Finite Group Databasebeta

← #27413099

Subgroups — #27413099

Order 13568.

870 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810664C211yesmaximal, characteristic
2C2×#107×D53C211yesmaximal, characteristic
3#20277336C211yesmaximal, characteristic
4#20277330C211yesmaximal, characteristic
5#20276201C211yesmaximal, characteristic, complemented
6#20276199C211yesmaximal, characteristic, complemented
7#20276198C211yesmaximal, characteristic, complemented
8#20276196C211yesmaximal, characteristic, complemented
9#20276190C211yesmaximal, characteristic, complemented
10#20276189C211yesmaximal, characteristic, complemented
11#20276187C211yesmaximal, characteristic, complemented
12#20274433C211yesmaximal, characteristic
13#20274430C211yesmaximal, characteristic
14#20274410C211yesmaximal, characteristic
15C53×#2083C211yesmaximal, characteristic, complemented
16C22×#11719912C2211yescharacteristic
17C22×C8×D53C2211yescharacteristic
18C2×#13133175C2211yescharacteristic
19C2×#13133175C2211yescharacteristic
20C2×#13133174C2211yescharacteristic
21C2×C4⋊C4×D53C2211yescharacteristic
22C2×C4⋊C4×D53C2211yescharacteristic
23#16810664—24no—
24C2×#13133151C2211yescharacteristic
25C2×#13133150C2211yescharacteristic
26C2×#13133150C2211yescharacteristic
27C2×#13133150C2211yescharacteristic
28#107×D53—24no—
29C2×#13133149C2211yescharacteristic
30C2×#13133148C2211yescharacteristic
31#16810522C2211yescharacteristic
32#16810521C2211yescharacteristic
33#16810520—21no—
34#16810519C2211yescharacteristic
35#16810518—21no—
36#16810518—21no—
37#16810518—21no—
38#16810494—21no—
39#16810494—21no—
40#16810492C2211yescharacteristic
41#16810489C2211yescharacteristic
42#16810488C2211yescharacteristic
43#16810482—21no—
44#16810480—21no—
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810378C2211yescharacteristic
52#16810377C2211yescharacteristic
53#16810377C2211yescharacteristic
54C2×#107×C53C2211yescharacteristic
55C2×#107×C53C2211yescharacteristic
56C53×#389C2211yescharacteristic
57C53×#384C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C53×#336C2211yescharacteristic
61C22×C4×D53—21no—
62C22×C4×D53—21no—
63C22×C4×D53C2311yescharacteristic
64C2×#11719912—22no—
65C2×#11719912—22no—
66C2×#11719912Q814yes—
67C22×#88322C2311yescharacteristic
68C22×#88322C2311yescharacteristic
69C2×C8×D53—22no—
70C2×C8×D53—22no—
71C2×C8×D53Q814yes—
72#13133175—22no—
73#13133175—22no—
74#13133175—22no—
75#13133175—22no—
76#13133174—22no—
77#13133174—22no—
78C2×C4⋊Dic53—21no—
79C2×C4⋊Dic53C2311yescharacteristic
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×#11719919D411yescharacteristic
90C2×#11719919D411yescharacteristic
91C4⋊C4×D53—22no—
92C4⋊C4×D53—22no—
93C4⋊C4×D53—22no—
94C4⋊C4×D53D414yes—
95C2×#11719921—21no—
96C2×#11719921—21no—
97C2×#11719921D411yescharacteristic
98C2×#11719921D411yescharacteristic
99C2×C4×Dic53—21no—
100C2×C4×Dic53—21no—
101C2×C4×Dic53C2311yescharacteristic
102#13133151—22no—
103#13133150—22no—
104#13133150—22no—
105#13133150—22no—
106#13133149—22no—
107#13133148—22no—
108#13133124—21no—
109#13133124—21no—
110#13133124—21no—
111#13133124—21no—
112#13133124—21no—
113#13133124—21no—
114#13133124—21no—
115C22×C8×C53C2311yescharacteristic
116C22×C8×C53C2311yescharacteristic
117C2×C4⋊C4×C53—21no—
118C2×C4⋊C4×C53—21no—
119C2×C4⋊C4×C53—21no—
120C2×C4⋊C4×C53C2311yescharacteristic
121C2×C4⋊C4×C53C2311yescharacteristic
122C2×C4⋊C4×C53C2311yescharacteristic
123C2×C4⋊C4×C53C2311yescharacteristic
124C2×C42×C53C2311yescharacteristic
125#107×C53—22no—
126#107×C53—22no—
127#95×C53—21no—
128C23×D53C2×D411yescharacteristic
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53C2411yescharacteristic
136C22×Dic53—21no—
137C22×Dic53—21no—
138C22×Dic53—21no—
139C22×Dic53—21no—
140C22×Dic53C2×D411yescharacteristic
141C22×Dic53C2×D411yescharacteristic
142C22×Dic53C2×D411yescharacteristic
143C2×C4×D53—24no—
144C2×C4×D53—42no—
145C2×C4×D53—81no—
146C2×C4×D53—81no—
147C2×C4×D53—81no—
148C2×C4×D53—81no—
149C2×C4×D53C2×D412yes—
150C2×C4×D53C2×Q812yes—
151C2×C4×D53C2×Q812yes—
152C2×C4×D53#4112yes—
153C2×C4×D53#4112yes—
154C2×C4×D53#4112yes—
155C2×C8×C53—21no—
156C2×C8×C53—21no—
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C2×C8×C53C2×Q812yes—
160C2×C8×C53C2×Q812yes—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—81no—
169C4⋊C4×C53—81no—
170C4⋊C4×C53—81no—
171C4⋊C4×C53C2×D412yes—
172C42×C53—21no—
173C42×C53—21no—
174#11719921—22no—
175#11719921—22no—
176#11719921—42no—
177#11719921—42no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—81no—
186C4⋊Dic53C2×D412yes—
187#11719919—24no—
188#11719919—22no—
189#11719919—22no—
190#11719919—42no—
191#11719919—81no—
192#11719919—81no—
193#11719919—81no—
194#11719919—81no—
195C4×Dic53—21no—
196C4×Dic53—21no—
197C4×Dic53—42no—
198C4×Dic53—42no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322C2×Q812yes—
204C2×#88322C2×Q812yes—
205#11719912—28no—
206C8×D53—28no—
207C23×C53C22×D411yescharacteristic
208C22×D53—21no—
209C22×D53—21no—
210C22×D53—21no—
211C22×D53—21no—
212C22×D53#12112yes—
213C22×D53#12312yes—
214C22×D53C4⋊Q812yes—
215C2×C4×C53—22no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—41no—
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—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53C22×D411yescharacteristic
234C2×C4×C53C22×Q811yescharacteristic
235C2×C4×C53C22×Q811yescharacteristic
236C2×C4×C53C2×#4111yescharacteristic
237C2×C4×C53C2×#4111yescharacteristic
238C2×C4×C53C2×#4111yescharacteristic
239C2×Dic53—22no—
240C2×Dic53—21no—
241C2×Dic53—21no—
242C2×Dic53—21no—
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
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#12112yes—
258C2×Dic53#12312yes—
259C2×Dic53C4⋊Q812yes—
260C4×D53—84no—
261C4×D53—84no—
262C4×D53C22⋊Q814yes—
263C4×D53C22⋊Q814yes—
264C4×D53#12514yes—
265C4×D53#12514yes—
266C8×C53—22no—
267C8×C53—22no—
268#88322—22no—
269#88322—22no—
270#23660—531nomaximal, Sylow
271C22×C53C2×#12111yescharacteristic
272C22×C53C2×#12311yescharacteristic
273C22×C53C2×C4⋊Q811yescharacteristic
274C22×C53#57611yescharacteristic
275C22×C53#57611yescharacteristic
276C22×C53#57811yescharacteristic
277C22×C53#57811yescharacteristic
278C2×D53—22no—
279C2×D53—22no—
280C2×D53—22no—
281C2×D53—22no—
282C2×D53—22no—
283C2×D53—22no—
284C2×D53#39814yes—
285C4×C53—41no—
286C4×C53—41no—
287C4×C53—42no—
288C4×C53—81no—
289C4×C53—81no—
290C4×C53—81no—
291C4×C53C2×C22⋊Q811yescharacteristic
292C4×C53C2×C22⋊Q811yescharacteristic
293C4×C53C2×#12511yescharacteristic
294C4×C53C2×#12511yescharacteristic
295Dic53—41no—
296Dic53—41no—
297Dic53—42no—
298Dic53—81no—
299Dic53—81no—
300Dic53—81no—
301Dic53#39814yes—
302C2×#428—531no—
303C22×#107—531no—
304#2317—531no—
305#2316—531no—
306#2085—531no—
307#2085—531no—
308#2085—531no—
309#2085—531no—
310#2083—531no—
311#2083—531no—
312#2083—531no—
313#2083D5311yescharacteristic, complemented
314#1736—531no—
315#1735—531no—
316#1729—531no—
317C2×C53C2×#39811yescharacteristic
318C2×C53#304111yescharacteristic
319C2×C53#304411yescharacteristic
320C2×C53#309811yescharacteristic
321C2×C53#309811yescharacteristic
322C2×C53#316411yescharacteristic
323C2×C53#316411yescharacteristic
324D53—24no—
325C22×M16—531no—
326C23×C8—531no—
327C2×#117—531no—
328C2×#117—531no—
329C2×#117—531no—
330C22×C4⋊C4—531no—
331C22×C4⋊C4—531no—
332#428—1064no—
333C2×#107—531no—
334C2×#107—531no—
335C2×#107—531no—
336C2×#107—531no—
337C2×#107—1064no—
338C2×#107C2×D5311yescharacteristic
339C2×#107C2×D5311yescharacteristic
340C2×#106—531no—
341C2×#106—531no—
342#389—531no—
343#389—531no—
344#389—531no—
345#389—1061no—
346#389—1061no—
347#389—1061no—
348#389—1061no—
349#389C2×D5311yescharacteristic
350#385—1061no—
351#385—1061no—
352#384—531no—
353#384C2×D5311yescharacteristic
354#382—531no—
355#382—531no—
356#380—1061no—
357#380—1061no—
358#340—531no—
359#340—531no—
360#340—531no—
361#340—531no—
362#340—531no—
363#340—531no—
364#340C2×D5311yescharacteristic
365#340C2×D5311yescharacteristic
366#336—531no—
367#336—531no—
368#336—531no—
369#336C2×D5311yescharacteristic
370C53#2366011yescharacteristic, complemented, Sylow
371C23×C4—531no—
372C23×C4—1061no—
373C23×C4—1061no—
374C2×M16—534no—
375C2×M16—1062no—
376C2×M16—1062no—
377C22×C8—531no—
378C22×C8—531no—
379C22×C8—534no—
380C22×C8—1062no—
381C22×C8—1062no—
382C22×C8C22×D5311yescharacteristic
383C22×C8C22×D5311yescharacteristic
384#117—1062no—
385#117—1062no—
386#117—1062no—
387#117—1062no—
388#117—1062no—
389#117—1062no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—531no—
394C2×C4⋊C4—534no—
395C2×C4⋊C4—531no—
396C2×C4⋊C4—531no—
397C2×C4⋊C4—531no—
398C2×C4⋊C4—531no—
399C2×C4⋊C4—531no—
400C2×C4⋊C4—1062no—
401C2×C4⋊C4—1062no—
402C2×C4⋊C4—1062no—
403C2×C4⋊C4—1061no—
404C2×C4⋊C4—1061no—
405C2×C4⋊C4—1061no—
406C2×C4⋊C4—1061no—
407C2×C4⋊C4—1061no—
408C2×C4⋊C4—1061no—
409C2×C4⋊C4—1061no—
410C2×C4⋊C4C22×D5311yescharacteristic
411C2×C4⋊C4C22×D5311yescharacteristic
412C2×C4⋊C4C22×D5311yescharacteristic
413C2×C4⋊C4C22×D5311yescharacteristic
414C2×C22⋊C4—531no—
415C2×C22⋊C4—531no—
416C2×C22⋊C4—1061no—
417C2×C22⋊C4—1061no—
418C2×C42—531no—
419C2×C42—1061no—
420C2×C42—1061no—
421C2×C42C22×D5311yescharacteristic
422#107—22no—
423#107—22no—
424#107—1062no—
425#107—1062no—
426#107—1062no—
427#107—1062no—
428#106—1062no—
429#106—1062no—
430#95—21no—
431#95—1061no—
432#95—1061no—
433#95—1061no—
434#95—1061no—
435#95—1061no—
436#95—1061no—
437#95—1061no—
438C24—531no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—531no—
444C22×C4—532no—
445C22×C4—532no—
446C22×C4—532no—
447C22×C4—532no—
448C22×C4—532no—
449C22×C4—532no—
450C22×C4—531no—
451C22×C4—531no—
452C22×C4—1064no—
453C22×C4—1061no—
454C22×C4—1061no—
455C22×C4—1061no—
456C22×C4—1061no—
457C22×C4—4241no—
458C22×C4—4241no—
459C22×C4—4241no—
460C22×C4—4241no—
461C22×C4—2122no—
462C22×C4D4×D5311yescharacteristic
463C22×C4D4×D5311yescharacteristic
464C22×C4C23×D5311yescharacteristic
465M16—1068no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—532no—
471C2×C8—532no—
472C2×C8—1068no—
473C2×C8—1061no—
474C2×C8—1061no—
475C2×C8—1061no—
476C2×C8—1061no—
477C2×C8Q8×D5312yes—
478C2×C8Q8×D5312yes—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—81no—
487C4⋊C4—81no—
488C4⋊C4—81no—
489C4⋊C4—532no—
490C4⋊C4—1064no—
491C4⋊C4—1062no—
492C4⋊C4—1062no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—1061no—
496C4⋊C4—1061no—
497C4⋊C4—1061no—
498C4⋊C4—1061no—
499C4⋊C4—1061no—
500C4⋊C4—4241no—
501C4⋊C4—4241no—
502C4⋊C4—4241no—
503C4⋊C4—4241no—
504C4⋊C4—4241no—
505C4⋊C4—2122no—
506C4⋊C4D4×D5312yes—
507C22⋊C4—1062no—
508C22⋊C4—1062no—
509C22⋊C4—2122no—
510C22⋊C4—2122no—
511C42—21no—
512C42—21no—
513C42—1061no—
514C42—1061no—
515C42—2122no—
516C42—2122no—
517C23—532no—
518C23—532no—
519C23—532no—
520C23—1061no—
521C23—1061no—
522C23—1061no—
523C23—1061no—
524C23C2×D4×D5311yescharacteristic
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—532no—
544C2×C4—532no—
545C2×C4—532no—
546C2×C4—534no—
547C2×C4—534no—
548C2×C4—534no—
549C2×C4—534no—
550C2×C4—1062no—
551C2×C4—1061no—
552C2×C4—1061no—
553C2×C4—1061no—
554C2×C4—1061no—
555C2×C4—1061no—
556C2×C4—1061no—
557C2×C4—4244no—
558C2×C4—4244no—
559C2×C4—2121no—
560C2×C4—2121no—
561C2×C4—2121no—
562C2×C4—2121no—
563C2×C4—2121no—
564C2×C4—2121no—
565C2×C4—2121no—
566C2×C4—2121no—
567C2×C4—2121no—
568C2×C4—2121no—
569C2×C4—2121no—
570C2×C4C2×D4×D5311yescharacteristic
571C2×C4C2×Q8×D5311yescharacteristic
572C2×C4C2×Q8×D5311yescharacteristic
573C2×C4#41×D5311yescharacteristic
574C2×C4#41×D5311yescharacteristic
575C2×C4#41×D5311yescharacteristic
576C8—22no—
577C8—22no—
578C8—1062no—
579C8—1062no—
580C22—534no—
581C22—1062no—
582C22—1062no—
583C22—1062no—
584C22—1062no—
585C22—1062no—
586C22—1062no—
587C22#121×D5311yescharacteristic
588C22#123×D5311yescharacteristic
589C22C4⋊Q8×D5311yescharacteristic
590C22#1681142411yescharacteristic
591C22#1681142711yescharacteristic
592C22#1681143611yescharacteristic
593C22#1681143911yescharacteristic
594C4—41no—
595C4—41no—
596C4—42no—
597C4—81no—
598C4—81no—
599C4—81no—
600C4—534no—
601C4—4241no—
602C4—4241no—
603C4—4241no—
604C4—2121no—
605C4—2121no—
606C4—2122no—
607C4C22⋊Q8×D5311yescharacteristic
608C4C22⋊Q8×D5311yescharacteristic
609C4#125×D5311yescharacteristic
610C4#125×D5311yescharacteristic
611C2—1064no—
612C2#398×D5311yescharacteristic
613C2#2028421311yescharacteristic
614C2#2028424011yescharacteristic
615C2#2028468811yescharacteristic
616C2#2028469411yescharacteristic
617C2#2028524811yescharacteristic
618C2#2028525011yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2083C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
hypercenter#2083D53
hypocenterC53#23660
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2083

Complements

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

NH ≅ G/Ntype
5. #20276201C2semidirect
6. #20276199C2semidirect
7. #20276198C2semidirect
8. #20276196C2semidirect
9. #20276190C2semidirect
10. #20276189C2semidirect
11. #20276187C2semidirect
15. C53×#2083C2semidirect
313. #2083D53semidirect
370. C53#23660semidirect

Maximal subgroups

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

subgroupindex
C2×#168106642
C2×#107×D532
#202773362
#202773302
#202762012
#202761992
#202761982
#202761962
#202761902
#202761892
#202761872
#202744332
#202744302
#202744102
C53×#20832
#2366053