Finite Group Databasebeta

← #27413719

Subgroups — #27413719

Order 13568.

820 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810731C211yesmaximal, characteristic, complemented
2#20277397C211yesmaximal, characteristic
3#20277397C211yesmaximal, characteristic
4#20276213C211yesmaximal, characteristic, complemented
5#20276148C211yesmaximal, characteristic
6#20276109C211yesmaximal, characteristic, complemented
7#20276095C211yesmaximal, characteristic, complemented
8#20276038C211yesmaximal, characteristic, complemented
9#20275919C211yesmaximal, characteristic
10#20275912C211yesmaximal, characteristic
11#20275725C211yesmaximal, characteristic
12#20274612C211yesmaximal, characteristic, complemented
13C4×#13133142C211yesmaximal, characteristic, complemented
14C4×#13133142C211yesmaximal, characteristic, complemented
15C2×C53×#433C211yesmaximal, characteristic
16C22×Dic212C2211yescharacteristic
17C2×#13133196C2211yescharacteristic
18C2×#13133196C2211yescharacteristic
19C2×C4×Dic106C2211yescharacteristic
20C2×C4×Dic106C2211yescharacteristic
21#16810731—24no—
22C2×#13133151C2211yescharacteristic
23C2×#13133149C2211yescharacteristic
24C2×#13133149C2211yescharacteristic
25C2×#13133142C2211yescharacteristic
26C2×#13133142C2211yescharacteristic
27C2×#13133142C2211yescharacteristic
28C2×#13133142C2211yescharacteristic
29C2×#13133142C2211yescharacteristic
30C2×#13133142C2211yescharacteristic
31#16810529C2211yescharacteristic
32#16810529C2211yescharacteristic
33#16810521C2211yescharacteristic
34#16810521C2211yescharacteristic
35#16810518—21no—
36#16810518—21no—
37#16810517—21no—
38#16810517—21no—
39#16810511—21no—
40#16810511—21no—
41#16810504—21no—
42#16810504—21no—
43#16810483C2211yescharacteristic
44#16810483C2211yescharacteristic
45C4×C4⋊Dic53C2211yescharacteristic
46C4×C4⋊Dic53C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52#16810386C2211yescharacteristic
53C2×#117×C53C2211yescharacteristic
54C53×#433—24no—
55C2×C4⋊C8×C53C2211yescharacteristic
56C2×C4⋊C8×C53C2211yescharacteristic
57C2×C22⋊C8×C53C2211yescharacteristic
58C2×C22⋊C8×C53C2211yescharacteristic
59C2×#97×C53C2211yescharacteristic
60C2×C4×C8×C53C2211yescharacteristic
61C22×Dic106C2311yescharacteristic
62C22×Dic106C2311yescharacteristic
63C2×Dic212—22no—
64C2×Dic212—22no—
65C2×Dic212—22no—
66#13133196—22no—
67#13133196—22no—
68#13133196—22no—
69#13133196—22no—
70C4×Dic106—22no—
71C4×Dic106—22no—
72C4×Dic106—22no—
73C4×Dic106—22no—
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×#11719919—21no—
85C2×#11719919—21no—
86C2×C22⋊Dic53—21no—
87C2×C22⋊Dic53—21no—
88C2×C22⋊Dic53—21no—
89C2×C22⋊Dic53—21no—
90C2×C4×Dic53—21no—
91C2×C4×Dic53—21no—
92C2×C4×Dic53—21no—
93C2×C4×Dic53—21no—
94#13133151—22no—
95#13133149—22no—
96#13133149—22no—
97#13133142—22no—
98#13133142—22no—
99#13133142—22no—
100#13133142—22no—
101#13133142—22no—
102#13133142—22no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109#13133124—21no—
110#13133124—21no—
111#13133124—21no—
112#13133124—21no—
113C23×C4×C53C2311yescharacteristic
114C22×C8×C53C2311yescharacteristic
115C22×C8×C53C2311yescharacteristic
116C22×C8×C53C2311yescharacteristic
117C22×C8×C53C2311yescharacteristic
118#117×C53—22no—
119#117×C53—22no—
120C2×C4⋊C4×C53D411yescharacteristic
121C2×C4⋊C4×C53D411yescharacteristic
122C2×C22⋊C4×C53D411yescharacteristic
123C2×C22⋊C4×C53D411yescharacteristic
124C2×C42×C53C2311yescharacteristic
125C2×C42×C53C2311yescharacteristic
126C4⋊C8×C53—22no—
127C4⋊C8×C53—22no—
128C22⋊C8×C53—22no—
129C22⋊C8×C53—22no—
130#97×C53D414yes—
131C4×C8×C53—22no—
132C24×C53C2×D411yescharacteristic
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2411yescharacteristic
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—21no—
153C2×Dic106—21no—
154C2×Dic106—21no—
155C2×Dic106—21no—
156C2×Dic106—21no—
157C2×Dic106—21no—
158C2×Dic106—21no—
159C2×Dic106—21no—
160C2×Dic106—21no—
161C2×Dic106—21no—
162C2×Dic106—21no—
163C2×Dic106#4112yes—
164C2×Dic106#4112yes—
165C2×C8×C53—21no—
166C2×C8×C53—21no—
167C2×C8×C53—21no—
168C2×C8×C53—21no—
169C2×C8×C53—21no—
170C2×C8×C53—21no—
171C2×C8×C53—21no—
172C2×C8×C53—21no—
173C2×C8×C53—21no—
174C2×C8×C53—21no—
175C2×C8×C53C2×D412yes—
176C2×C8×C53C2×D412yes—
177C4⋊C4×C53—22no—
178C4⋊C4×C53—22no—
179C22⋊C4×C53—22no—
180C22⋊C4×C53—22no—
181C42×C53—21no—
182C42×C53—21no—
183C42×C53C2×D412yes—
184C42×C53#4112yes—
185C22⋊Dic53—24no—
186C22⋊Dic53—24no—
187C22⋊Dic53—81no—
188C22⋊Dic53—81no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—21no—
199C4⋊Dic53#4112yes—
200C4⋊Dic53#4112yes—
201#11719919—24no—
202#11719919—24no—
203#11719919—42no—
204#11719919—42no—
205#11719919—81no—
206#11719919—81no—
207C4×Dic53—42no—
208C4×Dic53—42no—
209C4×Dic53—81no—
210C4×Dic53—81no—
211Dic212—28no—
212C23×C53—21no—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53—21no—
219C23×C53C22×D411yescharacteristic
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—21no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53#12112yes—
238C2×C4×C53#12112yes—
239C2×C4×C53C22×D411yescharacteristic
240C2×C4×C53C2×#4111yescharacteristic
241C2×C4×C53C2×#4111yescharacteristic
242C2×C4×C53C2×#4111yescharacteristic
243C2×C4×C53C2×#4111yescharacteristic
244C2×C4×C53C2×#4111yescharacteristic
245C2×Dic53—22no—
246C2×Dic53—22no—
247C2×Dic53—22no—
248C2×Dic53—22no—
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—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269Dic106—22no—
270Dic106—22no—
271Dic106—22no—
272Dic106—22no—
273Dic106—41no—
274Dic106—41no—
275Dic106—41no—
276Dic106—41no—
277C8×C53—22no—
278C8×C53—22no—
279C8×C53—41no—
280C8×C53—41no—
281#23713—531nomaximal, Sylow
282C22×C53—41no—
283C22×C53—41no—
284C22×C53—41no—
285C22×C53—41no—
286C22×C53—41no—
287C22×C53—41no—
288C22×C53—41no—
289C22×C53C2×#12111yescharacteristic
290C22×C53#52511yescharacteristic
291C22×C53#52511yescharacteristic
292C22×C53#57711yescharacteristic
293C22×C53#57711yescharacteristic
294C22×C53#57811yescharacteristic
295C22×C53#57811yescharacteristic
296C4×C53—22no—
297C4×C53—22no—
298C4×C53—41no—
299C4×C53—41no—
300C4×C53—41no—
301C4×C53#52511yescharacteristic
302C4×C53#52911yescharacteristic
303C4×C53#52911yescharacteristic
304C4×C53#53211yescharacteristic
305Dic53—42no—
306Dic53—42no—
307Dic53—42no—
308Dic53—42no—
309Dic53—81no—
310Dic53—81no—
311Dic53—81no—
312Dic53—81no—
313C2×#441—531no—
314C2×#433D5311yescharacteristic
315#2323—531no—
316#2323—531no—
317#2088—531no—
318#2077—531no—
319#2068—531no—
320#2066—531no—
321#2056—531no—
322#2036—531no—
323#2035—531no—
324#2001—531no—
325#1776—531no—
326C4×#103—531no—
327C4×#103—531no—
328C2×C53—81no—
329C2×C53#239211yescharacteristic
330C2×C53#306311yescharacteristic
331C2×C53#306311yescharacteristic
332C2×C53#312011yescharacteristic
333C2×C53#312111yescharacteristic
334C2×C53#312511yescharacteristic
335C2×C53#312711yescharacteristic
336C22×Q16—531no—
337C2×C22⋊Q8—531no—
338C2×C22⋊Q8—531no—
339C2×C4×Q8—531no—
340C2×C4×Q8—531no—
341C2×#117C2×D5311yescharacteristic
342#441—1064no—
343#433—24no—
344C2×#107—531no—
345C2×#106—531no—
346C2×#106—531no—
347C2×C4⋊C8C2×D5311yescharacteristic
348C2×C4⋊C8C2×D5311yescharacteristic
349C2×#103—531no—
350C2×#103—531no—
351C2×#103—531no—
352C2×#103—531no—
353C2×#103—531no—
354C2×#103—531no—
355C2×C22⋊C8C2×D5311yescharacteristic
356C2×C22⋊C8C2×D5311yescharacteristic
357C2×#97C2×D5311yescharacteristic
358C2×C4×C8C2×D5311yescharacteristic
359#391—531no—
360#391—531no—
361#389—531no—
362#389—531no—
363#389—1061no—
364#389—1061no—
365#388—1061no—
366#388—1061no—
367#388—1061no—
368#388—1061no—
369#387—1061no—
370#387—1061no—
371#380—531no—
372#380—531no—
373C4×C4⋊C4—531no—
374C4×C4⋊C4—531no—
375#340—531no—
376#340—531no—
377#340—531no—
378#340—531no—
379#340—531no—
380#340—531no—
381C53#2371311yescharacteristic, complemented, Sylow
382C22×Q8—531no—
383C22×Q8—531no—
384C23×C4C22×D5311yescharacteristic
385C2×Q16—1062no—
386C2×Q16—1062no—
387C2×Q16—1062no—
388C22×C8C22×D5311yescharacteristic
389C22×C8C22×D5311yescharacteristic
390C22×C8C22×D5311yescharacteristic
391C22×C8C22×D5311yescharacteristic
392C22⋊Q8—1062no—
393C22⋊Q8—1062no—
394C22⋊Q8—1062no—
395C22⋊Q8—1062no—
396C4×Q8—1062no—
397C4×Q8—1062no—
398C4×Q8—1062no—
399C4×Q8—1062no—
400#117—22no—
401#117—22no—
402C2×C4⋊C4—531no—
403C2×C4⋊C4—531no—
404C2×C4⋊C4—531no—
405C2×C4⋊C4—531no—
406C2×C4⋊C4—531no—
407C2×C4⋊C4—531no—
408C2×C4⋊C4—1061no—
409C2×C4⋊C4—1061no—
410C2×C4⋊C4—1061no—
411C2×C4⋊C4—1061no—
412C2×C4⋊C4—1061no—
413C2×C4⋊C4—1061no—
414C2×C4⋊C4D21211yescharacteristic
415C2×C4⋊C4D21211yescharacteristic
416C2×C22⋊C4—1061no—
417C2×C22⋊C4—1061no—
418C2×C22⋊C4—1061no—
419C2×C22⋊C4—1061no—
420C2×C22⋊C4D21211yescharacteristic
421C2×C22⋊C4D21211yescharacteristic
422C2×C42—1061no—
423C2×C42—1061no—
424C2×C42—1061no—
425C2×C42—1061no—
426C2×C42C22×D5311yescharacteristic
427C2×C42C22×D5311yescharacteristic
428#107—1062no—
429#106—1062no—
430#106—1062no—
431C4⋊C8—22no—
432C4⋊C8—22no—
433#103—1062no—
434#103—1062no—
435#103—1062no—
436#103—1062no—
437#103—1062no—
438#103—1062no—
439C22⋊C8—22no—
440C22⋊C8—22no—
441#97C22⋊D5314yes—
442C4×C8—22no—
443#95—1061no—
444#95—1061no—
445#95—1061no—
446#95—1061no—
447#95—1061no—
448#95—1061no—
449#95—1061no—
450#95—1061no—
451#95—1061no—
452#95—1061no—
453C24C2×D21211yescharacteristic
454C2×Q8—532no—
455C2×Q8—532no—
456C2×Q8—1061no—
457C2×Q8—1061no—
458C2×Q8—1061no—
459C2×Q8—1061no—
460C2×Q8—1061no—
461C2×Q8—1061no—
462C2×Q8—1061no—
463C2×Q8—1061no—
464C2×Q8—1061no—
465C2×Q8—1061no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—1061no—
473C22×C4—1061no—
474C22×C4—1061no—
475C22×C4—1061no—
476C22×C4—1061no—
477C22×C4—1061no—
478C22×C4—1061no—
479C22×C4—1061no—
480C22×C4C2×D21211yescharacteristic
481C22×C4C2×D21211yescharacteristic
482C22×C4C2×D21211yescharacteristic
483C22×C4C2×D21211yescharacteristic
484C22×C4C2×D21211yescharacteristic
485C22×C4C23×D5311yescharacteristic
486Q16—1068no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8C2×C22⋊D5312yes—
498C2×C8C2×C22⋊D5312yes—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—532no—
502C4⋊C4—532no—
503C4⋊C4—1064no—
504C4⋊C4—1064no—
505C4⋊C4—1061no—
506C4⋊C4—1061no—
507C4⋊C4—1061no—
508C4⋊C4—1061no—
509C4⋊C4—1061no—
510C4⋊C4—1061no—
511C4⋊C4—1061no—
512C4⋊C4—1061no—
513C4⋊C4—1061no—
514C4⋊C4—1061no—
515C4⋊C4—4241no—
516C4⋊C4—4241no—
517C4⋊C4—2122no—
518C4⋊C4—2122no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—1064no—
522C22⋊C4—1064no—
523C22⋊C4—4241no—
524C22⋊C4—4241no—
525C42—21no—
526C42—21no—
527C42—4241no—
528C42—4241no—
529C42—2122no—
530C42—2122no—
531C42#1171994512yes—
532C42C2×C22⋊D5312yes—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23C22×D21211yescharacteristic
541Q8—1062no—
542Q8—1062no—
543Q8—1062no—
544Q8—1062no—
545Q8—2121no—
546Q8—2121no—
547Q8—2121no—
548Q8—2121no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—1062no—
567C2×C4—1062no—
568C2×C4—1062no—
569C2×C4—1062no—
570C2×C4—2121no—
571C2×C4—2121no—
572C2×C4—2121no—
573C2×C4—2121no—
574C2×C4—2121no—
575C2×C4—2121no—
576C2×C4—2121no—
577C2×C4—2121no—
578C2×C4—2121no—
579C2×C4—2121no—
580C2×C4—2121no—
581C2×C4—2121no—
582C2×C4—2121no—
583C2×C4—2121no—
584C2×C4—2121no—
585C2×C4—2121no—
586C2×C4—2121no—
587C2×C4—2121no—
588C2×C4—2121no—
589C2×C4—2121no—
590C2×C4#1313319112yes—
591C2×C4#1313319112yes—
592C2×C4C22×C22⋊D5311yescharacteristic
593C2×C4#41×D5311yescharacteristic
594C2×C4#41×D5311yescharacteristic
595C2×C4#41×D5311yescharacteristic
596C2×C4#41×D5311yescharacteristic
597C2×C4C2×#1171994511yescharacteristic
598C8—22no—
599C8—22no—
600C8—41no—
601C8—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22C2×#1313319111yescharacteristic
610C22#1681113511yescharacteristic
611C22#1681113511yescharacteristic
612C22#1681143511yescharacteristic
613C22#1681143511yescharacteristic
614C22#1681144011yescharacteristic
615C22#1681144011yescharacteristic
616C4—22no—
617C4—22no—
618C4—41no—
619C4—41no—
620C4—41no—
621C4—4241no—
622C4—4241no—
623C4—4241no—
624C4—4241no—
625C4—2122no—
626C4—2122no—
627C4—2122no—
628C4—2122no—
629C4#1681113011yescharacteristic
630C4#1681115111yescharacteristic
631C4#1681115111yescharacteristic
632C4#1681116911yescharacteristic
633C2—81no—
634C2#2027795111yescharacteristic
635C2#2028440611yescharacteristic
636C2#2028440611yescharacteristic
637C2#2028493111yescharacteristic
638C2#2028494611yescharacteristic
639C2#2028499411yescharacteristic
640C2#2028501611yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#433C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenterC2×#433D53
hypocenterC53#23713
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#433

Complements

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

NH ≅ G/Ntype
1. C2×#16810731C2semidirect
4. #20276213C2semidirect
6. #20276109C2semidirect
7. #20276095C2semidirect
8. #20276038C2semidirect
12. #20274612C2semidirect
13. C4×#13133142C2semidirect
14. C4×#13133142C2semidirect
381. C53#23713semidirect

Maximal subgroups

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

subgroupindex
C2×#168107312
#202773972
#202762132
#202761482
#202761092
#202760952
#202760382
#202759192
#202759122
#202757252
#202746122
C4×#131331422
C2×C53×#4332
#2371353