Finite Group Databasebeta

← #27483735

Subgroups — #27483735

Order 13568.

850 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20280540C211yesmaximal, characteristic, complemented
2#20277307C211yesmaximal, characteristic
3#20276136C211yesmaximal, characteristic, complemented
4#20276131C211yesmaximal, characteristic, complemented
5#20275854C211yesmaximal, characteristic, complemented
6#20275849C211yesmaximal, characteristic, complemented
7#20275414C211yesmaximal, characteristic, complemented
8#20275402C211yesmaximal, characteristic, complemented
9#20275395C211yesmaximal, characteristic, complemented
10#20275393C211yesmaximal, characteristic, complemented
11#20274803C211yesmaximal, characteristic, complemented
12#20274802C211yesmaximal, characteristic, complemented
13#20274655C211yesmaximal, characteristic
14#20274483C211yesmaximal, characteristic, complemented
15C53×#1788C211yesmaximal, characteristic
16C2×#13133216C2211yescharacteristic
17C2×#13133176C2211yescharacteristic
18C22×C4⋊Dic53C2211yescharacteristic
19C2×#13133141C2211yescharacteristic
20C2×#13133141C2211yescharacteristic
21C2×#13133138C2211yescharacteristic
22C2×#13133138C2211yescharacteristic
23C2×C22⋊#88322C2211yescharacteristic
24C2×#13133127C2211yescharacteristic
25C2×C8×Dic53C2211yescharacteristic
26#16810565—21no—
27#16810559—21no—
28#16810544—21no—
29#16810535—21no—
30#16810530C2211yescharacteristic
31#16810530C2211yescharacteristic
32#16810526C2211yescharacteristic
33#16810526C2211yescharacteristic
34#16810522C2211yescharacteristic
35#16810519C2211yescharacteristic
36#16810503—21no—
37#16810503—21no—
38#16810488C2211yescharacteristic
39#16810483—21no—
40#16810483—21no—
41#16810483C2211yescharacteristic
42#16810386C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C22×#41×C53C2211yescharacteristic
53C2×#103×C53C2211yescharacteristic
54C2×#103×C53C2211yescharacteristic
55C2×#102×C53C2211yescharacteristic
56C2×#102×C53C2211yescharacteristic
57C2×C22⋊C8×C53C2211yescharacteristic
58C53×#380C2211yescharacteristic
59C22×Dic106—21no—
60C23×Dic53—21no—
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63#13133216—41no—
64#13133216—41no—
65#13133176—41no—
66#13133176—41no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—22no—
70C2×C4⋊Dic53—22no—
71C2×C4⋊Dic53D411yescharacteristic
72C2×C4⋊Dic53D411yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×C22⋊Dic53—21no—
80C2×C22⋊Dic53—21no—
81C2×C22⋊Dic53—21no—
82C2×C22⋊Dic53—21no—
83C2×C22⋊Dic53—21no—
84C2×C22⋊Dic53D411yescharacteristic
85C2×C22⋊Dic53D411yescharacteristic
86C2×C4×Dic53C2311yescharacteristic
87C2×C4×Dic53C2311yescharacteristic
88#13133141—22no—
89#13133141—41no—
90#13133138—22no—
91#13133138—41no—
92C22⋊#88322—22no—
93#13133127—41no—
94C8×Dic53—41no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104C2×#41×C53—22no—
105C2×#41×C53—22no—
106C2×#41×C53—22no—
107C2×#41×C53—22no—
108C2×#41×C53—41no—
109C2×#41×C53—41no—
110C22×Q8×C53C2311yescharacteristic
111C22×D4×C53—21no—
112C22×D4×C53C2311yescharacteristic
113C23×C4×C53—21no—
114C23×C4×C53C2311yescharacteristic
115C22×C8×C53C2311yescharacteristic
116C22×C8×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C22⋊C4×C53—21no—
120#103×C53—22no—
121#103×C53—41no—
122#102×C53—22no—
123#102×C53—41no—
124C22⋊C8×C53—22no—
125#95×C53—21no—
126C24×C53—21no—
127C24×C53C2×D411yescharacteristic
128#41×C53—44no—
129#41×C53—44no—
130#41×C53—44no—
131#41×C53—44no—
132C2×Q8×C53—21no—
133C2×Q8×C53—21no—
134C2×Q8×C53—21no—
135C2×Q8×C53—21no—
136C2×Q8×C53—21no—
137C2×Q8×C53#4112yes—
138C2×D4×C53—24no—
139C2×D4×C53—21no—
140C2×D4×C53—21no—
141C2×D4×C53—21no—
142C2×D4×C53—21no—
143C2×D4×C53—21no—
144C2×D4×C53—42no—
145C2×D4×C53—42no—
146C2×D4×C53#4112yes—
147C22×C4×C53—21no—
148C22×C4×C53—21no—
149C22×C4×C53—21no—
150C22×C4×C53—21no—
151C22×C4×C53—24no—
152C22×C4×C53—21no—
153C22×C4×C53—21no—
154C22×C4×C53—21no—
155C22×C4×C53—42no—
156C22×C4×C53—42no—
157C22×C4×C53C2×D411yescharacteristic
158C22×C4×C53#4112yes—
159C22×C4×C53#4112yes—
160C22×C4×C53C2411yescharacteristic
161C22×Dic53—21no—
162C22×Dic53—21no—
163C22×Dic53—21no—
164C22×Dic53—21no—
165C22×Dic53—42no—
166C22×Dic53—81no—
167C22×Dic53—81no—
168C22×Dic53C2×D411yescharacteristic
169C22×Dic53C2×D411yescharacteristic
170C22×Dic53C2×D411yescharacteristic
171C22×Dic53C2×D411yescharacteristic
172C2×Dic106—42no—
173C2×Dic106—81no—
174C2×Dic106—81no—
175C2×C8×C53—21no—
176C2×C8×C53—21no—
177C2×C8×C53—21no—
178C2×C8×C53—21no—
179C2×C8×C53—21no—
180C2×C8×C53—21no—
181C4⋊C4×C53—21no—
182C4⋊C4×C53—21no—
183C4⋊C4×C53—21no—
184C4⋊C4×C53—21no—
185C22⋊C4×C53—81no—
186C22⋊Dic53—41no—
187C22⋊Dic53—41no—
188C22⋊Dic53—81no—
189C22⋊Dic53—81no—
190C22⋊Dic53—81no—
191C22⋊Dic53—81no—
192C22⋊Dic53—81no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—21no—
199C4⋊Dic53—21no—
200C4⋊Dic53—21no—
201C4⋊Dic53—41no—
202C4⋊Dic53—41no—
203C4⋊Dic53—42no—
204C4⋊Dic53—81no—
205#11719919—42no—
206#11719919—81no—
207C4×Dic53—21no—
208C4×Dic53—21no—
209C4×Dic53—21no—
210C4×Dic53—21no—
211C2×#88322—21no—
212C2×#88322—21no—
213C2×#88322—21no—
214C2×#88322—21no—
215C2×#88322—21no—
216C2×#88322—21no—
217C23×C53—21no—
218C23×C53—21no—
219C23×C53—21no—
220C23×C53—21no—
221C23×C53—21no—
222C23×C53—21no—
223C23×C53—22no—
224C23×C53—41no—
225C23×C53—41no—
226C23×C53—41no—
227C23×C53—41no—
228C23×C53—41no—
229C23×C53—41no—
230C23×C53#12412yes—
231C23×C53C22×D411yescharacteristic
232Q8×C53—22no—
233Q8×C53—22no—
234Q8×C53—41no—
235Q8×C53—41no—
236D4×C53—22no—
237D4×C53—22no—
238D4×C53—48no—
239D4×C53—41no—
240D4×C53—41no—
241C2×C4×C53—21no—
242C2×C4×C53—21no—
243C2×C4×C53—22no—
244C2×C4×C53—22no—
245C2×C4×C53—22no—
246C2×C4×C53—22no—
247C2×C4×C53—22no—
248C2×C4×C53—48no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53—41no—
253C2×C4×C53—41no—
254C2×C4×C53—41no—
255C2×C4×C53—41no—
256C2×C4×C53—41no—
257C2×C4×C53#12412yes—
258C2×C4×C53C2×#4111yescharacteristic
259C2×C4×C53C2×#4111yescharacteristic
260C2×C4×C53C2×#4111yescharacteristic
261C2×C4×C53C2×#4111yescharacteristic
262C2×C4×C53H(5,𝔽2)11yescharacteristic
263C2×C4×C53H(5,𝔽2)11yescharacteristic
264C2×Dic53—21no—
265C2×Dic53—21no—
266C2×Dic53—21no—
267C2×Dic53—21no—
268C2×Dic53—21no—
269C2×Dic53—21no—
270C2×Dic53—21no—
271C2×Dic53—21no—
272C2×Dic53—21no—
273C2×Dic53—21no—
274C2×Dic53—21no—
275C2×Dic53—21no—
276C2×Dic53—41no—
277C2×Dic53—41no—
278C2×Dic53—41no—
279C2×Dic53—41no—
280C2×Dic53—41no—
281C2×Dic53—41no—
282C2×Dic53—41no—
283C2×Dic53—41no—
284C2×Dic53—41no—
285C2×Dic53—41no—
286C2×Dic53—41no—
287C2×Dic53—41no—
288C2×Dic53—84no—
289Dic106—84no—
290C8×C53—41no—
291C8×C53—41no—
292#88322—41no—
293#88322—41no—
294#29120—531nomaximal, Sylow
295C22×C53—22no—
296C22×C53—22no—
297C22×C53—22no—
298C22×C53—41no—
299C22×C53—41no—
300C22×C53—41no—
301C22×C53—41no—
302C22×C53—42no—
303C22×C53—42no—
304C22×C53—42no—
305C22×C53—81no—
306C22×C53—81no—
307C22×C53—81no—
308C22×C53—81no—
309C22×C53#52511yescharacteristic
310C22×C53C2×#12411yescharacteristic
311C22×C53#53411yescharacteristic
312C22×C53#57611yescharacteristic
313C22×C53#57711yescharacteristic
314C22×C53#57711yescharacteristic
315C22×C53#57811yescharacteristic
316C4×C53—22no—
317C4×C53—42no—
318C4×C53—81no—
319C4×C53—81no—
320C4×C53#55111yescharacteristic
321C4×C53#55111yescharacteristic
322C4×C53#55111yescharacteristic
323C4×C53#55111yescharacteristic
324Dic53—41no—
325Dic53—41no—
326Dic53—41no—
327Dic53—41no—
328Dic53—81no—
329Dic53—81no—
330Dic53—81no—
331Dic53—81no—
332#2628—531no—
333#2311—531no—
334#2074—531no—
335#2073—531no—
336#2024—531no—
337#2023—531no—
338#1936—531no—
339#1935—531no—
340#1934—531no—
341#1933—531no—
342#1824—531no—
343#1823—531no—
344#1788—531no—
345#1788D5311yescharacteristic
346#1750—531no—
347C2×C53—42no—
348C2×C53—82no—
349C2×C53#256111yescharacteristic
350C2×C53#312411yescharacteristic
351C2×C53#312511yescharacteristic
352C2×C53#313611yescharacteristic
353C2×C53#313711yescharacteristic
354C2×C53#319811yescharacteristic
355C2×C53#320111yescharacteristic
356C22×#41C2×D5311yescharacteristic
357C2×C4⋊Q8—531no—
358C2×#117—531no—
359C22×C4⋊C4—531no—
360C2×#103—531no—
361C2×#103—531no—
362C2×#103C2×D5311yescharacteristic
363C2×#103C2×D5311yescharacteristic
364C2×#102—531no—
365C2×#102—531no—
366C2×#102C2×D5311yescharacteristic
367C2×#102C2×D5311yescharacteristic
368C2×C22⋊C8—531no—
369C2×C22⋊C8C2×D5311yescharacteristic
370C2×#97—531no—
371C2×C4×C8—531no—
372#400—1061no—
373#399—1061no—
374#395—1061no—
375#393—1061no—
376#391—531no—
377#391—531no—
378#390—531no—
379#390—531no—
380#389—531no—
381#389—531no—
382#386—1061no—
383#386—1061no—
384#382—531no—
385#380—531no—
386#380—1061no—
387#380—1061no—
388#380C2×D5311yescharacteristic
389#340—531no—
390#340—531no—
391#340—531no—
392#340—531no—
393#340—531no—
394#340—531no—
395#340—531no—
396#340—531no—
397#336—531no—
398#336—531no—
399C53#2912011yescharacteristic, complemented, Sylow
400C2×#41—22no—
401C2×#41—22no—
402C2×#41—22no—
403C2×#41—22no—
404C2×#41—41no—
405C2×#41—41no—
406C22×Q8—1061no—
407C22×Q8C22×D5311yescharacteristic
408C22×D4—21no—
409C22×D4C22×D5311yescharacteristic
410C23×C4—21no—
411C23×C4—1061no—
412C23×C4C22×D5311yescharacteristic
413C22×C8—531no—
414C22×C8—531no—
415C22×C8C22×D5311yescharacteristic
416C22×C8C22×D5311yescharacteristic
417C4⋊Q8—2121no—
418C4⋊Q8—2121no—
419#117—2121no—
420#117—2121no—
421C2×C4⋊C4—531no—
422C2×C4⋊C4—531no—
423C2×C4⋊C4—531no—
424C2×C4⋊C4—531no—
425C2×C4⋊C4—531no—
426C2×C4⋊C4—531no—
427C2×C4⋊C4—1062no—
428C2×C4⋊C4—1062no—
429C2×C4⋊C4—1061no—
430C2×C4⋊C4—1061no—
431C2×C4⋊C4—1061no—
432C2×C4⋊C4—1061no—
433C2×C4⋊C4C22×D5311yescharacteristic
434C2×C4⋊C4C22×D5311yescharacteristic
435C2×C22⋊C4—21no—
436C2×C22⋊C4—531no—
437C2×C22⋊C4—531no—
438C2×C22⋊C4—1061no—
439C2×C22⋊C4—1061no—
440C2×C22⋊C4—1061no—
441C2×C22⋊C4—1061no—
442C2×C22⋊C4—1061no—
443C2×C42—531no—
444C2×C42—531no—
445#103—22no—
446#103—41no—
447#103—1062no—
448#103—2121no—
449#102—22no—
450#102—41no—
451#102—1062no—
452#102—2121no—
453C22⋊C8—22no—
454C22⋊C8—1062no—
455#97—2121no—
456C4×C8—2121no—
457#95—21no—
458#95—1061no—
459#95—1061no—
460#95—1061no—
461#95—1061no—
462#95—1061no—
463#95—1061no—
464#95—1061no—
465#95—1061no—
466#95—1061no—
467C24—21no—
468C24D4×D5311yescharacteristic
469#41—44no—
470#41—44no—
471#41—44no—
472#41—44no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—4241no—
479C2×Q8—4241no—
480C2×Q8—2122no—
481C2×Q8#1171994712yes—
482C2×D4—24no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4—42no—
489C2×D4—42no—
490C2×D4#1171994712yes—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—24no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—42no—
500C22×C4—42no—
501C22×C4—531no—
502C22×C4—531no—
503C22×C4—531no—
504C22×C4—531no—
505C22×C4—1061no—
506C22×C4—1061no—
507C22×C4—1061no—
508C22×C4—1061no—
509C22×C4—4241no—
510C22×C4—4241no—
511C22×C4—2122no—
512C22×C4#1171994512yes—
513C22×C4D4×D5311yescharacteristic
514C22×C4#1171994712yes—
515C22×C4C23×D5311yescharacteristic
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—1061no—
523C2×C8—1061no—
524C2×C8—1061no—
525C2×C8—1061no—
526C2×C8—1061no—
527C2×C8—1061no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—1061no—
533C4⋊C4—1061no—
534C4⋊C4—1061no—
535C4⋊C4—1061no—
536C4⋊C4—1061no—
537C4⋊C4—1061no—
538C4⋊C4—1061no—
539C4⋊C4—1061no—
540C4⋊C4—4241no—
541C4⋊C4—4241no—
542C4⋊C4—2121no—
543C4⋊C4—2121no—
544C4⋊C4—2122no—
545C4⋊C4—2122no—
546C22⋊C4—81no—
547C22⋊C4—4241no—
548C22⋊C4—4241no—
549C22⋊C4—4241no—
550C22⋊C4—4241no—
551C22⋊C4—4241no—
552C22⋊C4—2121no—
553C22⋊C4—2121no—
554C42—1061no—
555C42—1061no—
556C42—1061no—
557C42—1061no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—22no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23#1313320312yes—
572C23C2×D4×D5311yescharacteristic
573Q8—22no—
574Q8—22no—
575Q8—41no—
576Q8—41no—
577Q8—4244no—
578D4—22no—
579D4—22no—
580D4—48no—
581D4—41no—
582D4—41no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—48no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—1061no—
600C2×C4—1061no—
601C2×C4—1061no—
602C2×C4—1061no—
603C2×C4—1061no—
604C2×C4—1061no—
605C2×C4—1061no—
606C2×C4—1061no—
607C2×C4—1061no—
608C2×C4—1061no—
609C2×C4—1061no—
610C2×C4—1061no—
611C2×C4—4244no—
612C2×C4—2121no—
613C2×C4—2121no—
614C2×C4—2121no—
615C2×C4—2121no—
616C2×C4—2121no—
617C2×C4—2121no—
618C2×C4—2121no—
619C2×C4—2121no—
620C2×C4—2121no—
621C2×C4—2121no—
622C2×C4—2121no—
623C2×C4—2121no—
624C2×C4#1313320312yes—
625C2×C4C2×#1171994711yescharacteristic
626C2×C4C2×#1171994711yescharacteristic
627C2×C4C2×#1171994711yescharacteristic
628C2×C4C2×#1171994511yescharacteristic
629C2×C4#1313326211yescharacteristic
630C2×C4#1313326211yescharacteristic
631C8—41no—
632C8—41no—
633C8—2121no—
634C8—2121no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22—81no—
649C22#1681112911yescharacteristic
650C22C2×#1313320311yescharacteristic
651C22#1681118311yescharacteristic
652C22#1681142511yescharacteristic
653C22#1681143011yescharacteristic
654C22#1681143311yescharacteristic
655C22#1681143911yescharacteristic
656C4—22no—
657C4—42no—
658C4—81no—
659C4—81no—
660C4—4241no—
661C4—4241no—
662C4—4241no—
663C4—4241no—
664C4—2121no—
665C4—2121no—
666C4—2121no—
667C4—2121no—
668C4#1681129911yescharacteristic
669C4#1681129911yescharacteristic
670C4#1681130411yescharacteristic
671C4#1681130411yescharacteristic
672C2—42no—
673C2—82no—
674C2#2027987011yescharacteristic
675C2#2028497811yescharacteristic
676C2#2028499311yescharacteristic
677C2#2028507311yescharacteristic
678C2#2028508311yescharacteristic
679C2#2028556411yescharacteristic
680C2#2028559411yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1788C2
socleC23×C53C22×D4
prime radicalC22×#41×C53C22
hypercenter#1788D53
hypocenterC53#29120
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1788

Complements

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

NH ≅ G/Ntype
1. #20280540C2semidirect
3. #20276136C2semidirect
4. #20276131C2semidirect
5. #20275854C2semidirect
6. #20275849C2semidirect
7. #20275414C2semidirect
8. #20275402C2semidirect
9. #20275395C2semidirect
10. #20275393C2semidirect
11. #20274803C2semidirect
12. #20274802C2semidirect
14. #20274483C2semidirect
399. C53#29120semidirect

Maximal subgroups

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

subgroupindex
#202805402
#202773072
#202761362
#202761312
#202758542
#202758492
#202754142
#202754022
#202753952
#202753932
#202748032
#202748022
#202746552
#202744832
C53×#17882
#2912053