Finite Group Databasebeta

← #27432363

Subgroups — #27432363

Order 13568.

842 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810954C211yesmaximal, characteristic
2C2×#16810864C211yesmaximal, characteristic
3#20279200C211yesmaximal, characteristic
4#20277718C211yesmaximal, characteristic
5#20276198C211yesmaximal, characteristic, complemented
6#20276135C211yesmaximal, characteristic, complemented
7#20276095C211yesmaximal, characteristic, complemented
8#20276038C211yesmaximal, characteristic, complemented
9#20275485C211yesmaximal, characteristic, complemented
10#20275350C211yesmaximal, characteristic, complemented
11#20275055C211yesmaximal, characteristic, complemented
12#20275020C211yesmaximal, characteristic
13#20274731C211yesmaximal, characteristic, complemented
14C2×#16810386C211yesmaximal, characteristic
15C53×#1850C211yesmaximal, characteristic
16C22×Dic212C2211yescharacteristic
17C2×#13133196C2211yescharacteristic
18C2×#13133196C2211yescharacteristic
19C2×C4×Dic106C2211yescharacteristic
20C22×C4⋊Dic53C2211yescharacteristic
21C22×C4⋊Dic53C2211yescharacteristic
22#16810954—24no—
23#16810864—24no—
24C2×#13133151C2211yescharacteristic
25C2×#13133149C2211yescharacteristic
26C2×#13133149C2211yescharacteristic
27C2×#13133142C2211yescharacteristic
28C2×#13133142C2211yescharacteristic
29C2×#13133142C2211yescharacteristic
30C2×#13133142C2211yescharacteristic
31C2×#13133142C2211yescharacteristic
32C2×#13133142C2211yescharacteristic
33#16810563—21no—
34#16810563—21no—
35#16810529C2211yescharacteristic
36#16810529C2211yescharacteristic
37#16810529C2211yescharacteristic
38#16810521C2211yescharacteristic
39#16810520—21no—
40#16810510—21no—
41#16810495—21no—
42#16810489C2211yescharacteristic
43#16810489C2211yescharacteristic
44#16810484—21no—
45#16810484—21no—
46C4×C4⋊Dic53C2211yescharacteristic
47C22⋊C4×Dic53—21no—
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52#16810386C2211yescharacteristic
53#16810386C2211yescharacteristic
54C23×C8×C53C2211yescharacteristic
55C2×C4⋊C8×C53C2211yescharacteristic
56C2×C4⋊C8×C53C2211yescharacteristic
57C2×C22⋊C8×C53C2211yescharacteristic
58C4×C22⋊C4×C53C2211yescharacteristic
59C53×#336C2211yescharacteristic
60C53×#336C2211yescharacteristic
61C22×Dic106C2311yescharacteristic
62C22×Dic106C2311yescharacteristic
63C23×Dic53—21no—
64C23×Dic53—21no—
65C2×Dic212—22no—
66C2×Dic212—22no—
67C2×Dic212—41no—
68#13133196—22no—
69#13133196—22no—
70#13133196—22no—
71#13133196D414yes—
72C4×Dic106—22no—
73C4×Dic106—41no—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53—21no—
76C2×C4⋊Dic53—22no—
77C2×C4⋊Dic53—22no—
78C2×C4⋊Dic53—22no—
79C2×C4⋊Dic53—41no—
80C2×C4⋊Dic53C2311yescharacteristic
81C2×C4⋊Dic53C2311yescharacteristic
82C2×C4⋊Dic53C2311yescharacteristic
83C2×C4⋊Dic53C2311yescharacteristic
84C2×C4⋊Dic53C2311yescharacteristic
85C2×C4⋊Dic53C2311yescharacteristic
86C2×#11719919—21no—
87C2×#11719919—21no—
88C2×#11719919—21no—
89C2×#11719919—21no—
90C2×C22⋊Dic53—21no—
91C2×C22⋊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—41no—
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—
113#13133124—21no—
114#13133124—21no—
115C23×C4×C53C2311yescharacteristic
116C22×C8×C53—22no—
117C22×C8×C53—22no—
118C22×C8×C53—41no—
119C22×C8×C53C2311yescharacteristic
120C22×C8×C53C2311yescharacteristic
121C22×C8×C53C2311yescharacteristic
122C22×C8×C53C2311yescharacteristic
123C2×C22⋊C4×C53D411yescharacteristic
124C2×C22⋊C4×C53D411yescharacteristic
125C2×C42×C53C2311yescharacteristic
126C2×C42×C53C2311yescharacteristic
127C4⋊C8×C53—22no—
128C4⋊C8×C53—41no—
129C22⋊C8×C53—22no—
130#95×C53D411yescharacteristic
131#95×C53D411yescharacteristic
132C24×C53C2×D411yescharacteristic
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D412yes—
143C22×C4×C53#4112yes—
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—
153C22×Dic53—42no—
154C22×Dic53—42no—
155C22×Dic53—42no—
156C22×Dic53—42no—
157C22×Dic53—42no—
158C22×Dic53—81no—
159C2×Dic106—21no—
160C2×Dic106—21no—
161C2×Dic106—21no—
162C2×Dic106—21no—
163C2×Dic106—21no—
164C2×Dic106—21no—
165C2×Dic106—21no—
166C2×Dic106—21no—
167C2×Dic106—21no—
168C2×Dic106—21no—
169C2×Dic106—21no—
170C2×Dic106C2×D412yes—
171C2×C8×C53—21no—
172C2×C8×C53—21no—
173C2×C8×C53—21no—
174C2×C8×C53—21no—
175C2×C8×C53—21no—
176C2×C8×C53—21no—
177C2×C8×C53—21no—
178C2×C8×C53—21no—
179C2×C8×C53—21no—
180C2×C8×C53—21no—
181C2×C8×C53—21no—
182C2×C8×C53—44no—
183C2×C8×C53#4112yes—
184C22⋊C4×C53—22no—
185C22⋊C4×C53—22no—
186C42×C53—21no—
187C42×C53—21no—
188C42×C53—21no—
189C42×C53—21no—
190C22⋊Dic53—24no—
191C22⋊Dic53—42no—
192C4⋊Dic53—21no—
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—21no—
202C4⋊Dic53—42no—
203C4⋊Dic53—42no—
204C4⋊Dic53C2×D412yes—
205C4⋊Dic53#4112yes—
206#11719919—24no—
207#11719919—42no—
208#11719919—42no—
209#11719919—81no—
210C4×Dic53—42no—
211C4×Dic53—81no—
212Dic212—44no—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53#12112yes—
218C23×C53#13512yes—
219C23×C53#13712yes—
220C23×C53C22×D411yescharacteristic
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—22no—
224C2×C4×C53—22no—
225C2×C4×C53—22no—
226C2×C4×C53—22no—
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—21no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53#12112yes—
240C2×C4×C53C22×D411yescharacteristic
241C2×C4×C53C2×#4111yescharacteristic
242C2×C4×C53C2×#4111yescharacteristic
243C2×C4×C53C2×#4111yescharacteristic
244C2×C4×C53H(5,𝔽2)11yescharacteristic
245C2×C4×C532−1+411yescharacteristic
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—
269C2×Dic53—41no—
270C2×Dic53—84no—
271C2×Dic53—84no—
272Dic106—22no—
273Dic106—22no—
274Dic106—41no—
275Dic106—41no—
276Dic106—41no—
277Dic106—41no—
278Dic106—41no—
279Dic106—41no—
280C8×C53—22no—
281C8×C53—41no—
282C8×C53—41no—
283C8×C53—41no—
284#25208—531nomaximal, Sylow
285C22×C53—22no—
286C22×C53—22no—
287C22×C53—22no—
288C22×C53—22no—
289C22×C53—22no—
290C22×C53—22no—
291C22×C53#46414yes—
292C22×C53C2×#12111yescharacteristic
293C22×C53#54611yescharacteristic
294C22×C53#54711yescharacteristic
295C22×C53C2×#13511yescharacteristic
296C22×C53C2×#13711yescharacteristic
297C22×C53#57711yescharacteristic
298C22×C53#57811yescharacteristic
299C4×C53—22no—
300C4×C53—41no—
301C4×C53—41no—
302C4×C53—41no—
303C4×C53—41no—
304C4×C53#52911yescharacteristic
305C4×C53#53911yescharacteristic
306C4×C53#54611yescharacteristic
307C4×C53#54711yescharacteristic
308Dic53—42no—
309Dic53—42no—
310Dic53—42no—
311Dic53—81no—
312Dic53—81no—
313Dic53—81no—
314Dic53—81no—
315Dic53—81no—
316C2×#483—531no—
317C2×#467—531no—
318#2503—531no—
319#2360—531no—
320#2085—531no—
321#2074—531no—
322#2066—531no—
323#2056—531no—
324#1949—531no—
325#1927—531no—
326#1878—531no—
327#1872—531no—
328#1850D5311yescharacteristic
329#1805—531no—
330C2×#340—531no—
331C2×C53—24no—
332C2×C53#242511yescharacteristic
333C2×C53C2×#46411yescharacteristic
334C2×C53#304611yescharacteristic
335C2×C53#329611yescharacteristic
336C2×C53#331211yescharacteristic
337C2×C53#333611yescharacteristic
338C2×C53#334411yescharacteristic
339C22×Q16—531no—
340C23×C8C2×D5311yescharacteristic
341C2×C22⋊Q8—531no—
342C2×C22⋊Q8—531no—
343C2×C4×Q8—531no—
344C22×C4⋊C4—531no—
345C22×C4⋊C4—531no—
346#483—1064no—
347#467—1064no—
348C2×#107—531no—
349C2×#106—531no—
350C2×#106—531no—
351C2×C4⋊C8C2×D5311yescharacteristic
352C2×C4⋊C8C2×D5311yescharacteristic
353C2×#103—531no—
354C2×#103—531no—
355C2×#103—531no—
356C2×#103—531no—
357C2×#103—531no—
358C2×#103—531no—
359C2×C22⋊C8C2×D5311yescharacteristic
360#400—1061no—
361#400—1061no—
362#391—531no—
363#391—531no—
364#391—531no—
365#389—531no—
366#389—1061no—
367#387—1061no—
368#385—1061no—
369#382—531no—
370#382—531no—
371#381—1061no—
372#381—1061no—
373C4×C4⋊C4—531no—
374C4×C22⋊C4—1061no—
375C4×C22⋊C4C2×D5311yescharacteristic
376#340—531no—
377#340—531no—
378#340—531no—
379#340—531no—
380#340—531no—
381#340—531no—
382#336C2×D5311yescharacteristic
383#336C2×D5311yescharacteristic
384C53#2520811yescharacteristic, complemented, Sylow
385C22×Q8—531no—
386C22×Q8—531no—
387C23×C4—1061no—
388C23×C4—1061no—
389C23×C4C22×D5311yescharacteristic
390C2×Q16—1062no—
391C2×Q16—1062no—
392C2×Q16—2121no—
393C22×C8—22no—
394C22×C8—22no—
395C22×C8—41no—
396C22×C8C22×D5311yescharacteristic
397C22×C8C22×D5311yescharacteristic
398C22×C8C22×D5311yescharacteristic
399C22×C8C22×D5311yescharacteristic
400C22⋊Q8—534no—
401C22⋊Q8—1062no—
402C22⋊Q8—1062no—
403C22⋊Q8—1062no—
404C4×Q8—1062no—
405C4×Q8—2121no—
406C2×C4⋊C4—531no—
407C2×C4⋊C4—531no—
408C2×C4⋊C4—531no—
409C2×C4⋊C4—531no—
410C2×C4⋊C4—531no—
411C2×C4⋊C4—531no—
412C2×C4⋊C4—1062no—
413C2×C4⋊C4—1061no—
414C2×C4⋊C4—1062no—
415C2×C4⋊C4—1062no—
416C2×C4⋊C4—1061no—
417C2×C4⋊C4—1061no—
418C2×C4⋊C4—1061no—
419C2×C4⋊C4—1061no—
420C2×C4⋊C4—1061no—
421C2×C4⋊C4—2121no—
422C2×C22⋊C4—1061no—
423C2×C22⋊C4—1061no—
424C2×C22⋊C4D21211yescharacteristic
425C2×C22⋊C4D21211yescharacteristic
426C2×C42—1061no—
427C2×C42—1061no—
428C2×C42C22×D5311yescharacteristic
429C2×C42C22×D5311yescharacteristic
430#107—1062no—
431#106—1062no—
432#106—1062no—
433C4⋊C8—22no—
434C4⋊C8—41no—
435#103—1062no—
436#103—1062no—
437#103—1062no—
438#103—1062no—
439#103—1062no—
440#103—2121no—
441C22⋊C8—22no—
442#95—1061no—
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—
453#95—1061no—
454#95D21211yescharacteristic
455#95D21211yescharacteristic
456C24C2×D21211yescharacteristic
457C2×Q8—532no—
458C2×Q8—1061no—
459C2×Q8—1061no—
460C2×Q8—1061no—
461C2×Q8—1061no—
462C2×Q8—1061no—
463C2×Q8—1061no—
464C2×Q8—1061no—
465C2×Q8—1061no—
466C2×Q8—1061no—
467C2×Q8—1061no—
468C2×Q8—1061no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—1061no—
474C22×C4—1061no—
475C22×C4—1061no—
476C22×C4—1061no—
477C22×C4—1061no—
478C22×C4—1061no—
479C22×C4—1061no—
480C22×C4—1061no—
481C22×C4—4241no—
482C22×C4—2122no—
483C22×C4—2122no—
484C22×C4—2122no—
485C22×C4—2122no—
486C22×C4—2122no—
487C22×C4C2×D21211yescharacteristic
488C22×C4C2×D21211yescharacteristic
489C22×C4C2×D21211yescharacteristic
490C22×C4C2×D21211yescharacteristic
491C22×C4C2×D21211yescharacteristic
492C22×C4D4×D5312yes—
493C22×C4#1171994912yes—
494C22×C4C23×D5311yescharacteristic
495Q16—2124no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—44no—
508C2×C8#1171994712yes—
509C4⋊C4—532no—
510C4⋊C4—532no—
511C4⋊C4—1064no—
512C4⋊C4—1061no—
513C4⋊C4—1061no—
514C4⋊C4—1061no—
515C4⋊C4—1061no—
516C4⋊C4—1061no—
517C4⋊C4—1061no—
518C4⋊C4—1061no—
519C4⋊C4—1061no—
520C4⋊C4—1061no—
521C4⋊C4—1061no—
522C4⋊C4—4241no—
523C4⋊C4—2122no—
524C4⋊C4—2122no—
525C4⋊C4—2122no—
526C4⋊C4—2122no—
527C22⋊C4—22no—
528C22⋊C4—22no—
529C22⋊C4—1064no—
530C22⋊C4—2122no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—21no—
535C42—4241no—
536C42—2122no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23#1313319212yes—
542C23#1313323912yes—
543C23#1313325012yes—
544C23C22×D21211yescharacteristic
545Q8—1062no—
546Q8—1062no—
547Q8—2121no—
548Q8—2121no—
549Q8—2121no—
550Q8—2121no—
551Q8—2121no—
552Q8—2121no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—1062no—
572C2×C4—1062no—
573C2×C4—1062no—
574C2×C4—4244no—
575C2×C4—4244no—
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—2121no—
591C2×C4—2121no—
592C2×C4—2121no—
593C2×C4—2121no—
594C2×C4—2121no—
595C2×C4—2121no—
596C2×C4—2121no—
597C2×C4#1313319212yes—
598C2×C4C2×D4×D5311yescharacteristic
599C2×C4#41×D5311yescharacteristic
600C2×C4C2×#1171994711yescharacteristic
601C2×C4C2×#1171994911yescharacteristic
602C2×C4#1313326211yescharacteristic
603C2×C4#1313326411yescharacteristic
604C8—22no—
605C8—41no—
606C8—41no—
607C8—41no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22#1681084614yes—
615C22C2×#1313319211yescharacteristic
616C22#1681126811yescharacteristic
617C22#1681127711yescharacteristic
618C22C2×#1313323911yescharacteristic
619C22C2×#1313325011yescharacteristic
620C22#1681143511yescharacteristic
621C22#1681144011yescharacteristic
622C4—22no—
623C4—41no—
624C4—41no—
625C4—41no—
626C4—41no—
627C4—4241no—
628C4—4241no—
629C4—4241no—
630C4—4241no—
631C4—4241no—
632C4—2122no—
633C4—2122no—
634C4—2122no—
635C4#1681115911yescharacteristic
636C4#1681122211yescharacteristic
637C4#1681126011yescharacteristic
638C4#1681127111yescharacteristic
639C2—24no—
640C2#2027826411yescharacteristic
641C2C2×#1681084611yescharacteristic
642C2#2028426711yescharacteristic
643C2#2028655211yescharacteristic
644C2#2028676811yescharacteristic
645C2#2028708411yescharacteristic
646C2#2028717211yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1850C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#1850D53
hypocenterC53#25208
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1850

Complements

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

NH ≅ G/Ntype
5. #20276198C2semidirect
6. #20276135C2semidirect
7. #20276095C2semidirect
8. #20276038C2semidirect
9. #20275485C2semidirect
10. #20275350C2semidirect
11. #20275055C2semidirect
13. #20274731C2semidirect
384. C53#25208semidirect

Maximal subgroups

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

subgroupindex
C2×#168109542
C2×#168108642
#202792002
#202777182
#202761982
#202761352
#202760952
#202760382
#202754852
#202753502
#202750552
#202750202
#202747312
C2×#168103862
C53×#18502
#2520853