Finite Group Databasebeta

← #27432611

Subgroups — #27432611

Order 13568.

858 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
1C2×#16810961C211yesmaximal, characteristic, complemented
2#20279178C211yesmaximal, characteristic
3#20277724C211yesmaximal, characteristic, complemented
4#20276199C211yesmaximal, characteristic, complemented
5#20276126C211yesmaximal, characteristic, complemented
6#20276099C211yesmaximal, characteristic, complemented
7#20276028C211yesmaximal, characteristic, complemented
8#20275470C211yesmaximal, characteristic, complemented
9#20275345C211yesmaximal, characteristic, complemented
10#20275072C211yesmaximal, characteristic, complemented
11#20275033C211yesmaximal, characteristic
12#20274898C211yesmaximal, characteristic, complemented
13#20274728C211yesmaximal, characteristic, complemented
14#20274435C211yesmaximal, characteristic, complemented
15C2×C53×#468C211yesmaximal, characteristic
16C2×Q8×Dic53C2211yescharacteristic
17C2×#13133176C2211yescharacteristic
18C22×C4⋊Dic53C2211yescharacteristic
19#16810961—24no—
20C2×#13133150C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×#13133146C2211yescharacteristic
23C2×#13133146C2211yescharacteristic
24C2×#13133141C2211yescharacteristic
25C2×#13133141C2211yescharacteristic
26C2×#13133138C2211yescharacteristic
27C2×#13133138C2211yescharacteristic
28C2×C22⋊#88322C2211yescharacteristic
29#16810537—21no—
30#16810537—21no—
31#16810530C2211yescharacteristic
32#16810526C2211yescharacteristic
33#16810526C2211yescharacteristic
34#16810522—21no—
35#16810522C2211yescharacteristic
36#16810511—21no—
37#16810511—21no—
38#16810509—21no—
39#16810498—21no—
40#16810492C2211yescharacteristic
41#16810488C2211yescharacteristic
42C4⋊C4×Dic53C2211yescharacteristic
43C22⋊C4×Dic53—21no—
44C4×C22⋊Dic53C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810377C2211yescharacteristic
52#16810377C2211yescharacteristic
53C22×SD16×C53C2211yescharacteristic
54C22×M16×C53C2211yescharacteristic
55C2×C22⋊Q8×C53C2211yescharacteristic
56C2×#121×C53C2211yescharacteristic
57C53×#468—24no—
58C2×#107×C53C2211yescharacteristic
59C2×#103×C53C2211yescharacteristic
60C2×#102×C53C2211yescharacteristic
61C23×Dic53—21no—
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64Q8×Dic53—22no—
65Q8×Dic53—41no—
66#13133176—22no—
67#13133176—22no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—22no—
72C2×C4⋊Dic53—41no—
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×Dic53—21no—
87C2×C4×Dic53—21no—
88C2×C4×Dic53—21no—
89C2×C4×Dic53C2311yescharacteristic
90C2×C4×Dic53C2311yescharacteristic
91#13133150—22no—
92#13133148—22no—
93#13133146—22no—
94#13133146—41no—
95#13133141—22no—
96#13133141—22no—
97#13133138—22no—
98#13133138—41no—
99C22⋊#88322—22no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124D411yescharacteristic
107#13133124D411yescharacteristic
108C22×Q8×C53C2311yescharacteristic
109C22×D4×C53—21no—
110C22×D4×C53C2311yescharacteristic
111C23×C4×C53C2311yescharacteristic
112C2×SD16×C53—22no—
113C2×SD16×C53—22no—
114C2×SD16×C53—41no—
115C2×M16×C53—22no—
116C2×M16×C53—22no—
117C2×M16×C53—41no—
118C22×C8×C53C2311yescharacteristic
119C22×C8×C53C2311yescharacteristic
120C22⋊Q8×C53—22no—
121C22⋊Q8×C53D414yes—
122#121×C53—22no—
123#121×C53—22no—
124C2×C4⋊C4×C53—21no—
125C2×C4⋊C4×C53C2311yescharacteristic
126C2×C4⋊C4×C53C2311yescharacteristic
127C2×C22⋊C4×C53—21no—
128C2×C22⋊C4×C53—21no—
129#107×C53—22no—
130#103×C53—22no—
131#102×C53—22no—
132C24×C53—21no—
133C24×C53C2×D411yescharacteristic
134C2×Q8×C53—21no—
135C2×Q8×C53—21no—
136C2×Q8×C53—21no—
137C2×Q8×C53—21no—
138C2×Q8×C53—21no—
139C2×Q8×C53C2×D412yes—
140C2×D4×C53—21no—
141C2×D4×C53—21no—
142C2×D4×C53—21no—
143C2×D4×C53—21no—
144C2×D4×C53—21no—
145C2×D4×C53—21no—
146C2×D4×C53—42no—
147C2×D4×C53—81no—
148C2×D4×C53—81no—
149C22×C4×C53—21no—
150C22×C4×C53—21no—
151C22×C4×C53—21no—
152C22×C4×C53—21no—
153C22×C4×C53—21no—
154C22×C4×C53—21no—
155C22×C4×C53—21no—
156C22×C4×C53C2×D411yescharacteristic
157C22×C4×C53C2×D412yes—
158C22×C4×C53#4112yes—
159C22×C4×C53C2411yescharacteristic
160C22×Dic53—21no—
161C22×Dic53—21no—
162C22×Dic53—21no—
163C22×Dic53—21no—
164C22×Dic53—42no—
165C22×Dic53—42no—
166C22×Dic53—81no—
167C22×Dic53C2×D411yescharacteristic
168C22×Dic53C2×D411yescharacteristic
169C22×Dic53C2×D411yescharacteristic
170C22×Dic53C2×D411yescharacteristic
171SD16×C53—44no—
172M16×C53—44no—
173C2×C8×C53—21no—
174C2×C8×C53—21no—
175C2×C8×C53—21no—
176C2×C8×C53—21no—
177C2×C8×C53—21no—
178C2×C8×C53#4112yes—
179C4⋊C4×C53—24no—
180C4⋊C4×C53—21no—
181C4⋊C4×C53—21no—
182C4⋊C4×C53—21no—
183C4⋊C4×C53C2×D412yes—
184C22⋊C4×C53—24no—
185C22⋊C4×C53—42no—
186C22⋊Dic53—22no—
187C22⋊Dic53—41no—
188C22⋊Dic53—42no—
189C22⋊Dic53—42no—
190C22⋊Dic53—42no—
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—42no—
201C4⋊Dic53—42no—
202C4⋊Dic53—81no—
203C4⋊Dic53#4112yes—
204#11719919—81no—
205#11719919—81no—
206C4×Dic53—21no—
207C4×Dic53—21no—
208C4×Dic53—21no—
209C4×Dic53—21no—
210C4×Dic53—42no—
211C4×Dic53—42no—
212C4×Dic53—81no—
213C2×#88322—21no—
214C2×#88322—21no—
215C2×#88322—21no—
216C2×#88322—21no—
217C2×#88322—21no—
218C2×#88322—21no—
219C23×C53—21no—
220C23×C53—21no—
221C23×C53—21no—
222C23×C53—21no—
223C23×C53—21no—
224C23×C53—21no—
225C23×C53—41no—
226C23×C53—41no—
227C23×C53—41no—
228C23×C53—41no—
229C23×C53—41no—
230C23×C53—41no—
231C23×C53—41no—
232C23×C53#12112yes—
233C23×C53C22×D411yescharacteristic
234Q8×C53—22no—
235Q8×C53—22no—
236Q8×C53—41no—
237Q8×C53—41no—
238D4×C53—41no—
239D4×C53—41no—
240D4×C53—41no—
241D4×C53—41no—
242D4×C53—84no—
243C2×C4×C53—21no—
244C2×C4×C53—21no—
245C2×C4×C53—22no—
246C2×C4×C53—22no—
247C2×C4×C53—22no—
248C2×C4×C53—22no—
249C2×C4×C53—22no—
250C2×C4×C53—22no—
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—41no—
258C2×C4×C53#12112yes—
259C2×C4×C53C22×D411yescharacteristic
260C2×C4×C53C2×#4111yescharacteristic
261C2×C4×C53C2×#4111yescharacteristic
262C2×C4×C53C2×#4111yescharacteristic
263C2×C4×C53H(5,𝔽2)11yescharacteristic
264C2×C4×C532−1+411yescharacteristic
265C2×Dic53—21no—
266C2×Dic53—21no—
267C2×Dic53—22no—
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—21no—
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—
289C2×Dic53#13712yes—
290C8×C53—22no—
291C8×C53—41no—
292#88322—41no—
293#88322—41no—
294#25224—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—81no—
303C22×C53—81no—
304C22×C53—81no—
305C22×C53—81no—
306C22×C53—81no—
307C22×C53—81no—
308C22×C53—81no—
309C22×C53C2×#12111yescharacteristic
310C22×C53#54611yescharacteristic
311C22×C53#54711yescharacteristic
312C22×C53C2×#13711yescharacteristic
313C22×C53#57511yescharacteristic
314C22×C53#57611yescharacteristic
315C22×C53#57711yescharacteristic
316C4×C53—22no—
317C4×C53—42no—
318C4×C53—42no—
319C4×C53—81no—
320C4×C53#52911yescharacteristic
321C4×C53#53911yescharacteristic
322C4×C53#54611yescharacteristic
323C4×C53#54711yescharacteristic
324Dic53—22no—
325Dic53—41no—
326Dic53—41no—
327Dic53—41no—
328Dic53—42no—
329Dic53—81no—
330Dic53—81no—
331Dic53—81no—
332C2×#485—531no—
333C2×#468D5311yescharacteristic
334#2500—531no—
335#2360—531no—
336#2085—531no—
337#2072—531no—
338#2067—531no—
339#2055—531no—
340#1947—531no—
341#1926—531no—
342#1880—531no—
343#1874—531no—
344#1851—531no—
345#1805—531no—
346#1736—531no—
347C2×C53—42no—
348C2×C53—161no—
349C2×C53#242511yescharacteristic
350C2×C53#303711yescharacteristic
351C2×C53#304511yescharacteristic
352C2×C53#331911yescharacteristic
353C2×C53#332911yescharacteristic
354C2×C53#334611yescharacteristic
355C2×C53#335111yescharacteristic
356C22×SD16C2×D5311yescharacteristic
357C22×M16C2×D5311yescharacteristic
358C2×C22⋊Q8C2×D5311yescharacteristic
359C2×#121C2×D5311yescharacteristic
360C2×C4×Q8—531no—
361C2×#117—531no—
362C22×C4⋊C4—531no—
363#485—1064no—
364#468—24no—
365C2×#107—531no—
366C2×#107C2×D5311yescharacteristic
367C2×#106—531no—
368C2×C4⋊C8—531no—
369C2×C4⋊C8—531no—
370C2×#103—531no—
371C2×#103—531no—
372C2×#103C2×D5311yescharacteristic
373C2×#102—531no—
374C2×#102—531no—
375C2×#102C2×D5311yescharacteristic
376C2×C22⋊C8—531no—
377#394—1061no—
378#394—1061no—
379#391—531no—
380#390—531no—
381#390—531no—
382#389—531no—
383#389—1061no—
384#388—1061no—
385#388—1061no—
386#387—1061no—
387#385—1061no—
388#384—531no—
389#382—531no—
390C4×C4⋊C4—531no—
391C4×C22⋊C4—531no—
392C4×C22⋊C4—1061no—
393#340—531no—
394#340—531no—
395#340—531no—
396#340—531no—
397#340—531no—
398#340—531no—
399#336—531no—
400#336—531no—
401C53#2522411yescharacteristic, complemented, Sylow
402C22×Q8C22×D5311yescharacteristic
403C22×D4—21no—
404C22×D4C22×D5311yescharacteristic
405C23×C4—1061no—
406C23×C4C22×D5311yescharacteristic
407C2×SD16—22no—
408C2×SD16—22no—
409C2×SD16—41no—
410C2×M16—22no—
411C2×M16—22no—
412C2×M16—41no—
413C22×C8—531no—
414C22×C8—531no—
415C22×C8C22×D5311yescharacteristic
416C22×C8C22×D5311yescharacteristic
417C22⋊Q8—22no—
418C22⋊Q8C22⋊D5314yes—
419#121—22no—
420#121—22no—
421C4×Q8—1062no—
422C4×Q8—2121no—
423#117—1062no—
424#117—1062no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—531no—
427C2×C4⋊C4—531no—
428C2×C4⋊C4—531no—
429C2×C4⋊C4—531no—
430C2×C4⋊C4—1062no—
431C2×C4⋊C4—1061no—
432C2×C4⋊C4—1061no—
433C2×C4⋊C4—1061no—
434C2×C4⋊C4—1061no—
435C2×C4⋊C4—1061no—
436C2×C4⋊C4—2121no—
437C2×C4⋊C4C22×D5311yescharacteristic
438C2×C4⋊C4C22×D5311yescharacteristic
439C2×C22⋊C4—21no—
440C2×C22⋊C4—21no—
441C2×C22⋊C4—531no—
442C2×C22⋊C4—531no—
443C2×C22⋊C4—1061no—
444C2×C22⋊C4—1061no—
445C2×C22⋊C4—1061no—
446C2×C22⋊C4—1061no—
447C2×C22⋊C4—1061no—
448C2×C42—531no—
449C2×C42—531no—
450C2×C42—1061no—
451C2×C42—1061no—
452C2×C42—1061no—
453#107—22no—
454#107—1062no—
455#106—1062no—
456C4⋊C8—1062no—
457C4⋊C8—2121no—
458#103—22no—
459#103—1062no—
460#103—1062no—
461#102—22no—
462#102—1062no—
463#102—2121no—
464C22⋊C8—1062no—
465#95—531no—
466#95—531no—
467#95—1061no—
468#95—1061no—
469#95—1061no—
470#95—1061no—
471#95—1061no—
472#95—1061no—
473C24—21no—
474C24D4×D5311yescharacteristic
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8C2×C22⋊D5312yes—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—42no—
488C2×D4—81no—
489C2×D4—81no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—531no—
498C22×C4—531no—
499C22×C4—531no—
500C22×C4—531no—
501C22×C4—1061no—
502C22×C4—1061no—
503C22×C4—1061no—
504C22×C4—1061no—
505C22×C4—4241no—
506C22×C4—2122no—
507C22×C4—2122no—
508C22×C4D4×D5311yescharacteristic
509C22×C4#1171994712yes—
510C22×C4C2×C22⋊D5312yes—
511C22×C4C23×D5311yescharacteristic
512SD16—44no—
513M16—44no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—1061no—
520C2×C8—1061no—
521C2×C8—1061no—
522C2×C8—1061no—
523C2×C8—1061no—
524C2×C8—1061no—
525C2×C8#1171994712yes—
526C4⋊C4—24no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—532no—
531C4⋊C4—1061no—
532C4⋊C4—1061no—
533C4⋊C4—1061no—
534C4⋊C4—1061no—
535C4⋊C4—1061no—
536C4⋊C4—1061no—
537C4⋊C4—1061no—
538C4⋊C4—4241no—
539C4⋊C4—4241no—
540C4⋊C4—4241no—
541C4⋊C4—2122no—
542C4⋊C4—2122no—
543C4⋊C4C2×C22⋊D5312yes—
544C22⋊C4—24no—
545C22⋊C4—42no—
546C22⋊C4—1062no—
547C22⋊C4—4241no—
548C22⋊C4—4241no—
549C22⋊C4—2122no—
550C22⋊C4—2121no—
551C22⋊C4—2122no—
552C22⋊C4—2122no—
553C42—1061no—
554C42—1061no—
555C42—1061no—
556C42—1061no—
557C42—4241no—
558C42—2122no—
559C42—2122no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23#1313318912yes—
574C23C2×D4×D5311yescharacteristic
575Q8—22no—
576Q8—22no—
577Q8—41no—
578Q8—41no—
579D4—41no—
580D4—41no—
581D4—41no—
582D4—41no—
583D4—84no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—532no—
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—1062no—
612C2×C4—4244no—
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#1313318912yes—
625C2×C4C22×C22⋊D5311yescharacteristic
626C2×C4#41×D5311yescharacteristic
627C2×C4C2×#1171994711yescharacteristic
628C2×C4C2×#1171994711yescharacteristic
629C2×C4#1313326211yescharacteristic
630C2×C4#1313326411yescharacteristic
631C8—22no—
632C8—41no—
633C8—2121no—
634C8—2121no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22—81no—
649C22C2×#1313318911yescharacteristic
650C22#1681126211yescharacteristic
651C22#1681127311yescharacteristic
652C22#137×D5311yescharacteristic
653C22#1681142211yescharacteristic
654C22#1681142711yescharacteristic
655C22#1681143311yescharacteristic
656C4—22no—
657C4—42no—
658C4—42no—
659C4—81no—
660C4—1062no—
661C4—4241no—
662C4—4241no—
663C4—4241no—
664C4—2121no—
665C4—2121no—
666C4—2121no—
667C4—2122no—
668C4#1681115411yescharacteristic
669C4#1681121811yescharacteristic
670C4#1681126711yescharacteristic
671C4#1681127211yescharacteristic
672C2—42no—
673C2—161no—
674C2#2027827211yescharacteristic
675C2#2028418411yescharacteristic
676C2#2028425811yescharacteristic
677C2#2028686611yescharacteristic
678C2#2028700411yescharacteristic
679C2#2028719511yescharacteristic
680C2#2028728111yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#468C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
hypercenterC2×#468D53
hypocenterC53#25224
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#468

Complements

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

NH ≅ G/Ntype
1. C2×#16810961C2semidirect
3. #20277724C2semidirect
4. #20276199C2semidirect
5. #20276126C2semidirect
6. #20276099C2semidirect
7. #20276028C2semidirect
8. #20275470C2semidirect
9. #20275345C2semidirect
10. #20275072C2semidirect
12. #20274898C2semidirect
13. #20274728C2semidirect
14. #20274435C2semidirect
401. C53#25224semidirect

Maximal subgroups

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

subgroupindex
C2×#168109612
#202791782
#202777242
#202761992
#202761262
#202760992
#202760282
#202754702
#202753452
#202750722
#202750332
#202748982
#202747282
#202744352
C2×C53×#4682
#2522453