Finite Group Databasebeta

← #27446625

Subgroups — #27446625

Order 13568.

882 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810907C211yesmaximal, characteristic, complemented
2C22×#13133148C211yesmaximal, characteristic
3#20278418C211yesmaximal, characteristic
4#20277720C211yesmaximal, characteristic
5#20276255C211yesmaximal, characteristic, complemented
6#20276219C211yesmaximal, characteristic, complemented
7#20276097C211yesmaximal, characteristic, complemented
8#20276046C211yesmaximal, characteristic, complemented
9#20275475C211yesmaximal, characteristic, complemented
10#20275291C211yesmaximal, characteristic, complemented
11#20275136C211yesmaximal, characteristic
12#20275020C211yesmaximal, characteristic
13#20274984C211yesmaximal, characteristic, complemented
14#20274892C211yesmaximal, characteristic
15C53×#1853C211yesmaximal, characteristic
16C23×#88322C2211yescharacteristic
17C2×#13133216C2211yescharacteristic
18C2×#13133196C2211yescharacteristic
19C2×#13133194—21no—
20C2×#13133182C2211yescharacteristic
21C22×C4⋊Dic53C2211yescharacteristic
22#16810907—24no—
23C2×#13133148—24no—
24C2×#13133148C2211yescharacteristic
25C2×#13133148C2211yescharacteristic
26C2×#13133148C2211yescharacteristic
27C2×#13133148C2211yescharacteristic
28C2×#13133146C2211yescharacteristic
29C2×#13133146C2211yescharacteristic
30C2×#13133142C2211yescharacteristic
31C2×#13133142C2211yescharacteristic
32C2×#13133140C2211yescharacteristic
33C2×#13133140C2211yescharacteristic
34#16810557—21no—
35#16810544—21no—
36#16810528C2211yescharacteristic
37#16810522C2211yescharacteristic
38#16810519C2211yescharacteristic
39#16810519C2211yescharacteristic
40#16810518—21no—
41#16810504—21no—
42#16810496—21no—
43#16810495—21no—
44#16810492C2211yescharacteristic
45C4⋊C4×Dic53C2211yescharacteristic
46C4×C22⋊Dic53—21no—
47C4×C22⋊Dic53C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810377C2211yescharacteristic
53#16810377C2211yescharacteristic
54C22×C4⋊C4×C53C2211yescharacteristic
55C2×#106×C53C2211yescharacteristic
56C2×#106×C53C2211yescharacteristic
57C2×C22⋊C8×C53C2211yescharacteristic
58C53×#380C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C53×#340C2211yescharacteristic
61C22×Dic106—21no—
62C22×Dic106C2311yescharacteristic
63C23×Dic53—21no—
64C22×#88322—22no—
65C22×#88322—22no—
66C22×#88322—22no—
67C22×#88322C2311yescharacteristic
68C22×#88322C2311yescharacteristic
69#13133216—22no—
70#13133216—41no—
71#13133196—22no—
72#13133196Q814yescomplemented
73#13133194—44no—
74#13133182—22no—
75#13133182—41no—
76C2×C4⋊Dic53—21no—
77C2×C4⋊Dic53—21no—
78C2×C4⋊Dic53—22no—
79C2×C4⋊Dic53—41no—
80C2×C4⋊Dic53C2311yescharacteristic
81C2×C4⋊Dic53C2311yescharacteristic
82C2×C4⋊Dic53C2311yescharacteristic
83C2×#11719919—21no—
84C2×#11719919—21no—
85C2×#11719919—21no—
86C2×#11719919—21no—
87C2×#11719919—21no—
88C2×#11719919—21no—
89C2×#11719919—21no—
90C2×C22⋊Dic53—21no—
91C2×C22⋊Dic53D411yescharacteristic
92C2×C22⋊Dic53D411yescharacteristic
93C2×C4×Dic53—21no—
94C2×C4×Dic53—21no—
95C2×C4×Dic53C2311yescharacteristic
96C2×C4×Dic53C2311yescharacteristic
97#13133148—22no—
98#13133148—22no—
99#13133148—22no—
100#13133148D414yes—
101#13133146—22no—
102#13133146—22no—
103#13133142—22no—
104#13133142—41no—
105#13133140—22no—
106#13133140—22no—
107#13133124—21no—
108#13133124—21no—
109#13133124—21no—
110#13133124—21no—
111#13133124—21no—
112#13133124—21no—
113#13133124D411yescharacteristic
114#13133124D411yescharacteristic
115C23×C4×C53—21no—
116C23×C4×C53C2311yescharacteristic
117C22×C8×C53C2311yescharacteristic
118C22×C8×C53C2311yescharacteristic
119C2×C4⋊C4×C53—21no—
120C2×C4⋊C4×C53—22no—
121C2×C4⋊C4×C53—41no—
122C2×C4⋊C4×C53C2311yescharacteristic
123C2×C4⋊C4×C53C2311yescharacteristic
124C2×C4⋊C4×C53C2311yescharacteristic
125C2×C4⋊C4×C53C2311yescharacteristic
126C2×C22⋊C4×C53—21no—
127#106×C53—22no—
128#106×C53—41no—
129C22⋊C8×C53—22no—
130#95×C53—21no—
131C24×C53C2×D411yescharacteristic
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—42no—
141C22×C4×C53—42no—
142C22×C4×C53—81no—
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×Q812yes—
145C22×C4×C53#4112yes—
146C22×C4×C53C2411yescharacteristic
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—42no—
152C22×Dic53—42no—
153C22×Dic53—81no—
154C22×Dic53C2×D411yescharacteristic
155C22×Dic53C2×D411yescharacteristic
156C22×Dic53C2×D411yescharacteristic
157C22×Dic53C2×D411yescharacteristic
158C2×Dic106—21no—
159C2×Dic106—21no—
160C2×Dic106—21no—
161C2×Dic106—21no—
162C2×Dic106—21no—
163C2×Dic106—42no—
164C2×Dic106—42no—
165C2×Dic106—81no—
166C2×Dic106C2×Q812yes—
167C2×C8×C53—21no—
168C2×C8×C53—21no—
169C2×C8×C53—21no—
170C2×C8×C53—21no—
171C2×C8×C53—21no—
172C2×C8×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53—21no—
177C4⋊C4×C53—21no—
178C4⋊C4×C53—21no—
179C4⋊C4×C53—21no—
180C4⋊C4×C53—42no—
181C4⋊C4×C53C2×D412yes—
182C22⋊C4×C53—42no—
183C22⋊Dic53—24no—
184C22⋊Dic53—22no—
185C22⋊Dic53SD1614yes—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—42no—
191C4⋊Dic53—42no—
192C4⋊Dic53C2×D412yes—
193C4⋊Dic53C2×Q812yes—
194#11719919—24no—
195#11719919—42no—
196#11719919—42no—
197#11719919—42no—
198#11719919—42no—
199#11719919—81no—
200#11719919—81no—
201C4×Dic53—21no—
202C4×Dic53—21no—
203C4×Dic53—21no—
204C4×Dic53—21no—
205C4×Dic53—42no—
206C4×Dic53—81no—
207C2×#88322—28no—
208C2×#88322—21no—
209C2×#88322—21no—
210C2×#88322—21no—
211C2×#88322—21no—
212C2×#88322C2×D412yes—
213C2×#88322#4112yes—
214C23×C53—21no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53C22⋊Q812yes—
219C23×C53C2×SD1612yes—
220C23×C53#13712yes—
221C23×C53C22×D411yescharacteristic
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—22no—
225C2×C4×C53—22no—
226C2×C4×C53—22no—
227C2×C4×C53—22no—
228C2×C4×C53—22no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—84no—
241C2×C4×C53C22⋊Q812yes—
242C2×C4×C53C22×D411yescharacteristic
243C2×C4×C53C22×Q811yescharacteristic
244C2×C4×C53C2×#4111yescharacteristic
245C2×C4×C53C2×#4111yescharacteristic
246C2×C4×C53H(5,𝔽2)11yescharacteristic
247C2×C4×C532−1+411yescharacteristic
248C2×Dic53—21no—
249C2×Dic53—21no—
250C2×Dic53—21no—
251C2×Dic53—21no—
252C2×Dic53—22no—
253C2×Dic53—22no—
254C2×Dic53—22no—
255C2×Dic53—21no—
256C2×Dic53—21no—
257C2×Dic53—21no—
258C2×Dic53—21no—
259C2×Dic53—21no—
260C2×Dic53—21no—
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×Dic53C2×SD1612yes—
272C2×Dic53C2×SD1612yes—
273Dic106—22no—
274Dic106—22no—
275Dic106—41no—
276Dic106—41no—
277Dic106—84no—
278C8×C53—41no—
279C8×C53—41no—
280#88322—22no—
281#88322—22no—
282#26340—531nomaximal, Sylow
283C22×C53—22no—
284C22×C53—22no—
285C22×C53—22no—
286C22×C53—22no—
287C22×C53—22no—
288C22×C53—22no—
289C22×C53#47514yescomplemented
290C22×C53C2×C22⋊Q811yescharacteristic
291C22×C53D4211yescharacteristic
292C22×C53#54711yescharacteristic
293C22×C53C22×SD1611yescharacteristic
294C22×C53C2×#13711yescharacteristic
295C22×C53#57711yescharacteristic
296C22×C53#57711yescharacteristic
297C4×C53—22no—
298C4×C53—42no—
299C4×C53—81no—
300C4×C53—81no—
301C4×C53—81no—
302C4×C53#52511yescharacteristic
303C4×C53#53911yescharacteristic
304C4×C53D4×Q811yescharacteristic
305C4×C53#55411yescharacteristic
306Dic53—22no—
307Dic53—22no—
308Dic53—42no—
309Dic53—41no—
310Dic53—42no—
311Dic53—41no—
312Dic53—42no—
313Dic53—81no—
314C2×#475—531no—
315C22×#106—531no—
316#2440—531no—
317#2360—531no—
318#2096—531no—
319#2088—531no—
320#2066—531no—
321#2058—531no—
322#1948—531no—
323#1919—531no—
324#1892—531no—
325#1872—531no—
326#1865—531no—
327#1853D5311yescharacteristic
328#1850—531no—
329C2×C53—24no—
330C2×C53#244911yescharacteristic
331C2×C53C2×#47511yescharacteristic
332C2×C53#308411yescharacteristic
333C2×C53#326611yescharacteristic
334C2×C53D4×SD1611yescharacteristic
335C2×C53#333511yescharacteristic
336C2×C53#334411yescharacteristic
337C23×C8—531no—
338C2×C4⋊Q8—531no—
339C2×C22⋊Q8—531no—
340C2×C22⋊Q8—1061no—
341C2×C4×Q8—531no—
342C22×C4⋊C4—531no—
343C22×C4⋊C4C2×D5311yescharacteristic
344#475—1064no—
345C2×#106—531no—
346C2×#106—531no—
347C2×#106—531no—
348C2×#106—531no—
349C2×#106—1064no—
350C2×#106C2×D5311yescharacteristic
351C2×#106C2×D5311yescharacteristic
352C2×C4⋊C8—531no—
353C2×C4⋊C8—531no—
354C2×#103—531no—
355C2×#103—531no—
356C2×#103—531no—
357C2×#103—531no—
358C2×C22⋊C8C2×D5311yescharacteristic
359#398—1061no—
360#395—1061no—
361#391—531no—
362#389—531no—
363#389—531no—
364#389—531no—
365#389—1061no—
366#387—1061no—
367#385—1061no—
368#385—1061no—
369#384—531no—
370#380C2×D5311yescharacteristic
371C4×C4⋊C4—531no—
372C4×C22⋊C4—531no—
373C4×C22⋊C4—1061no—
374#340—531no—
375#340—531no—
376#340—531no—
377#340—531no—
378#340C2×D5311yescharacteristic
379#340C2×D5311yescharacteristic
380#336—531no—
381#336—531no—
382C53#2634011yescharacteristic, complemented, Sylow
383C22×Q8—531no—
384C22×Q8—1061no—
385C23×C4—21no—
386C23×C4—1061no—
387C23×C4C22×D5311yescharacteristic
388C22×C8—531no—
389C22×C8—531no—
390C22×C8—1062no—
391C22×C8—1062no—
392C22×C8—1062no—
393C22×C8C22×D5311yescharacteristic
394C22×C8C22×D5311yescharacteristic
395C4⋊Q8—1062no—
396C4⋊Q8—2121no—
397C22⋊Q8—534no—
398C22⋊Q8—1062no—
399C22⋊Q8—2124no—
400C4×Q8—1062no—
401C4×Q8—2121no—
402C2×C4⋊C4—22no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—41no—
405C2×C4⋊C4—531no—
406C2×C4⋊C4—531no—
407C2×C4⋊C4—531no—
408C2×C4⋊C4—1062no—
409C2×C4⋊C4—1061no—
410C2×C4⋊C4—1061no—
411C2×C4⋊C4—1061no—
412C2×C4⋊C4—1061no—
413C2×C4⋊C4—1061no—
414C2×C4⋊C4—1061no—
415C2×C4⋊C4—1061no—
416C2×C4⋊C4—1061no—
417C2×C4⋊C4—1061no—
418C2×C4⋊C4—2121no—
419C2×C4⋊C4C22×D5311yescharacteristic
420C2×C4⋊C4C22×D5311yescharacteristic
421C2×C4⋊C4C22×D5311yescharacteristic
422C2×C4⋊C4C22×D5311yescharacteristic
423C2×C22⋊C4—21no—
424C2×C22⋊C4—531no—
425C2×C22⋊C4—531no—
426C2×C22⋊C4—1061no—
427C2×C42—531no—
428C2×C42—531no—
429C2×C42—1061no—
430C2×C42—1061no—
431#106—22no—
432#106—41no—
433#106—534no—
434#106—1062no—
435#106—1062no—
436#106—1062no—
437C4⋊C8—1062no—
438C4⋊C8—1062no—
439#103—1062no—
440#103—1062no—
441#103—1062no—
442#103—2121no—
443C22⋊C8—22no—
444#95—21no—
445#95—531no—
446#95—531no—
447#95—1061no—
448#95—1061no—
449#95—1061no—
450#95—1061no—
451#95—1061no—
452#95—1061no—
453C24D4×D5311yescharacteristic
454C2×Q8—532no—
455C2×Q8—1061no—
456C2×Q8—1061no—
457C2×Q8—1061no—
458C2×Q8—1061no—
459C2×Q8—1061no—
460C2×Q8—4241no—
461C2×Q8—2122no—
462C2×Q8—2122no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—81no—
474C22×C4—531no—
475C22×C4—531no—
476C22×C4—531no—
477C22×C4—531no—
478C22×C4—1061no—
479C22×C4—1061no—
480C22×C4—1061no—
481C22×C4—1061no—
482C22×C4—4241no—
483C22×C4—2122no—
484C22×C4—2122no—
485C22×C4#1171994512yes—
486C22×C4D4×D5311yescharacteristic
487C22×C4Q8×D5312yes—
488C22×C4C23×D5311yescharacteristic
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—532no—
496C2×C8—532no—
497C2×C8—1068no—
498C2×C8—1061no—
499C2×C8—1061no—
500C2×C8—1061no—
501C2×C8—1061no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—42no—
510C4⋊C4—532no—
511C4⋊C4—532no—
512C4⋊C4—1064no—
513C4⋊C4—1061no—
514C4⋊C4—1061no—
515C4⋊C4—1061no—
516C4⋊C4—1061no—
517C4⋊C4—4241no—
518C4⋊C4—4241no—
519C4⋊C4—2122no—
520C4⋊C4—2122no—
521C4⋊C4—2122no—
522C4⋊C4—2122no—
523C4⋊C4—2122no—
524C4⋊C4—2122no—
525C4⋊C4D4×D5312yes—
526C22⋊C4—42no—
527C22⋊C4—534no—
528C22⋊C4—1064no—
529C22⋊C4—1062no—
530C42—1061no—
531C42—1061no—
532C42—1061no—
533C42—1061no—
534C42—4241no—
535C42—2122no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23#1313319512yes—
541C23SD16×D5312yes—
542C23#1313324612yes—
543C23C2×D4×D5311yescharacteristic
544Q8—1062no—
545Q8—1062no—
546Q8—4244no—
547Q8—2121no—
548Q8—2121no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—84no—
568C2×C4—532no—
569C2×C4—532no—
570C2×C4—1061no—
571C2×C4—1061no—
572C2×C4—1061no—
573C2×C4—1061no—
574C2×C4—1061no—
575C2×C4—1062no—
576C2×C4—1062no—
577C2×C4—1061no—
578C2×C4—1061no—
579C2×C4—1061no—
580C2×C4—1061no—
581C2×C4—1061no—
582C2×C4—1062no—
583C2×C4—4244no—
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#1313319512yes—
594C2×C4C2×D4×D5311yescharacteristic
595C2×C4C2×Q8×D5311yescharacteristic
596C2×C4#41×D5311yescharacteristic
597C2×C4C2×#1171994511yescharacteristic
598C2×C4#1313326211yescharacteristic
599C2×C4#1313326411yescharacteristic
600C8—41no—
601C8—41no—
602C8—1062no—
603C8—1062no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22#1681090714yescomplemented
611C22C2×#1313319511yescharacteristic
612C22#1681125511yescharacteristic
613C22#1681127111yescharacteristic
614C22C2×SD16×D5311yescharacteristic
615C22C2×#1313324611yescharacteristic
616C22#1681143211yescharacteristic
617C22#1681143311yescharacteristic
618C4—22no—
619C4—42no—
620C4—81no—
621C4—81no—
622C4—81no—
623C4—1062no—
624C4—1062no—
625C4—4241no—
626C4—2121no—
627C4—2122no—
628C4—2122no—
629C4—2121no—
630C4—2122no—
631C4#1681113211yescharacteristic
632C4#1681121511yescharacteristic
633C4#1681128611yescharacteristic
634C4#1681132711yescharacteristic
635C2—24no—
636C2#2027852611yescharacteristic
637C2C2×#1681090711yescharacteristic
638C2#2028457611yescharacteristic
639C2#2028622911yescharacteristic
640C2#2028640411yescharacteristic
641C2#2028707011yescharacteristic
642C2#2028717411yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1853C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#1853D53
hypocenterC53#26340
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1853

Complements

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

NH ≅ G/Ntype
1. C2×#16810907C2semidirect
5. #20276255C2semidirect
6. #20276219C2semidirect
7. #20276097C2semidirect
8. #20276046C2semidirect
9. #20275475C2semidirect
10. #20275291C2semidirect
13. #20274984C2semidirect
72. #13133196Q8semidirect
289. C22×C53#475semidirect
382. C53#26340semidirect
610. C22#16810907semidirect

Maximal subgroups

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

subgroupindex
C2×#168109072
C22×#131331482
#202784182
#202777202
#202762552
#202762192
#202760972
#202760462
#202754752
#202752912
#202751362
#202750202
#202749842
#202748922
C53×#18532
#2634053