Finite Group Databasebeta

← #27387161

Subgroups — #27387161

Order 13568.

890 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810720C211yesmaximal, characteristic
2#20277362C211yesmaximal, characteristic, complemented
3#20277076C211yesmaximal, characteristic
4#20275355C211yesmaximal, characteristic, complemented
5#20275350C211yesmaximal, characteristic, complemented
6#20275304C211yesmaximal, characteristic, complemented
7#20275289C211yesmaximal, characteristic, complemented
8#20275049C211yesmaximal, characteristic
9#20275047C211yesmaximal, characteristic
10#20274978C211yesmaximal, characteristic
11#20274614C211yesmaximal, characteristic, complemented
12#20274534C211yesmaximal, characteristic, complemented
13#102×Dic53C211yesmaximal, characteristic
14#20274465C211yesmaximal, characteristic, complemented
15C53×#1862C211yesmaximal, characteristic
16C22×#11719923C2211yescharacteristic
17C2×#13133182C2211yescharacteristic
18C2×D4×Dic53C2211yescharacteristic
19C2×D4×Dic53C2211yescharacteristic
20#16810720C418yescomplemented
21C2×#13133150C2211yescharacteristic
22C2×#13133149C2211yescharacteristic
23C2×#13133142C2211yescharacteristic
24C2×#13133142C2211yescharacteristic
25C2×#13133140C2211yescharacteristic
26C2×#13133138C2211yescharacteristic
27C2×#13133127C2211yescharacteristic
28C2×C8×Dic53C2211yescharacteristic
29#16810529C2211yescharacteristic
30#16810528C2211yescharacteristic
31#16810522C2211yescharacteristic
32#16810510—21no—
33#16810504—21no—
34#16810490—21no—
35#16810488—21no—
36C4⋊C4×Dic53C2211yescharacteristic
37C4⋊C4×Dic53C2211yescharacteristic
38C4⋊C4×Dic53C2211yescharacteristic
39C4×C4⋊Dic53C2211yescharacteristic
40C22⋊C4×Dic53—21no—
41C22⋊C4×Dic53—21no—
42C4×C22⋊Dic53—21no—
43#16810469C2211yescharacteristic
44C2×#13133124—21no—
45#16810386C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810378C2211yescharacteristic
51#16810377C2211yescharacteristic
52#16810377C2211yescharacteristic
53C2×C4×D4×C53C2211yescharacteristic
54C2×#107×C53C2211yescharacteristic
55C2×#102×C53C2211yescharacteristic
56C2×#102×C53C2211yescharacteristic
57C53×#389C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#336C2211yescharacteristic
60C22×Dic106C2311yescharacteristic
61C23×Dic53—21no—
62C23×Dic53—21no—
63C2×#11719923—22no—
64C2×#11719923—22no—
65C2×#11719923C2×C414yes—
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68#13133182—22no—
69#13133182C2×C414yescomplemented
70D4×Dic53—22no—
71D4×Dic53—22no—
72D4×Dic53—22no—
73D4×Dic53C2×C414yes—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×#11719919—21no—
79C2×C22⋊Dic53—21no—
80C2×C22⋊Dic53—21no—
81C2×C4×Dic53—21no—
82C2×C4×Dic53—21no—
83C2×C4×Dic53—21no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53—21no—
86C2×C4×Dic53C2311yescharacteristic
87C2×C4×Dic53C2311yescharacteristic
88#13133150C2×C414yescomplemented
89#13133149—22no—
90#13133142—22no—
91#13133142C2×C414yescomplemented
92#13133140—22no—
93#13133138—22no—
94#13133127C2×C414yescomplemented
95C8×Dic53—22no—
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#13133124—21no—
105#13133124D411yescharacteristic
106#13133124D411yescharacteristic
107#13133124D411yescharacteristic
108#13133124D411yescharacteristic
109C22×D4×C53C2311yescharacteristic
110C23×C4×C53—21no—
111C22×C8×C53C2311yescharacteristic
112C22×C8×C53C2311yescharacteristic
113C4×D4×C53—22no—
114C4×D4×C53—22no—
115C2×C4⋊C4×C53—21no—
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C22⋊C4×C53—21no—
120C2×C42×C53C2311yescharacteristic
121#107×C53—22no—
122#102×C53—22no—
123#102×C53C2×C414yes—
124#95×C53—21no—
125C24×C53—21no—
126C2×D4×C53—21no—
127C2×D4×C53—21no—
128C2×D4×C53—21no—
129C2×D4×C53C22×C412yes—
130C2×D4×C53#4112yes—
131C2×D4×C53#4112yes—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—24no—
135C22×C4×C53—21no—
136C22×C4×C53—81no—
137C22×C4×C53—81no—
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2411yescharacteristic
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—24no—
145C22×Dic53—24no—
146C22×Dic53—21no—
147C22×Dic53—81no—
148C22×Dic53—81no—
149C22×Dic53—81no—
150C22×Dic53—81no—
151C22×Dic53C2×D411yescharacteristic
152C22×Dic53C2×D411yescharacteristic
153C22×Dic53C2×D411yescharacteristic
154C22×Dic53C2×D411yescharacteristic
155C2×Dic106—21no—
156C2×Dic106—21no—
157C2×Dic106—21no—
158C2×Dic106—21no—
159C2×Dic106—21no—
160C2×Dic106C22×C412yes—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C2×C8×C53—21no—
165C2×C8×C53—21no—
166C2×C8×C53C22×C412yes—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—81no—
172C4⋊C4×C53C22×C412yes—
173C4⋊C4×C53#4112yes—
174C22⋊C4×C53—24no—
175C42×C53—21no—
176C42×C53—21no—
177C22⋊Dic53—24no—
178C22⋊Dic53—24no—
179#11719923—28no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—81no—
185C4⋊Dic53C22×C412yes—
186C4⋊Dic53#4112yes—
187#11719919—24no—
188C4×Dic53—21no—
189C4×Dic53—21no—
190C4×Dic53—24no—
191C4×Dic53—81no—
192C4×Dic53—81no—
193C4×Dic53—81no—
194C4×Dic53—81no—
195C4×Dic53C22×C412yes—
196C4×Dic53#4112yes—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322C22×C412yes—
203C23×C53—22no—
204C23×C53—22no—
205C23×C53—22no—
206C23×C53—41no—
207C23×C53—41no—
208C23×C53—41no—
209C23×C53—41no—
210C23×C53C22×D411yescharacteristic
211D4×C53—22no—
212D4×C53—22no—
213D4×C53—22no—
214D4×C53#11714yes—
215C2×C4×C53—22no—
216C2×C4×C53—21no—
217C2×C4×C53—22no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—84no—
231C2×C4×C53C23×C411yescharacteristic
232C2×C4×C53C2×#4111yescharacteristic
233C2×C4×C53C2×#4111yescharacteristic
234C2×C4×C53C2×#4111yescharacteristic
235C2×C4×C53C2×#4111yescharacteristic
236C2×C4×C53C2×#4111yescharacteristic
237C2×Dic53—21no—
238C2×Dic53—21no—
239C2×Dic53—21no—
240C2×Dic53—21no—
241C2×Dic53—22no—
242C2×Dic53—22no—
243C2×Dic53—21no—
244C2×Dic53—22no—
245C2×Dic53—21no—
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—21no—
249C2×Dic53—21no—
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—84no—
260C2×Dic53—84no—
261C2×Dic53C4×D412yes—
262C2×Dic53C4×D412yes—
263Dic106—22no—
264Dic106—22no—
265Dic106—41no—
266Dic106—41no—
267C8×C53—22no—
268C8×C53—41no—
269#88322—22no—
270#88322—41no—
271#21125—531nomaximal, Sylow
272C22×C53—24no—
273C22×C53—42no—
274C22×C53—42no—
275C22×C53—42no—
276C22×C53—42no—
277C22×C53—42no—
278C22×C53—42no—
279C22×C53C2×C4×D411yescharacteristic
280C22×C53#52511yescharacteristic
281C22×C53#52511yescharacteristic
282C22×C53#57611yescharacteristic
283C22×C53#57711yescharacteristic
284C22×C53#57711yescharacteristic
285C22×C53#57811yescharacteristic
286C4×C53—41no—
287C4×C53—42no—
288C4×C53—41no—
289C4×C53—42no—
290C4×C53—81no—
291C4×C53C2×#11711yescharacteristic
292C4×C53C4×#4111yescharacteristic
293C4×C53#52911yescharacteristic
294C4×C53#52911yescharacteristic
295Dic53—22no—
296Dic53—22no—
297Dic53—41no—
298Dic53—42no—
299Dic53—41no—
300Dic53—42no—
301Dic53—42no—
302Dic53—81no—
303C2×#440—531no—
304#2320—531no—
305#2275—531no—
306#1928—531no—
307#1927—531no—
308#1921—531no—
309#1918—531no—
310#1877—531no—
311#1877—531no—
312#1864—531no—
313#1862D5311yescharacteristic
314#1776—531no—
315C4×#103—531no—
316C4×#102—531no—
317#1746—531no—
318C2×C53—44no—
319C2×C53#229011yescharacteristic
320C2×C53#294611yescharacteristic
321C2×C53#294711yescharacteristic
322C2×C53#311811yescharacteristic
323C2×C53#312011yescharacteristic
324C2×C53#312311yescharacteristic
325C2×C53#312411yescharacteristic
326C22×SD16—531no—
327C2×C4×Q8—531no—
328C2×C4×D4—531no—
329C2×C4×D4—531no—
330C2×C4×D4C2×D5311yescharacteristic
331#440—538no—
332C2×#107—531no—
333C2×#107C2×D5311yescharacteristic
334C2×#106—531no—
335C2×#103—531no—
336C2×#103—531no—
337C2×#103—531no—
338C2×#102—531no—
339C2×#102C2×D5311yescharacteristic
340C2×#102C2×D5311yescharacteristic
341C2×#97—531no—
342C2×C4×C8—531no—
343#391—531no—
344#391—531no—
345#389—531no—
346#389C2×D5311yescharacteristic
347#387—1061no—
348#387—1061no—
349#383—1061no—
350#382—1061no—
351C4×C4⋊C4—531no—
352C4×C4⋊C4—531no—
353C4×C4⋊C4—531no—
354C4×C4⋊C4—531no—
355C4×C22⋊C4—1061no—
356C4×C22⋊C4—1061no—
357C4×C22⋊C4—1061no—
358#376—531no—
359C2×#95—1061no—
360#340—531no—
361#340—531no—
362#340—531no—
363#340—531no—
364#340—531no—
365#340C2×D5311yescharacteristic
366#336—531no—
367#336—531no—
368#336—531no—
369#336C2×D5311yescharacteristic
370C53#2112511yescharacteristic, complemented, Sylow
371C22×Q8—531no—
372C22×D4C22×D5311yescharacteristic
373C23×C4—21no—
374C23×C4—1061no—
375C23×C4—1061no—
376C2×SD16—534no—
377C2×SD16—1062no—
378C2×SD16—1062no—
379C22×C8—531no—
380C22×C8—531no—
381C22×C8C22×D5311yescharacteristic
382C22×C8C22×D5311yescharacteristic
383C4×Q8—534no—
384C4×Q8—1062no—
385C4×D4—22no—
386C4×D4—22no—
387C4×D4—534no—
388C4×D4—1062no—
389C4×D4—1062no—
390C4×D4—1062no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—531no—
393C2×C4⋊C4—531no—
394C2×C4⋊C4—531no—
395C2×C4⋊C4—1061no—
396C2×C4⋊C4—1061no—
397C2×C4⋊C4C22×D5311yescharacteristic
398C2×C4⋊C4C22×D5311yescharacteristic
399C2×C4⋊C4C22×D5311yescharacteristic
400C2×C22⋊C4—21no—
401C2×C22⋊C4—1061no—
402C2×C22⋊C4—1061no—
403C2×C42—531no—
404C2×C42—531no—
405C2×C42—1061no—
406C2×C42—1061no—
407C2×C42—1061no—
408C2×C42—1061no—
409C2×C42—1061no—
410C2×C42C22×D5311yescharacteristic
411#107—22no—
412#107—534no—
413#106—1062no—
414#103—534no—
415#103—1062no—
416#103—1062no—
417#102—22no—
418#102—1062no—
419#102C4×D5314yes—
420#97—534no—
421C4×C8—1062no—
422#95—21no—
423#95—531no—
424#95—531no—
425#95—531no—
426#95—531no—
427#95—1061no—
428#95—1061no—
429#95—1061no—
430#95—1061no—
431#95—1061no—
432#95—1061no—
433#95—1061no—
434#95—1061no—
435#95—1061no—
436C24—21no—
437C2×Q8—532no—
438C2×Q8—1061no—
439C2×Q8—1061no—
440C2×Q8—1061no—
441C2×Q8—1061no—
442C2×Q8—1061no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4C2×C4×D5312yes—
447C2×D4#1171994712yes—
448C2×D4#1171994912yes—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—24no—
452C22×C4—21no—
453C22×C4—81no—
454C22×C4—81no—
455C22×C4—531no—
456C22×C4—531no—
457C22×C4—531no—
458C22×C4—531no—
459C22×C4—1064no—
460C22×C4—1064no—
461C22×C4—1061no—
462C22×C4—1061no—
463C22×C4—1061no—
464C22×C4—1061no—
465C22×C4—4241no—
466C22×C4—4241no—
467C22×C4—4241no—
468C22×C4—4241no—
469C22×C4D4×D5311yescharacteristic
470C22×C4D4×D5311yescharacteristic
471C22×C4C23×D5311yescharacteristic
472SD16—1068no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—532no—
479C2×C8—1061no—
480C2×C8—1061no—
481C2×C8—1061no—
482C2×C8—1061no—
483C2×C8—1061no—
484C2×C8C2×C4×D5312yes—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—81no—
490C4⋊C4—532no—
491C4⋊C4—532no—
492C4⋊C4—1064no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—1061no—
496C4⋊C4—1061no—
497C4⋊C4—4241no—
498C4⋊C4C2×C4×D5312yes—
499C4⋊C4#1171994712yes—
500C22⋊C4—24no—
501C22⋊C4—1064no—
502C22⋊C4—1064no—
503C42—21no—
504C42—21no—
505C42—532no—
506C42—532no—
507C42—1064no—
508C42—1061no—
509C42—1061no—
510C42—4241no—
511C42—4241no—
512C42—4241no—
513C42—4241no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23C2×D4×D5311yescharacteristic
522Q8—1062no—
523Q8—1062no—
524Q8—2121no—
525Q8—2121no—
526D4—22no—
527D4—22no—
528D4—22no—
529D4#1313317514yes—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—84no—
546C2×C4—532no—
547C2×C4—532no—
548C2×C4—1061no—
549C2×C4—1061no—
550C2×C4—1061no—
551C2×C4—1061no—
552C2×C4—1062no—
553C2×C4—1062no—
554C2×C4—1061no—
555C2×C4—1061no—
556C2×C4—1062no—
557C2×C4—1061no—
558C2×C4—1061no—
559C2×C4—1061no—
560C2×C4—1061no—
561C2×C4—4244no—
562C2×C4—4244no—
563C2×C4—2121no—
564C2×C4—2121no—
565C2×C4—2121no—
566C2×C4—2121no—
567C2×C4—2121no—
568C2×C4—2121no—
569C2×C4—2121no—
570C2×C4—2121no—
571C2×C4—2121no—
572C2×C4C22×C4×D5311yescharacteristic
573C2×C4#41×D5311yescharacteristic
574C2×C4#41×D5311yescharacteristic
575C2×C4C2×#1171994711yescharacteristic
576C2×C4C2×#1171994711yescharacteristic
577C2×C4C2×#1171994911yescharacteristic
578C8—22no—
579C8—41no—
580C8—1062no—
581C8—2121no—
582C22—24no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22C4×D4×D5311yescharacteristic
590C22#1681112911yescharacteristic
591C22#1681113611yescharacteristic
592C22#1681142411yescharacteristic
593C22#1681143011yescharacteristic
594C22#1681143311yescharacteristic
595C22#1681143911yescharacteristic
596C4—42no—
597C4—41no—
598C4—42no—
599C4—41no—
600C4—81no—
601C4—1062no—
602C4—1062no—
603C4—4241no—
604C4—2121no—
605C4—2122no—
606C4—2121no—
607C4—2122no—
608C4—2122no—
609C4C2×#1313317511yescharacteristic
610C4C4×#1171994711yescharacteristic
611C4#1681115011yescharacteristic
612C4#1681115911yescharacteristic
613C2—44no—
614C2#2027715111yescharacteristic
615C2#2028354011yescharacteristic
616C2#2028355511yescharacteristic
617C2#2028489211yescharacteristic
618C2#2028492211yescharacteristic
619C2#2028496911yescharacteristic
620C2#2028498211yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1862C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
cosocleC22×C4×C53C24
hypercenter#1862D53
hypocenterC53#21125
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1862

Complements

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

NH ≅ G/Ntype
2. #20277362C2semidirect
4. #20275355C2semidirect
5. #20275350C2semidirect
6. #20275304C2semidirect
7. #20275289C2semidirect
11. #20274614C2semidirect
12. #20274534C2semidirect
14. #20274465C2semidirect
20. #16810720C4semidirect
69. #13133182C2×C4semidirect
88. #13133150C2×C4semidirect
91. #13133142C2×C4semidirect
94. #13133127C2×C4semidirect
370. C53#21125semidirect

Maximal subgroups

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

subgroupindex
C2×#168107202
#202773622
#202770762
#202753552
#202753502
#202753042
#202752892
#202750492
#202750472
#202749782
#202746142
#202745342
#102×Dic532
#202744652
C53×#18622
#2112553