Finite Group Databasebeta

← #27469457

Subgroups — #27469457

Order 13568.

986 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810939C211yesmaximal, characteristic, complemented
2C2×#16810947C211yesmaximal, characteristic, complemented
3C2×#16810874C211yesmaximal, characteristic
4C2×#16810666C211yesmaximal, characteristic, complemented
5#20279186C211yesmaximal, characteristic, complemented
6#20276249C211yesmaximal, characteristic, complemented
7#20276248C211yesmaximal, characteristic, complemented
8#20276243C211yesmaximal, characteristic, complemented
9#20276239C211yesmaximal, characteristic, complemented
10#20275590C211yesmaximal, characteristic, complemented
11#20275580C211yesmaximal, characteristic, complemented
12#20274884C211yesmaximal, characteristic, complemented
13#20274871C211yesmaximal, characteristic, complemented
14#20274499C211yesmaximal, characteristic, complemented
15C53×#2500C211yesmaximal, characteristic, complemented
16C22×#11719922C2211yescharacteristic
17C22×#11719917C2211yescharacteristic
18C2×#13133201—22no—
19C2×#13133191C2211yescharacteristic
20C22×C4⋊Dic53C2211yescharacteristic
21#16810947—24no—
22#16810939—24no—
23#16810874—24no—
24#16810666—24no—
25C2×#13133150C2211yescharacteristic
26C2×#13133150C2211yescharacteristic
27C2×#13133150C2211yescharacteristic
28C2×#13133148C2211yescharacteristic
29C2×#13133148C2211yescharacteristic
30C2×#13133148C2211yescharacteristic
31C2×#13133148C2211yescharacteristic
32C2×#13133148C2211yescharacteristic
33C2×#13133148C2211yescharacteristic
34C2×#13133138C2211yescharacteristic
35C2×#13133138C2211yescharacteristic
36C2×#13133138C2211yescharacteristic
37C2×#13133137C2211yescharacteristic
38C2×#13133137C2211yescharacteristic
39C2×#13133137C2211yescharacteristic
40C2×C22⋊#88322C2211yescharacteristic
41C2×#13133128C2211yescharacteristic
42C2×C4×#88322C2211yescharacteristic
43#16810560—22no—
44#16810525C2211yescharacteristic
45#16810525C2211yescharacteristic
46#16810493C2211yescharacteristic
47#16810493C2211yescharacteristic
48#16810378C2211yescharacteristic
49#16810378C2211yescharacteristic
50C2×#121×C53C2211yescharacteristic
51C2×#117×C53C2211yescharacteristic
52C53×#394—22no—
53C53×#390C2211yescharacteristic
54C53×#390C2211yescharacteristic
55C53×#388—22no—
56C53×#384C2211yescharacteristic
57C53×#382C2211yescharacteristic
58C4×C22⋊C4×C53C2211yescharacteristic
59C22×D212C2311yescharacteristic
60C22×C22⋊D53—21no—
61C23×Dic53—21no—
62C2×#11719922—22no—
63C2×#11719922—22no—
64C2×#11719922—22no—
65C2×#11719917—22no—
66C2×#11719917—22no—
67C2×#11719917D414yes—
68C22×#88322C2311yescharacteristic
69C22×#88322C2311yescharacteristic
70C22×#88322C2311yescharacteristic
71C22×#88322C2311yescharacteristic
72#13133201—48no—
73#13133191—22no—
74#13133191—22no—
75C2×C4⋊Dic53—21no—
76C2×C4⋊Dic53—22no—
77C2×C4⋊Dic53—22no—
78C2×C4⋊Dic53—22no—
79C2×C4⋊Dic53—22no—
80C2×C4⋊Dic53C2311yescharacteristic
81C2×C4⋊Dic53C2311yescharacteristic
82C2×C4⋊Dic53C2311yescharacteristic
83C2×#11719921—22no—
84C2×#11719921—22no—
85C2×#11719921—21no—
86#13133150—22no—
87#13133150—22no—
88#13133150—22no—
89#13133148—22no—
90#13133148—22no—
91#13133148—22no—
92#13133148—22no—
93#13133148—22no—
94#13133148—22no—
95#13133138—22no—
96#13133138—22no—
97#13133138—22no—
98#13133137—22no—
99#13133137—22no—
100#13133137—22no—
101C22⋊#88322D414yes—
102#13133128—22no—
103C4×#88322—22no—
104C22×D4×C53—21no—
105C22×D4×C53C2311yescharacteristic
106C23×C4×C53C2311yescharacteristic
107#121×C53—22no—
108#121×C53—22no—
109#117×C53—22no—
110#117×C53—22no—
111C2×C4⋊C4×C53—22no—
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C22⋊C4×C53—21no—
116C2×C22⋊C4×C53—21no—
117C2×C22⋊C4×C53—22no—
118C2×C22⋊C4×C53—22no—
119C2×C22⋊C4×C53D412yes—
120C2×C42×C53—21no—
121C2×C42×C53C2311yescharacteristic
122C2×C42×C53C2311yescharacteristic
123#95×C53—22no—
124#95×C53D412yes—
125C24×C53—21no—
126C24×C53C2×D411yescharacteristic
127C23×D53—21no—
128C2×D4×C53—21no—
129C2×D4×C53—21no—
130C2×D4×C53—21no—
131C2×D4×C53—21no—
132C2×D4×C53—21no—
133C2×D4×C53—21no—
134C2×D4×C53—42no—
135C2×D4×C53—82no—
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×C53—21no—
143C22×C4×C53C2×D412yes—
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D412yes—
146C22×C4×C53C2×D412yes—
147C22×C4×C53C2×D412yes—
148C22×C4×C53C2411yescharacteristic
149C2×C22⋊D53—44no—
150C2×C22⋊D53—42no—
151C22×Dic53—21no—
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53—44no—
155C22×Dic53—42no—
156C2×D212—21no—
157C2×D212—21no—
158C2×D212—21no—
159C2×D212—21no—
160C2×D212—21no—
161C2×D212—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—82no—
169C22⋊C4×C53—24no—
170C22⋊C4×C53—42no—
171C22⋊C4×C53—42no—
172C22⋊C4×C53—44no—
173C22⋊C4×C53—82no—
174C42×C53—21no—
175C42×C53—21no—
176C42×C53—21no—
177C42×C53—21no—
178C42×C53—42no—
179#11719922—44no—
180#11719921—44no—
181#11719921—42no—
182#11719921—44no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—44no—
190C4⋊Dic53—42no—
191C4⋊Dic53—44no—
192#11719917—28no—
193C2×#88322—21no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322C2×D412yes—
200C2×#88322C2×D412yes—
201C2×#88322C2×D412yes—
202C2×#88322C2×D412yes—
203C2×#88322#4112yes—
204C2×#88322#4112yes—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53—22no—
208C23×C53—22no—
209C23×C53—41no—
210C23×C53—41no—
211C23×C53—41no—
212C23×C53—42no—
213C23×C53—42no—
214C23×C53#12712yes—
215C23×C53C22×D411yescharacteristic
216C22×D53—41no—
217C22×D53—41no—
218C22×D53—41no—
219C22×D53—42no—
220C22×D53—42no—
221D4×C53—41no—
222D4×C53—41no—
223D4×C53—41no—
224D4×C53—41no—
225D4×C53—84no—
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—22no—
233C2×C4×C53—22no—
234C2×C4×C53—22no—
235C2×C4×C53—22no—
236C2×C4×C53—22no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—42no—
241C2×C4×C53—42no—
242C2×C4×C53—42no—
243C2×C4×C53#12712yes—
244C2×C4×C53AGL(1,ℤ/8)14yes—
245C2×C4×C53C22×D411yescharacteristic
246C2×C4×C53C22×D411yescharacteristic
247C2×C4×C53C2×#4111yescharacteristic
248C2×C4×C53C2×#4111yescharacteristic
249C2×C4×C532−1+411yescharacteristic
250C2×C4×C532−1+411yescharacteristic
251C22⋊D53—84no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—42no—
256C2×Dic53—42no—
257C2×Dic53—42no—
258C2×Dic53—84no—
259D212—41no—
260D212—41no—
261D212—41no—
262D212—41no—
263#88322—22no—
264#88322—22no—
265#88322—22no—
266#88322#12314yescomplemented
267#27941—531nomaximal, Sylow
268C22×C53—22no—
269C22×C53—22no—
270C22×C53—22no—
271C22×C53—42no—
272C22×C53—42no—
273C22×C53—82no—
274C22×C53—82no—
275C22×C53—81no—
276C22×C53—81no—
277C22×C53—81no—
278C22×C53C2×#12711yescharacteristic
279C22×C53#54711yescharacteristic
280C22×C53#54711yescharacteristic
281C22×C53C2×AGL(1,ℤ/8)12yes—
282C22×C53#57712yes—
283C2×D53—82no—
284C2×D53—82no—
285C2×D53—81no—
286C2×D53—81no—
287C2×D53—81no—
288C4×C53—24no—
289C4×C53—22no—
290C4×C53—42no—
291C4×C53—81no—
292C4×C53—81no—
293C4×C53—81no—
294C4×C53C2×#12311yescharacteristic
295C4×C53#53611yescharacteristic
296C4×C53#54711yescharacteristic
297C4×C53#54711yescharacteristic
298Dic53—81no—
299Dic53—81no—
300Dic53—81no—
301C2×#481—531no—
302C2×#482—531no—
303C2×#469—531no—
304C2×#428—531no—
305#2501—531no—
306#2500D5311yescharacteristic, complemented
307#2094—531no—
308#2094—531no—
309#2093—531no—
310#2092—531no—
311#1972—531no—
312#1970—531no—
313#1847—531no—
314#1844—531no—
315#1753—531no—
316C2×C53—42no—
317C2×C53—161no—
318C2×C53#250111yescharacteristic
319C2×C53#315111yescharacteristic
320C2×C53#315411yescharacteristic
321C2×C53#334512yes—
322C2×C53#335412yes—
323D53—161no—
324C22×D8—531no—
325C22×M16—531no—
326C2×#123—1062no—
327C2×#121—531no—
328C2×#121C2×D5311yescharacteristic
329C2×#117C2×D5311yescharacteristic
330C22×C4⋊C4—531no—
331#482—1064no—
332#481—1064no—
333#469—1064no—
334#428—1064no—
335C2×#107—531no—
336C2×#107—531no—
337C2×#107—531no—
338C2×#106—531no—
339C2×#106—531no—
340C2×#106—531no—
341C2×#106—531no—
342C2×#106—531no—
343C2×#106—531no—
344C2×#102—531no—
345C2×#102—531no—
346C2×#102—531no—
347C2×#102—531no—
348C2×#102—531no—
349C2×#102—531no—
350C2×C22⋊C8—531no—
351C2×#97—531no—
352C2×C4×C8—531no—
353#399—1062no—
354#394—22no—
355#390—531no—
356#390—531no—
357#390C2×D5311yescharacteristic
358#390C2×D5311yescharacteristic
359#388—22no—
360#384—531no—
361#384—531no—
362#384C2×D5311yescharacteristic
363#382C2×D5311yescharacteristic
364C4×C22⋊C4C2×D5311yescharacteristic
365#336—531no—
366#336—531no—
367C53#2794111yescharacteristic, complemented, Sylow
368C22×D4—21no—
369C22×D4—531no—
370C22×D4—1061no—
371C22×D4C22×D5311yescharacteristic
372C23×C4—1061no—
373C23×C4C22×D5311yescharacteristic
374C2×D8—1062no—
375C2×D8—1062no—
376C2×D8—1062no—
377C2×M16—534no—
378C2×M16—1062no—
379C2×M16—1062no—
380C22×C8—531no—
381C22×C8—531no—
382C22×C8—531no—
383C22×C8—531no—
384#123—2128no—
385#121—22no—
386#121—22no—
387#121—1062no—
388#121—1062no—
389#117—22no—
390#117—22no—
391C2×C4⋊C4—22no—
392C2×C4⋊C4—531no—
393C2×C4⋊C4—531no—
394C2×C4⋊C4—531no—
395C2×C4⋊C4—1062no—
396C2×C4⋊C4—1062no—
397C2×C4⋊C4—1061no—
398C2×C4⋊C4—1062no—
399C2×C4⋊C4—1062no—
400C2×C4⋊C4C22×D5311yescharacteristic
401C2×C4⋊C4C22×D5311yescharacteristic
402C2×C4⋊C4C22×D5311yescharacteristic
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—22no—
406C2×C22⋊C4—22no—
407C2×C22⋊C4—1061no—
408C2×C22⋊C4—1062no—
409C2×C22⋊C4—1062no—
410C2×C22⋊C4C22⋊D5312yes—
411C2×C42—21no—
412C2×C42C22×D5311yescharacteristic
413C2×C42C22×D5311yescharacteristic
414#107—1062no—
415#107—1062no—
416#107—1062no—
417#106—1062no—
418#106—1062no—
419#106—1062no—
420#106—1062no—
421#106—1062no—
422#106—1062no—
423#102—1062no—
424#102—1062no—
425#102—1062no—
426#102—1062no—
427#102—1062no—
428#102—1062no—
429C22⋊C8—534no—
430#97—1062no—
431C4×C8—1062no—
432#95—22no—
433#95C22⋊D5312yes—
434C24—21no—
435C24—1061no—
436C24C2×C22⋊D5311yescharacteristic
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—42no—
444C2×D4—82no—
445C2×D4—1061no—
446C2×D4—1061no—
447C2×D4—1061no—
448C2×D4—1061no—
449C2×D4—1061no—
450C2×D4—1061no—
451C2×D4—2124no—
452C2×D4—2122no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—1061no—
461C22×C4—1061no—
462C22×C4—1061no—
463C22×C4—2124no—
464C22×C4—2122no—
465C22×C4D4×D5312yes—
466C22×C4D4×D5312yes—
467C22×C4C2×C22⋊D5312yes—
468C22×C4C2×C22⋊D5311yescharacteristic
469C22×C4C2×C22⋊D5312yes—
470C22×C4C23×D5311yescharacteristic
471D8—2124no—
472M16—1068no—
473C2×C8—532no—
474C2×C8—532no—
475C2×C8—532no—
476C2×C8—532no—
477C2×C8—532no—
478C2×C8—532no—
479C2×C8—1061no—
480C2×C8—1061no—
481C2×C8—1061no—
482C2×C8—1061no—
483C2×C8—1061no—
484C2×C8—1061no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—82no—
492C4⋊C4—1061no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—1061no—
496C4⋊C4—1061no—
497C4⋊C4—1061no—
498C4⋊C4—2124no—
499C4⋊C4—2122no—
500C4⋊C4—2124no—
501C22⋊C4—24no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—44no—
505C22⋊C4—82no—
506C22⋊C4—2124no—
507C22⋊C4—2122no—
508C22⋊C4—2124no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—42no—
514C23—21no—
515C23—21no—
516C23—22no—
517C23—22no—
518C23—41no—
519C23—41no—
520C23—42no—
521C23—41no—
522C23—42no—
523C23—2121no—
524C23—2121no—
525C23—2122no—
526C23—2121no—
527C23—2122no—
528C23#1313321312yes—
529C23C22×C22⋊D5311yescharacteristic
530D4—41no—
531D4—41no—
532D4—41no—
533D4—41no—
534D4—84no—
535D4—4244no—
536D4—2121no—
537D4—2121no—
538D4—2121no—
539D4—2121no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—4244no—
558C2×C4—2122no—
559C2×C4—2122no—
560C2×C4—2122no—
561C2×C4—2121no—
562C2×C4—2121no—
563C2×C4—2121no—
564C2×C4#1313321312yes—
565C2×C4#1313324214yes—
566C2×C4C2×D4×D5311yescharacteristic
567C2×C4C2×D4×D5311yescharacteristic
568C2×C4#41×D5311yescharacteristic
569C2×C4#41×D5311yescharacteristic
570C2×C4#1313326411yescharacteristic
571C2×C4#1313326411yescharacteristic
572C8—534no—
573C8—1062no—
574C8—1062no—
575C8—1062no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—42no—
580C22—42no—
581C22—81no—
582C22—82no—
583C22—81no—
584C22—82no—
585C22—81no—
586C22—4241no—
587C22—4242no—
588C22—4241no—
589C22—4242no—
590C22—4241no—
591C22C2×#1313321311yescharacteristic
592C22#1681127211yescharacteristic
593C22#1681127211yescharacteristic
594C22C2×#1313324212yes—
595C22#1681143412yes—
596C4—22no—
597C4—24no—
598C4—42no—
599C4—81no—
600C4—81no—
601C4—81no—
602C4—4241no—
603C4—4241no—
604C4—4241no—
605C4#123×D5311yescharacteristic
606C4#1681119511yescharacteristic
607C4#1681127111yescharacteristic
608C4#1681127111yescharacteristic
609C2—42no—
610C2—161no—
611C2—8481no—
612C2#2027918411yescharacteristic
613C2#2028516811yescharacteristic
614C2#2028518711yescharacteristic
615C2#2028717912yes—
616C2#2028730412yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2500C2
socleC23×C53C22×D4
prime radicalC2×#16810874C2
hypercenter#2500D53
hypocenterC53#27941
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2500

Complements

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

NH ≅ G/Ntype
1. C2×#16810939C2semidirect
2. C2×#16810947C2semidirect
4. C2×#16810666C2semidirect
5. #20279186C2semidirect
6. #20276249C2semidirect
7. #20276248C2semidirect
8. #20276243C2semidirect
9. #20276239C2semidirect
10. #20275590C2semidirect
11. #20275580C2semidirect
12. #20274884C2semidirect
13. #20274871C2semidirect
14. #20274499C2semidirect
15. C53×#2500C2semidirect
266. #88322#123semidirect
306. #2500D53semidirect
367. C53#27941semidirect

Maximal subgroups

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

subgroupindex
C2×#168109392
C2×#168109472
C2×#168108742
C2×#168106662
#202791862
#202762492
#202762482
#202762432
#202762392
#202755902
#202755802
#202748842
#202748712
#202744992
C53×#25002
#2794153