⋊Dic53C2211yescharacteristic
43C4×C4⋊Dic53C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52C2×#117×C53C2211yescharacteristic
53C2×C4⋊C8×C53C2211yescharacteristic
54C2×C4⋊C8×C53C2211yescharacteristic
55C2×C22⋊C8×C53C2211yescharacteristic
56C2×C22⋊C8×C53C2211yescharacteristic
57C53×#336C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C22×D212C2311yescharacteristic
60C22×C22⋊D53—21no—
61C22×C4×D53—21no—
62#13133191—22no—
63#13133191—22no—
64C4×D212—22no—
65C4×D212—22no—
66C2×C4⋊Dic53—21no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×#11719919—21no—
76C2×C22⋊Dic53—21no—
77C2×C22⋊Dic53—21no—
78C2×C22⋊Dic53—21no—
79C2×#11719921—21no—
80C2×#11719921—21no—
81C2×#11719921—21no—
82C2×#11719921—21no—
83C2×C4×Dic53—21no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53—21no—
86#13133151—22no—
87#13133151—22no—
88#13133151—41no—
89#13133149—41no—
90#13133139—22no—
91#13133139—22no—
92#13133139—41no—
93#13133139—41no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103C23×C4×C53C2311yescharacteristic
104C22×C8×C53C2311yescharacteristic
105C22×C8×C53C2311yescharacteristic
106C22×C8×C53C2311yescharacteristic
107C22×C8×C53C2311yescharacteristic
108#117×C53—22no—
109#117×C53—22no—
110C2×C4⋊C4×C53D411yescharacteristic
111C2×C4⋊C4×C53D411yescharacteristic
112C2×C22⋊C4×C53D411yescharacteristic
113C2×C22⋊C4×C53D411yescharacteristic
114C2×C42×C53C2311yescharacteristic
115C2×C42×C53C2311yescharacteristic
116C4⋊C8×C53—22no—
117C4⋊C8×C53—22no—
118C22⋊C8×C53—41no—
119C22⋊C8×C53—41no—
120C24×C53C2×D411yescharacteristic
121C23×D53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2411yescharacteristic
134C2×C22⋊D53—24no—
135C2×C22⋊D53—81no—
136C2×C22⋊D53—81no—
137C22×Dic53—21no—
138C22×Dic53—21no—
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C2×D212—21no—
145C2×D212—21no—
146C2×D212—21no—
147C2×D212—21no—
148C2×D212—21no—
149C2×D212#4112yes—
150C2×C4×D53—42no—
151C2×C4×D53—81no—
152C2×C4×D53—81no—
153C2×C8×C53—21no—
154C2×C8×C53—21no—
155C2×C8×C53—21no—
156C2×C8×C53—21no—
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C2×C8×C53—21no—
165C4⋊C4×C53—22no—
166C4⋊C4×C53—22no—
167C22⋊C4×C53—22no—
168C22⋊C4×C53—22no—
169C42×C53—21no—
170C42×C53—21no—
171C42×C53#4112yes—
172C42×C53#4112yes—
173C22⋊Dic53—81no—
174C22⋊Dic53—81no—
175C22⋊Dic53—81no—
176#11719921—24no—
177#11719921—42no—
178#11719921—81no—
179#11719921—81no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—81no—
194C4⋊Dic53—81no—
195C4⋊Dic53#4112yes—
196#11719919—81no—
197C4×Dic53—81no—
198C4×Dic53—81no—
199C4×Dic53—81no—
200C23×C53—21no—
201C23×C53—21no—
202C23×C53—21no—
203C23×C53—21no—
204C23×C53—21no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53C22×D411yescharacteristic
208C22×D53—22no—
209C22×D53—41no—
210C22×D53—41no—
211C22×D53—41no—
212C22×D53—41no—
213C22×D53—41no—
214C22×D53—41no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
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—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53#12312yes—
233C2×C4×C53#12312yes—
234C2×C4×C53C2×#4111yescharacteristic
235C2×C4×C53C2×#4111yescharacteristic
236C2×C4×C53C2×#4111yescharacteristic
237C2×C4×C53C2×#4111yescharacteristic
238C2×C4×C53H(5,𝔽2)11yescharacteristic
239C2×C4×C532−1+411yescharacteristic
240C22⋊D53—84no—
241C2×Dic53—22no—
242C2×Dic53—41no—
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—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262D212—22no—
263D212—22no—
264D212—41no—
265D212—41no—
266C4×D53—84no—
267C8×C53—41no—
268C8×C53—41no—
269C8×C53—41no—
270C8×C53—41no—
271#27636—531nomaximal, Sylow
272C22×C53—41no—
273C22×C53—41no—
274C22×C53—41no—
275C22×C53—41no—
276C22×C53—41no—
277C22×C53—41no—
278C22×C53—41no—
279C22×C53C2×#12311yescharacteristic
280C22×C53#52511yescharacteristic
281C22×C53#53711yescharacteristic
282C22×C53#57611yescharacteristic
283C22×C53#57611yescharacteristic
284C22×C53#57711yescharacteristic
285C22×C53#57811yescharacteristic
286C2×D53—42no—
287C2×D53—42no—
288C2×D53—42no—
289C2×D53—81no—
290C2×D53—81no—
291C2×D53—81no—
292C2×D53—81no—
293C4×C53—22no—
294C4×C53—22no—
295C4×C53—41no—
296C4×C53—41no—
297C4×C53—41no—
298C4×C53#55211yescharacteristic
299C4×C53#55311yescharacteristic
300C4×C53#55511yescharacteristic
301C4×C53#55611yescharacteristic
302Dic53—42no—
303Dic53—81no—
304Dic53—81no—
305Dic53—81no—
306Dic53—81no—
307Dic53—81no—
308Dic53—81no—
309#2324—531no—
310#2314—531no—
311#2077—531no—
312#2076—531no—
313#2033—531no—
314#2033—531no—
315#1949—531no—
316#1947—531no—
317#1926—531no—
318#1925—531no—
319#1921—531no—
320#1918—531no—
321#1804—531no—
322#1802—531no—
323#1798D5311yescharacteristic, complemented
324C2×C53—81no—
325C2×C53#248511yescharacteristic
326C2×C53#309611yescharacteristic
327C2×C53#309811yescharacteristic
328C2×C53#312011yescharacteristic
329C2×C53#312211yescharacteristic
330C2×C53#325711yescharacteristic
331C2×C53#326211yescharacteristic
332D53—82no—
333C2×#121—531no—
334C2×C4×D4—531no—
335C2×#117C2×D5311yescharacteristic
336C2×#107—531no—
337C2×#107—531no—
338C2×#107—531no—
339C2×#106—531no—
340C2×C4⋊C8C2×D5311yescharacteristic
341C2×C4⋊C8C2×D5311yescharacteristic
342C2×#102—531no—
343C2×#102—531no—
344C2×#102—531no—
345C2×#102—531no—
346C2×C22⋊C8C2×D5311yescharacteristic
347C2×C22⋊C8C2×D5311yescharacteristic
348#390—531no—
349#390—1061no—
350#389—531no—
351#388—1061no—
352#388—1061no—
353#388—1061no—
354#387—1061no—
355#387—1061no—
356#386—1061no—
357#385—1061no—
358#384—531no—
359#382—531no—
360#380—531no—
361#380—531no—
362#380—531no—
363C4×C4⋊C4—531no—
364C4×C4⋊C4—531no—
365C4×C4⋊C4—531no—
366#340—531no—
367#340—531no—
368#340—531no—
369#340—531no—
370#340—531no—
371#340—531no—
372#340—531no—
373#340—531no—
374#336C2×D5311yescharacteristic
375#336C2×D5311yescharacteristic
376C53#2763611yescharacteristic, complemented, Sylow
377C22×D4—531no—
378C22×D4—1061no—
379C23×C4—1061no—
380C23×C4C22×D5311yescharacteristic
381C22×C8C22×D5311yescharacteristic
382C22×C8C22×D5311yescharacteristic
383C22×C8C22×D5311yescharacteristic
384C22×C8C22×D5311yescharacteristic
385#121—1062no—
386#121—1062no—
387C4×D4—1062no—
388C4×D4—1062no—
389#117—22no—
390#117—22no—
391C2×C4⋊C4—531no—
392C2×C4⋊C4—531no—
393C2×C4⋊C4—531no—
394C2×C4⋊C4—531no—
395C2×C4⋊C4—531no—
396C2×C4⋊C4—531no—
397C2×C4⋊C4—531no—
398C2×C4⋊C4—1061no—
399C2×C4⋊C4—1061no—
400C2×C4⋊C4—1061no—
401C2×C4⋊C4D21211yescharacteristic
402C2×C4⋊C4D21211yescharacteristic
403C2×C22⋊C4—1061no—
404C2×C22⋊C4—1061no—
405C2×C22⋊C4—1061no—
406C2×C22⋊C4—1061no—
407C2×C22⋊C4—1061no—
408C2×C22⋊C4—1061no—
409C2×C22⋊C4—1061no—
410C2×C22⋊C4D21211yescharacteristic
411C2×C22⋊C4D21211yescharacteristic
412C2×C42—1061no—
413C2×C42—1061no—
414C2×C42—1061no—
415C2×C42C22×D5311yescharacteristic
416C2×C42C22×D5311yescharacteristic
417#107—1062no—
418#107—1062no—
419#107—2121no—
420#106—2121no—
421C4⋊C8—22no—
422C4⋊C8—22no—
423#102—1062no—
424#102—1062no—
425#102—2121no—
426#102—2121no—
427C22⋊C8—41no—
428C22⋊C8—41no—
429#95—1061no—
430#95—1061no—
431#95—1061no—
432#95—1061no—
433#95—1061no—
434#95—1061no—
435#95—1061no—
436#95—1061no—
437#95—1061no—
438C24—1061no—
439C24C2×D21211yescharacteristic
440C2×D4—532no—
441C2×D4—1064no—
442C2×D4—1061no—
443C2×D4—1061no—
444C2×D4—1061no—
445C2×D4—1061no—
446C2×D4—1061no—
447C2×D4—4241no—
448C2×D4—4241no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—1061no—
456C22×C4—1061no—
457C22×C4—1061no—
458C22×C4—1061no—
459C22×C4—1061no—
460C22×C4—1061no—
461C22×C4—1061no—
462C22×C4—4241no—
463C22×C4—4241no—
464C22×C4—2122no—
465C22×C4C2×D21211yescharacteristic
466C22×C4C2×D21211yescharacteristic
467C22×C4C2×D21211yescharacteristic
468C22×C4C2×D21211yescharacteristic
469C22×C4C2×D21211yescharacteristic
470C22×C4C23×D5311yescharacteristic
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—532no—
486C4⋊C4—1061no—
487C4⋊C4—1061no—
488C4⋊C4—1061no—
489C4⋊C4—1061no—
490C4⋊C4—1061no—
491C4⋊C4—1061no—
492C4⋊C4—1061no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—1061no—
496C4⋊C4—1061no—
497C4⋊C4—1061no—
498C4⋊C4—1061no—
499C4⋊C4—4241no—
500C4⋊C4—4241no—
501C4⋊C4—4241no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—1064no—
505C22⋊C4—4241no—
506C22⋊C4—4241no—
507C22⋊C4—4241no—
508C22⋊C4—4241no—
509C22⋊C4—4241no—
510C22⋊C4—2122no—
511C42—21no—
512C42—21no—
513C42—4241no—
514C42—4241no—
515C42—4241no—
516C42#1171994712yes—
517C42#1171994712yes—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—1062no—
526C23—2121no—
527C23—2121no—
528C23—2121no—
529C23—2121no—
530C23—2121no—
531C23—2121no—
532C23C22×D21211yescharacteristic
533D4—1062no—
534D4—1062no—
535D4—4244no—
536D4—2121no—
537D4—2121no—
538C2×C4—21no—
539C2×C4—21no—
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—21no—
550C2×C4—21no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—1062no—
556C2×C4—4244no—
557C2×C4—2121no—
558C2×C4—2121no—
559C2×C4—2121no—
560C2×C4—2121no—
561C2×C4—2121no—
562C2×C4—2121no—
563C2×C4—2121no—
564C2×C4—2121no—
565C2×C4—2121no—
566C2×C4—2121no—
567C2×C4—2121no—
568C2×C4—2121no—
569C2×C4—2121no—
570C2×C4—2121no—
571C2×C4—2121no—
572C2×C4—2121no—
573C2×C4—2121no—
574C2×C4—2121no—
575C2×C4—2121no—
576C2×C4—2121no—
577C2×C4#1313320112yes—
578C2×C4#1313320112yes—
579C2×C4#41×D5311yescharacteristic
580C2×C4#41×D5311yescharacteristic
581C2×C4C2×#1171994711yescharacteristic
582C2×C4C2×#1171994711yescharacteristic
583C2×C4#1313326211yescharacteristic
584C2×C4#1313326511yescharacteristic
585C8—41no—
586C8—41no—
587C8—41no—
588C8—41no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—4241no—
597C22—4241no—
598C22—4241no—
599C22—4241no—
600C22—2122no—
601C22—2122no—
602C22—2122no—
603C22C2×#1313320111yescharacteristic
604C22#1681113511yescharacteristic
605C22#1681120411yescharacteristic
606C22#1681142811yescharacteristic
607C22#1681142811yescharacteristic
608C22#1681143511yescharacteristic
609C22#1681144011yescharacteristic
610C4—22no—
611C4—22no—
612C4—41no—
613C4—41no—
614C4—41no—
615C4—4241no—
616C4—4241no—
617C4—4241no—
618C4—4241no—
619C4—4241no—
620C4—4241no—
621C4—2122no—
622C4#1681130711yescharacteristic
623C4#1681132011yescharacteristic
624C4#1681133711yescharacteristic
625C4#1681134411yescharacteristic
626C2—81no—
627C2—4242no—
628C2#2027901311yescharacteristic
629C2#2028467411yescharacteristic
630C2#2028469511yescharacteristic
631C2#2028493111yescharacteristic
632C2#2028495911yescharacteristic
633C2#2028612811yescharacteristic
634C2#2028617611yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1798C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenter#1798D53
hypocenterC53#27636
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#117 ◁ #1798

Complements

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

NH ≅ G/Ntype
2. #20277324C2semidirect
3. #20276148C2semidirect
4. #20276143C2semidirect
7. #20275485C2semidirect
8. #20275474C2semidirect
9. #20275343C2semidirect
10. #20275336C2semidirect
11. #20275308C2semidirect
12. #20275287C2semidirect
13. #20274726C2semidirect
14. #20274716C2semidirect
15. C53×#1798C2semidirect
323. #1798D53semidirect
376. C53#27636semidirect

Maximal subgroups

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

subgroupindex
#202774082
#202773242
#202761482
#202761432
#202758982
#202754852
#202754742
#202753432
#202753362
#202753082
#202752872
#202747262
#202747162
C53×#17982
#2763653