Finite Group Databasebeta

← #27466019

Subgroups — #27466019

Order 13568.

832 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810665C211yesmaximal, characteristic, complemented
2#20279023C211yesmaximal, characteristic
3#20279012C211yesmaximal, characteristic
4#20276213C211yesmaximal, characteristic, complemented
5#20276203C211yesmaximal, characteristic, complemented
6#20276148C211yesmaximal, characteristic, complemented
7#20276143C211yesmaximal, characteristic, complemented
8#20275448C211yesmaximal, characteristic, complemented
9#20275443C211yesmaximal, characteristic, complemented
10#20274811C211yesmaximal, characteristic, complemented
11#20274806C211yesmaximal, characteristic, complemented
12#20274748C211yesmaximal, characteristic, complemented
13#20274748C211yesmaximal, characteristic, complemented
14#20274748C211yesmaximal, characteristic, complemented
15C2×C53×#436C211yesmaximal, characteristic
16C2×#13133198—21no—
17C2×#13133176C2211yescharacteristic
18C22×C4⋊Dic53C2211yescharacteristic
19#16810665—24no—
20C2×#13133151C2211yescharacteristic
21C2×#13133151C2211yescharacteristic
22C2×#13133151C2211yescharacteristic
23C2×#13133149C2211yescharacteristic
24C2×#13133149C2211yescharacteristic
25C2×#13133149C2211yescharacteristic
26#16810553—21no—
27#16810551—21no—
28#16810548—21no—
29#16810537—21no—
30#16810521C2211yescharacteristic
31#16810521C2211yescharacteristic
32#16810521C2211yescharacteristic
33#16810521C2211yescharacteristic
34#16810511—21no—
35#16810504—21no—
36#16810484—21no—
37#16810483C2211yescharacteristic
38#16810483C2211yescharacteristic
39#16810483C2211yescharacteristic
40#16810483C2211yescharacteristic
41#16810483C2211yescharacteristic
42#16810483C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52#16810386C2211yescharacteristic
53C22×M16×C53C2211yescharacteristic
54C22×M16×C53C2211yescharacteristic
55C2×C4×D4×C53C2211yescharacteristic
56C53×#436—24no—
57C2×C4⋊C8×C53C2211yescharacteristic
58C2×C22⋊C8×C53C2211yescharacteristic
59C2×C22⋊C8×C53C2211yescharacteristic
60C2×C4×C8×C53C2211yescharacteristic
61C23×Dic53—21no—
62#13133198—44no—
63#13133176—22no—
64#13133176—22no—
65C2×C4⋊Dic53—21no—
66C2×C4⋊Dic53—22no—
67C2×C4⋊Dic53—22no—
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×C22⋊Dic53—21no—
81C2×C22⋊Dic53—21no—
82C2×C22⋊Dic53—21no—
83C2×C22⋊Dic53—21no—
84C2×C22⋊Dic53—21no—
85C2×C22⋊Dic53—21no—
86C2×C22⋊Dic53—21no—
87C2×C4×Dic53—21no—
88#13133151—22no—
89#13133151—22no—
90#13133151—22no—
91#13133149—22no—
92#13133149—22no—
93#13133149—22no—
94#13133124—21no—
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—
104C22×D4×C53D411yescharacteristic
105C23×C4×C53C2311yescharacteristic
106C23×C4×C53C2311yescharacteristic
107C2×M16×C53—22no—
108C2×M16×C53—22no—
109C2×M16×C53—22no—
110C2×M16×C53—22no—
111C2×M16×C53—22no—
112C2×M16×C53D414yes—
113C22×C8×C53C2311yescharacteristic
114C22×C8×C53C2311yescharacteristic
115C22×C8×C53C2311yescharacteristic
116C22×C8×C53C2311yescharacteristic
117C4×D4×C53—22no—
118C4×D4×C53—22no—
119C2×C4⋊C4×C53D411yescharacteristic
120C2×C22⋊C4×C53D411yescharacteristic
121C2×C22⋊C4×C53D411yescharacteristic
122C2×C42×C53C2311yescharacteristic
123C4⋊C8×C53—22no—
124C22⋊C8×C53—22no—
125C22⋊C8×C53—22no—
126C4×C8×C53—22no—
127C24×C53C2×D411yescharacteristic
128C24×C53C2×D411yescharacteristic
129C2×D4×C53—22no—
130C2×D4×C53—22no—
131C2×D4×C53—41no—
132C22×C4×C53—21no—
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×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D412yes—
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53#4112yes—
148C22×C4×C53C2411yescharacteristic
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53—21no—
155C22×Dic53—21no—
156C22×Dic53—21no—
157C22×Dic53—42no—
158C22×Dic53—42no—
159C22×Dic53—42no—
160M16×C53—28no—
161M16×C53—44no—
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C2×C8×C53—21no—
165C2×C8×C53—21no—
166C2×C8×C53—21no—
167C2×C8×C53—21no—
168C2×C8×C53—21no—
169C2×C8×C53—21no—
170C2×C8×C53C2×D412yes—
171C2×C8×C53C2×D412yes—
172C2×C8×C53#4112yes—
173C2×C8×C53#4112yes—
174C4⋊C4×C53—22no—
175C22⋊C4×C53—22no—
176C22⋊C4×C53—22no—
177C42×C53—21no—
178C42×C53—21no—
179C22⋊Dic53—42no—
180C22⋊Dic53—42no—
181C22⋊Dic53—42no—
182C22⋊Dic53—42no—
183C22⋊Dic53—42no—
184C22⋊Dic53—81no—
185C22⋊Dic53—81no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
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—21no—
200C4⋊Dic53—21no—
201C4⋊Dic53—21no—
202C4⋊Dic53—42no—
203#11719919—42no—
204#11719919—42no—
205#11719919—42no—
206#11719919—81no—
207C4×Dic53—42no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53—21no—
219C23×C53—21no—
220C23×C53—21no—
221C23×C53#12112yes—
222C23×C53C22×D411yescharacteristic
223D4×C53—44no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—22no—
229C2×C4×C53—22no—
230C2×C4×C53—22no—
231C2×C4×C53—22no—
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—41no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53#12112yes—
243C2×C4×C53AGL(1,ℤ/8)12yes—
244C2×C4×C53C22×D411yescharacteristic
245C2×C4×C53C2×#4111yescharacteristic
246C2×C4×C53C2×#4111yescharacteristic
247C2×C4×C53C2×#4111yescharacteristic
248C2×C4×C532−1+411yescharacteristic
249C2×C4×C532−1+411yescharacteristic
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—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—84no—
275C8×C53—22no—
276C8×C53—41no—
277C8×C53—41no—
278C8×C53#12314yescomplemented
279#27670—531nomaximal, Sylow
280C22×C53—22no—
281C22×C53—22no—
282C22×C53—22no—
283C22×C53—41no—
284C22×C53—41no—
285C22×C53—41no—
286C22×C53—41no—
287C22×C53—41no—
288C22×C53—41no—
289C22×C53—41no—
290C22×C53—41no—
291C22×C53—41no—
292C22×C53—41no—
293C22×C53—41no—
294C22×C53C2×#12111yescharacteristic
295C22×C53#54711yescharacteristic
296C22×C53#54711yescharacteristic
297C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
298C22×C53#57511yescharacteristic
299C22×C53#57711yescharacteristic
300C22×C53#57811yescharacteristic
301C4×C53—22no—
302C4×C53—22no—
303C4×C53—41no—
304C4×C53—41no—
305C4×C53C2×#12311yescharacteristic
306C4×C53#53611yescharacteristic
307C4×C53#55211yescharacteristic
308C4×C53#55211yescharacteristic
309Dic53—81no—
310Dic53—81no—
311Dic53—81no—
312Dic53—81no—
313Dic53—81no—
314Dic53—81no—
315Dic53—81no—
316Dic53—81no—
317C2×#436D5311yescharacteristic
318C2×#428—531no—
319#2486—531no—
320#2485—531no—
321#2088—531no—
322#2086—531no—
323#2077—531no—
324#2076—531no—
325#1942—531no—
326#1941—531no—
327#1825—531no—
328#1824—531no—
329#1809—531no—
330#1809—531no—
331#1809—531no—
332C2×C53—42no—
333C2×C53—81no—
334C2×C53#248611yescharacteristic
335C2×C53#305511yescharacteristic
336C2×C53#306311yescharacteristic
337C2×C53#335111yescharacteristic
338C2×C53#335211yescharacteristic
339C2×C53#335311yescharacteristic
340C2×C53#335411yescharacteristic
341C22×M16C2×D5311yescharacteristic
342C22×M16C2×D5311yescharacteristic
343C2×#123—1061no—
344C2×C4×D4C2×D5311yescharacteristic
345C2×#117—531no—
346C22×C4⋊C4—531no—
347#436—24no—
348#428—1064no—
349C2×#107—531no—
350C2×#107—531no—
351C2×#107—531no—
352C2×#106—531no—
353C2×#106—531no—
354C2×#106—531no—
355C2×C4⋊C8C2×D5311yescharacteristic
356C2×C22⋊C8C2×D5311yescharacteristic
357C2×C22⋊C8C2×D5311yescharacteristic
358C2×C4×C8C2×D5311yescharacteristic
359#398—1061no—
360#397—1061no—
361#397—1061no—
362#394—1061no—
363#389—531no—
364#389—531no—
365#389—531no—
366#389—531no—
367#388—1061no—
368#387—1061no—
369#381—1061no—
370#380—531no—
371#380—531no—
372#380—531no—
373#380—531no—
374#380—531no—
375#380—531no—
376#340—531no—
377#340—531no—
378#340—531no—
379#340—531no—
380#340—531no—
381#340—531no—
382#340—531no—
383#340—531no—
384#340—531no—
385#340—531no—
386C53#2767011yescharacteristic, complemented, Sylow
387C22×D4D21211yescharacteristic
388C23×C4—1061no—
389C23×C4C22×D5311yescharacteristic
390C23×C4C22×D5311yescharacteristic
391C2×M16—22no—
392C2×M16—22no—
393C2×M16—22no—
394C2×M16—22no—
395C2×M16—22no—
396C2×M16C22⋊D5314yes—
397C22×C8C22×D5311yescharacteristic
398C22×C8C22×D5311yescharacteristic
399C22×C8C22×D5311yescharacteristic
400C22×C8C22×D5311yescharacteristic
401#123—2124no—
402C4×D4—22no—
403C4×D4—22no—
404#117—1062no—
405#117—1062no—
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—531no—
413C2×C4⋊C4—531no—
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⋊C4D21211yescharacteristic
422C2×C22⋊C4—1061no—
423C2×C22⋊C4—1061no—
424C2×C22⋊C4—1061no—
425C2×C22⋊C4—1061no—
426C2×C22⋊C4—1061no—
427C2×C22⋊C4—1061no—
428C2×C22⋊C4—1061no—
429C2×C22⋊C4D21211yescharacteristic
430C2×C22⋊C4D21211yescharacteristic
431C2×C42—1061no—
432C2×C42C22×D5311yescharacteristic
433#107—1062no—
434#107—1062no—
435#107—1062no—
436#106—1062no—
437#106—1062no—
438#106—1062no—
439C4⋊C8—22no—
440C22⋊C8—22no—
441C22⋊C8—22no—
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
454C24C2×D21211yescharacteristic
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—41no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—1061no—
469C22×C4—1061no—
470C22×C4—1061no—
471C22×C4—1061no—
472C22×C4—1061no—
473C22×C4—1061no—
474C22×C4—1061no—
475C22×C4—1061no—
476C22×C4—2122no—
477C22×C4—2122no—
478C22×C4—2122no—
479C22×C4C2×D21211yescharacteristic
480C22×C4C2×D21211yescharacteristic
481C22×C4C2×D21211yescharacteristic
482C22×C4C2×D21211yescharacteristic
483C22×C4#1171994512yes—
484C22×C4C2×C22⋊D5312yes—
485C22×C4C23×D5311yescharacteristic
486M16—28no—
487M16—44no—
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#1171994712yes—
497C2×C8#1171994712yes—
498C2×C8C2×C22⋊D5312yes—
499C2×C8C2×C22⋊D5312yes—
500C4⋊C4—22no—
501C4⋊C4—1061no—
502C4⋊C4—1061no—
503C4⋊C4—1061no—
504C4⋊C4—1061no—
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—1061no—
516C4⋊C4—1061no—
517C4⋊C4—4241no—
518C4⋊C4—2122no—
519C4⋊C4—2122no—
520C4⋊C4—2122no—
521C4⋊C4—2122no—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—4241no—
525C22⋊C4—4241no—
526C22⋊C4—2122no—
527C22⋊C4—2122no—
528C22⋊C4—2122no—
529C22⋊C4—2122no—
530C22⋊C4—2122no—
531C42—21no—
532C42—21no—
533C42—2122no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23#1313319112yes—
548C23C22×D21211yescharacteristic
549D4—44no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—4244no—
569C2×C4—2121no—
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—2121no—
591C2×C4—2121no—
592C2×C4—2121no—
593C2×C4#1313319112yes—
594C2×C4#1313324512yes—
595C2×C4C22×C22⋊D5311yescharacteristic
596C2×C4C2×#1171994711yescharacteristic
597C2×C4C2×#1171994711yescharacteristic
598C2×C4C2×#1171994511yescharacteristic
599C2×C4#1313326411yescharacteristic
600C2×C4#1313326411yescharacteristic
601C8—22no—
602C8—41no—
603C8—41no—
604C8#1313319814yescomplemented
605C22—22no—
606C22—22no—
607C22—22no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22C2×#1313319111yescharacteristic
620C22#1681127711yescharacteristic
621C22#1681127711yescharacteristic
622C22C2×#1313324511yescharacteristic
623C22#1681142311yescharacteristic
624C22#1681143511yescharacteristic
625C22#1681144011yescharacteristic
626C4—22no—
627C4—22no—
628C4—41no—
629C4—41no—
630C4—4241no—
631C4—4241no—
632C4—4241no—
633C4—4241no—
634C4—4241no—
635C4—4241no—
636C4—4241no—
637C4—4241no—
638C4C2×#1313319811yescharacteristic
639C4#1681119411yescharacteristic
640C4#1681131311yescharacteristic
641C4#1681131311yescharacteristic
642C2—42no—
643C2—81no—
644C2#2027902311yescharacteristic
645C2#2028433411yescharacteristic
646C2#2028440611yescharacteristic
647C2#2028727711yescharacteristic
648C2#2028728911yescharacteristic
649C2#2028730011yescharacteristic
650C2#2028731111yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#436C2
socleC23×C53C22×D4
prime radicalC22×D4×C53D4
hypercenterC2×#436D53
hypocenterC53#27670
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#436

Complements

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

NH ≅ G/Ntype
1. C2×#16810665C2semidirect
4. #20276213C2semidirect
5. #20276203C2semidirect
6. #20276148C2semidirect
7. #20276143C2semidirect
8. #20275448C2semidirect
9. #20275443C2semidirect
10. #20274811C2semidirect
11. #20274806C2semidirect
12. #20274748C2semidirect
13. #20274748C2semidirect
14. #20274748C2semidirect
278. C8×C53#123semidirect
386. C53#27670semidirect
604. C8#13133198semidirect

Maximal subgroups

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

subgroupindex
C2×#168106652
#202790232
#202790122
#202762132
#202762032
#202761482
#202761432
#202754482
#202754432
#202748112
#202748062
#202747482
C2×C53×#4362
#2767053