Finite Group Databasebeta

← #27452148

Subgroups — #27452148

Order 13568.

848 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279091C211yesmaximal, characteristic, complemented
2#20277735C211yesmaximal, characteristic
3#20276028C211yesmaximal, characteristic, complemented
4#20275901C211yesmaximal, characteristic
5#20275849C211yesmaximal, characteristic, complemented
6#20275661C211yesmaximal, characteristic, complemented
7#20275393C211yesmaximal, characteristic, complemented
8#20275352C211yesmaximal, characteristic, complemented
9#20275318C211yesmaximal, characteristic, complemented
10#20275296C211yesmaximal, characteristic, complemented
11#20275051C211yesmaximal, characteristic, complemented
12#20275023C211yesmaximal, characteristic, complemented
13#20274802C211yesmaximal, characteristic, complemented
14#102×Dic53C211yesmaximal, characteristic, complemented
15C53×#2000C211yesmaximal, characteristic
16C2×#13133216C2211yescharacteristic
17C2×D4×Dic53—21no—
18C2×D4×Dic53C2211yescharacteristic
19C22×C4⋊Dic53C2211yescharacteristic
20C2×#13133148C2211yescharacteristic
21C2×#13133146C2211yescharacteristic
22C2×#13133141C2211yescharacteristic
23C2×#13133141C2211yescharacteristic
24C2×#13133138C2211yescharacteristic
25C2×#13133138C2211yescharacteristic
26C2×C22⋊#88322C2211yescharacteristic
27C2×C8×Dic53C2211yescharacteristic
28#16810561—21no—
29#16810535—21no—
30#16810530C2211yescharacteristic
31#16810530C2211yescharacteristic
32#16810528—21no—
33#16810526C2211yescharacteristic
34#16810522C2211yescharacteristic
35#16810517—21no—
36#16810516—21no—
37#16810503—21no—
38#16810498—21no—
39#16810483C2211yescharacteristic
40C4⋊C4×Dic53C2211yescharacteristic
41C4⋊C4×Dic53C2211yescharacteristic
42C4×C22⋊Dic53C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C22×SD16×C53C2211yescharacteristic
53C2×C22⋊Q8×C53C2211yescharacteristic
54C2×#121×C53C2211yescharacteristic
55C2×#103×C53C2211yescharacteristic
56C2×#102×C53C2211yescharacteristic
57C2×C22⋊C8×C53C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C22×Dic106—21no—
60C23×Dic53—21no—
61C23×Dic53—21no—
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64#13133216—22no—
65#13133216—41no—
66D4×Dic53—22no—
67D4×Dic53—41no—
68D4×Dic53—44no—
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×C22⋊Dic53—21no—
79C2×C22⋊Dic53—21no—
80C2×C22⋊Dic53—21no—
81C2×C22⋊Dic53—21no—
82C2×C22⋊Dic53D411yescharacteristic
83C2×C22⋊Dic53D411yescharacteristic
84C2×C4×Dic53—21no—
85C2×C4×Dic53—21no—
86C2×C4×Dic53C2311yescharacteristic
87C2×C4×Dic53C2311yescharacteristic
88#13133148—22no—
89#13133146—41no—
90#13133141—22no—
91#13133141—41no—
92#13133138—22no—
93#13133138—41no—
94C22⋊#88322—41no—
95C8×Dic53—41no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124D411yescharacteristic
105#13133124D411yescharacteristic
106C22×Q8×C53C2311yescharacteristic
107C22×D4×C53—21no—
108C22×D4×C53C2311yescharacteristic
109C23×C4×C53C2311yescharacteristic
110C2×SD16×C53—22no—
111C2×SD16×C53—41no—
112C2×SD16×C53—41no—
113C22×C8×C53C2311yescharacteristic
114C22×C8×C53C2311yescharacteristic
115C22⋊Q8×C53—22no—
116C22⋊Q8×C53—41no—
117#121×C53—22no—
118#121×C53—22no—
119C2×C4⋊C4×C53—21no—
120C2×C4⋊C4×C53C2311yescharacteristic
121C2×C4⋊C4×C53C2311yescharacteristic
122C2×C22⋊C4×C53—21no—
123C2×C22⋊C4×C53—21no—
124#103×C53—22no—
125#102×C53—22no—
126C22⋊C8×C53—41no—
127C24×C53—21no—
128C24×C53C2×D411yescharacteristic
129C2×Q8×C53—21no—
130C2×Q8×C53—21no—
131C2×Q8×C53—21no—
132C2×Q8×C53—21no—
133C2×Q8×C53—21no—
134C2×Q8×C53—21no—
135C2×D4×C53—24no—
136C2×D4×C53—21no—
137C2×D4×C53—21no—
138C2×D4×C53—21no—
139C2×D4×C53—21no—
140C2×D4×C53—21no—
141C2×D4×C53—42no—
142C2×D4×C53—42no—
143C2×D4×C53#4112yes—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53—21no—
148C22×C4×C53—21no—
149C22×C4×C53—21no—
150C22×C4×C53—21no—
151C22×C4×C53—21no—
152C22×C4×C53—21no—
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2411yescharacteristic
155C22×Dic53—21no—
156C22×Dic53—21no—
157C22×Dic53—21no—
158C22×Dic53—21no—
159C22×Dic53—42no—
160C22×Dic53—42no—
161C22×Dic53—42no—
162C22×Dic53—42no—
163C22×Dic53—81no—
164C22×Dic53—81no—
165C22×Dic53C2×D411yescharacteristic
166C22×Dic53C2×D411yescharacteristic
167C22×Dic53C2×D411yescharacteristic
168C22×Dic53C2×D411yescharacteristic
169C2×Dic106—42no—
170C2×Dic106—42no—
171C2×Dic106—81no—
172SD16×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—21no—
179C4⋊C4×C53—21no—
180C4⋊C4×C53—21no—
181C4⋊C4×C53—42no—
182C4⋊C4×C53#4112yes—
183C4⋊C4×C53#4112yes—
184C22⋊C4×C53—24no—
185C22⋊C4×C53—42no—
186C22⋊Dic53—41no—
187C22⋊Dic53—42no—
188C22⋊Dic53—42no—
189C22⋊Dic53—81no—
190C22⋊Dic53—81no—
191C22⋊Dic53SD1614yescomplemented
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—21no—
199C4⋊Dic53—42no—
200C4⋊Dic53—42no—
201C4⋊Dic53#4112yes—
202#11719919—42no—
203C4×Dic53—21no—
204C4×Dic53—21no—
205C4×Dic53—21no—
206C4×Dic53—21no—
207C4×Dic53—42no—
208C4×Dic53—81no—
209C2×#88322—21no—
210C2×#88322—21no—
211C2×#88322—21no—
212C2×#88322—21no—
213C2×#88322—21no—
214C2×#88322—21no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53—21no—
219C23×C53—21no—
220C23×C53—21no—
221C23×C53—22no—
222C23×C53—41no—
223C23×C53—41no—
224C23×C53—41no—
225C23×C53—41no—
226C23×C53—41no—
227C23×C53—41no—
228C23×C53C2×SD1612yes—
229C23×C53C22×D411yescharacteristic
230Q8×C53—41no—
231Q8×C53—41no—
232Q8×C53—41no—
233Q8×C53—41no—
234D4×C53—22no—
235D4×C53—22no—
236D4×C53—48no—
237D4×C53—41no—
238D4×C53—41no—
239C2×C4×C53—21no—
240C2×C4×C53—22no—
241C2×C4×C53—22no—
242C2×C4×C53—21no—
243C2×C4×C53—21no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—41no—
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×C53C2×#4111yescharacteristic
256C2×C4×C53C2×#4111yescharacteristic
257C2×C4×C53C2×#4111yescharacteristic
258C2×C4×C53C2×#4111yescharacteristic
259C2×C4×C53H(5,𝔽2)11yescharacteristic
260C2×C4×C53H(5,𝔽2)11yescharacteristic
261C2×Dic53—21no—
262C2×Dic53—21no—
263C2×Dic53—21no—
264C2×Dic53—21no—
265C2×Dic53—21no—
266C2×Dic53—22no—
267C2×Dic53—21no—
268C2×Dic53—21no—
269C2×Dic53—21no—
270C2×Dic53—21no—
271C2×Dic53—21no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275C2×Dic53—41no—
276C2×Dic53—41no—
277C2×Dic53—41no—
278C2×Dic53—41no—
279C2×Dic53—41no—
280C2×Dic53—41no—
281C2×Dic53—41no—
282C2×Dic53—41no—
283C2×Dic53—84no—
284C2×Dic53—84no—
285C2×Dic53C2×SD1612yes—
286C2×Dic53C2×SD1612yes—
287Dic106—84no—
288C8×C53—41no—
289C8×C53—41no—
290#88322—41no—
291#88322—41no—
292#26722—531nomaximal, Sylow
293C22×C53—22no—
294C22×C53—22no—
295C22×C53—22no—
296C22×C53—41no—
297C22×C53—41no—
298C22×C53—41no—
299C22×C53—41no—
300C22×C53—42no—
301C22×C53—42no—
302C22×C53—42no—
303C22×C53—81no—
304C22×C53—81no—
305C22×C53—81no—
306C22×C53—81no—
307C22×C53#52511yescharacteristic
308C22×C53#54611yescharacteristic
309C22×C53#54611yescharacteristic
310C22×C53C22×SD1611yescharacteristic
311C22×C53#57511yescharacteristic
312C22×C53#57611yescharacteristic
313C22×C53#57811yescharacteristic
314C4×C53—42no—
315C4×C53—41no—
316C4×C53—42no—
317C4×C53—81no—
318C4×C53#52911yescharacteristic
319C4×C53#53811yescharacteristic
320C4×C53#55511yescharacteristic
321C4×C53#55911yescharacteristic
322Dic53—22no—
323Dic53—22no—
324Dic53—42no—
325Dic53—41no—
326Dic53—41no—
327Dic53—81no—
328Dic53—81no—
329Dic53—81no—
330#2492—531no—
331#2361—531no—
332#2055—531no—
333#2034—531no—
334#2023—531no—
335#2000D5311yescharacteristic
336#1987—531no—
337#1933—531no—
338#1927—531no—
339#1923—531no—
340#1919—531no—
341#1877—531no—
342#1872—531no—
343#1823—531no—
344C4×#102—531no—
345C2×C53—42no—
346C2×C53—82no—
347C2×C53#245911yescharacteristic
348C2×C53#310111yescharacteristic
349C2×C53#312411yescharacteristic
350C2×C53#329511yescharacteristic
351C2×C53#329811yescharacteristic
352C2×C53#331311yescharacteristic
353C2×C53#332211yescharacteristic
354C22×SD16C2×D5311yescharacteristic
355C2×C4⋊Q8—531no—
356C2×C22⋊Q8C2×D5311yescharacteristic
357C2×#121C2×D5311yescharacteristic
358C2×C4×D4—531no—
359C2×C4×D4—1061no—
360C22×C4⋊C4—531no—
361C2×#106—531no—
362C2×C4⋊C8—531no—
363C2×#103—531no—
364C2×#103—531no—
365C2×#103C2×D5311yescharacteristic
366C2×#102—531no—
367C2×#102—531no—
368C2×#102C2×D5311yescharacteristic
369C2×C22⋊C8—531no—
370C2×C22⋊C8C2×D5311yescharacteristic
371C2×C4×C8—531no—
372#400—1061no—
373#393—1061no—
374#391—531no—
375#391—531no—
376#391—1061no—
377#390—531no—
378#389—531no—
379#388—1061no—
380#388—1061no—
381#386—1061no—
382#385—1061no—
383#380—531no—
384C4×C4⋊C4—531no—
385C4×C4⋊C4—531no—
386C4×C22⋊C4—531no—
387#340—531no—
388#340—531no—
389#340—531no—
390#340—531no—
391#340—531no—
392#340—531no—
393#340—531no—
394#340C2×D5311yescharacteristic
395#336—531no—
396#336—531no—
397C53#2672211yescharacteristic, complemented, Sylow
398C22×Q8—1061no—
399C22×Q8C22×D5311yescharacteristic
400C22×D4—21no—
401C22×D4C22×D5311yescharacteristic
402C23×C4—1061no—
403C23×C4—1061no—
404C23×C4C22×D5311yescharacteristic
405C2×SD16—22no—
406C2×SD16—41no—
407C2×SD16—41no—
408C22×C8—531no—
409C22×C8—531no—
410C22×C8C22×D5311yescharacteristic
411C22×C8C22×D5311yescharacteristic
412C4⋊Q8—1062no—
413C4⋊Q8—2121no—
414C22⋊Q8—22no—
415C22⋊Q8—41no—
416#121—22no—
417#121—22no—
418C4×D4—1062no—
419C4×D4—2121no—
420C4×D4—2124no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—531no—
423C2×C4⋊C4—531no—
424C2×C4⋊C4—531no—
425C2×C4⋊C4—531no—
426C2×C4⋊C4—1062no—
427C2×C4⋊C4—1061no—
428C2×C4⋊C4—1061no—
429C2×C4⋊C4—1061no—
430C2×C4⋊C4—2121no—
431C2×C4⋊C4C22×D5311yescharacteristic
432C2×C4⋊C4C22×D5311yescharacteristic
433C2×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—531no—
436C2×C22⋊C4—531no—
437C2×C22⋊C4—1061no—
438C2×C22⋊C4—1061no—
439C2×C22⋊C4—1061no—
440C2×C22⋊C4—1061no—
441C2×C42—531no—
442C2×C42—531no—
443C2×C42—1061no—
444C2×C42—1061no—
445#106—1062no—
446C4⋊C8—2121no—
447#103—22no—
448#103—1062no—
449#103—2121no—
450#102—22no—
451#102—1062no—
452#102—2121no—
453C22⋊C8—41no—
454C22⋊C8—2121no—
455C4×C8—2121no—
456#95—531no—
457#95—531no—
458#95—1061no—
459#95—1061no—
460#95—1061no—
461#95—1061no—
462#95—1061no—
463#95—1061no—
464#95—1061no—
465#95—1061no—
466C24—21no—
467C24D4×D5311yescharacteristic
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—4241no—
475C2×Q8—2122no—
476C2×Q8—2122no—
477C2×D4—24no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—42no—
484C2×D4—42no—
485C2×D4#1171994712yes—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—531no—
496C22×C4—531no—
497C22×C4—531no—
498C22×C4—531no—
499C22×C4—1061no—
500C22×C4—1061no—
501C22×C4—1061no—
502C22×C4—1061no—
503C22×C4—4241no—
504C22×C4—4241no—
505C22×C4—2122no—
506C22×C4—2122no—
507C22×C4—2122no—
508C22×C4—2122no—
509C22×C4D4×D5311yescharacteristic
510C22×C4C23×D5311yescharacteristic
511SD16—44no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—1061no—
519C2×C8—1061no—
520C2×C8—1061no—
521C2×C8—1061no—
522C2×C8—1061no—
523C2×C8—1061no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—42no—
527C4⋊C4—532no—
528C4⋊C4—1061no—
529C4⋊C4—1061no—
530C4⋊C4—1061no—
531C4⋊C4—1061no—
532C4⋊C4—1061no—
533C4⋊C4—1061no—
534C4⋊C4—1061no—
535C4⋊C4—2122no—
536C4⋊C4—2122no—
537C4⋊C4—2122no—
538C4⋊C4#1171994712yes—
539C4⋊C4#1171994712yes—
540C22⋊C4—24no—
541C22⋊C4—42no—
542C22⋊C4—534no—
543C22⋊C4—4241no—
544C22⋊C4—4241no—
545C22⋊C4—2122no—
546C22⋊C4—2121no—
547C22⋊C4—2122no—
548C42—1061no—
549C42—1061no—
550C42—1061no—
551C42—1061no—
552C42—4241no—
553C42—2122no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—22no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23SD16×D5312yes—
568C23C2×D4×D5311yescharacteristic
569Q8—41no—
570Q8—41no—
571Q8—41no—
572Q8—41no—
573Q8—4244no—
574D4—22no—
575D4—22no—
576D4—48no—
577D4—41no—
578D4—41no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—532no—
596C2×C4—532no—
597C2×C4—1061no—
598C2×C4—1061no—
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—1062no—
608C2×C4—4244no—
609C2×C4—4244no—
610C2×C4—2121no—
611C2×C4—2121no—
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#41×D5311yescharacteristic
622C2×C4C2×#1171994711yescharacteristic
623C2×C4C2×#1171994711yescharacteristic
624C2×C4C2×#1171994711yescharacteristic
625C2×C4#1313326211yescharacteristic
626C2×C4#1313326211yescharacteristic
627C8—41no—
628C8—41no—
629C8—2121no—
630C8—2121no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22#1681112911yescharacteristic
646C22#1681126011yescharacteristic
647C22#1681126211yescharacteristic
648C22C2×SD16×D5311yescharacteristic
649C22#1681141911yescharacteristic
650C22#1681142511yescharacteristic
651C22#1681143911yescharacteristic
652C4—42no—
653C4—41no—
654C4—42no—
655C4—81no—
656C4—1062no—
657C4—1062no—
658C4—4241no—
659C4—4241no—
660C4—4241no—
661C4—2121no—
662C4—2122no—
663C4—2121no—
664C4#1681115411yescharacteristic
665C4#1681120811yescharacteristic
666C4#1681133711yescharacteristic
667C4#1681135911yescharacteristic
668C2—42no—
669C2—82no—
670C2#2027864711yescharacteristic
671C2#2028471811yescharacteristic
672C2#2028497811yescharacteristic
673C2#2028653411yescharacteristic
674C2#2028657311yescharacteristic
675C2#2028677611yescharacteristic
676C2#2028690911yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2000C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
hypercenter#2000D53
hypocenterC53#26722
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2000

Complements

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

NH ≅ G/Ntype
1. #20279091C2semidirect
3. #20276028C2semidirect
5. #20275849C2semidirect
6. #20275661C2semidirect
7. #20275393C2semidirect
8. #20275352C2semidirect
9. #20275318C2semidirect
10. #20275296C2semidirect
11. #20275051C2semidirect
12. #20275023C2semidirect
13. #20274802C2semidirect
14. #102×Dic53C2semidirect
191. C22⋊Dic53SD16semidirect
397. C53#26722semidirect

Maximal subgroups

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

subgroupindex
#202790912
#202777352
#202760282
#202759012
#202758492
#202756612
#202753932
#202753522
#202753182
#202752962
#202750512
#202750232
#202748022
#102×Dic532
C53×#20002
#2672253