Finite Group Databasebeta

← #27385319

Subgroups — #27385319

Order 13568.

2258 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811135C211yesmaximal, characteristic
2C22×C4⋊C4×D53C211yesmaximal, characteristic, complemented
3C2×#16810625C211yesmaximal, characteristic, complemented
4C2×#16810622C211yesmaximal, characteristic, complemented
5C2×#16810586C211yesmaximal, characteristic, complemented
6#20275031C214yesmaximal, complemented
7#20275021C214yesmaximal, complemented
8#20274436C211yesmaximal, characteristic, complemented
9C53×#1736C211yesmaximal, characteristic, complemented
10C22×#11719945C2211yescharacteristic
11C23×C4×D53—21no—
12C23×C4×D53C2211yescharacteristic
13C22×#11719917C2211yescharacteristic
14#16811135—24no—
15#16811135C418yescomplemented
16C2×#13133201—21no—
17C2×#13133197C2212yes—
18C2×#13133192C2212yes—
19C2×#13133187—21no—
20C2×C4×D212—22no—
21C22×C4⋊Dic53C2211yescharacteristic
22C22×#11719919—21no—
23C2×C4⋊C4×D53—22no—
24C2×C4⋊C4×D53—24no—
25C2×C4⋊C4×D53—24no—
26C2×C4⋊C4×D53C2214yes—
27#16810625—24no—
28#16810622—24no—
29C2×#13133142C2212yescomplemented
30C2×#13133140C2212yescomplemented
31C2×#13133139C2212yescomplemented
32C2×#13133137C2212yescomplemented
33#16810586—24no—
34C2×#13133129C2214yes—
35#16810385C2214yescomplemented
36C22×M16×C53C2211yescharacteristic
37C2×#117×C53C2211yescharacteristic
38C22×C4⋊C4×C53C2211yescharacteristic
39C53×#340C2214yescomplemented
40C24×D53D411yescharacteristic
41C2×#11719945—22no—
42C2×#11719945—22no—
43C2×#11719945—22no—
44C2×#11719945—22no—
45C2×#11719945—22no—
46C2×#11719945C2×C414yescomplemented
47C22×Dic106C2311yescharacteristic
48C22×D212—22no—
49C22×D212C2311yescharacteristic
50C22×C22⋊D53—21no—
51C23×Dic53—21no—
52C23×Dic53D411yescharacteristic
53C22×C4×D53—22no—
54C22×C4×D53—22no—
55C22×C4×D53—22no—
56C22×C4×D53—22no—
57C22×C4×D53—28no—
58C22×C4×D53—24no—
59C22×C4×D53—21no—
60C22×C4×D53—48no—
61C22×C4×D53C2×C414yes—
62C22×C4×D53D412yes—
63C22×C4×D53D414yes—
64C22×C4×D53C2312yes—
65C2×#11719917—22no—
66C2×#11719917—22no—
67C2×#11719917—22no—
68C22×#88322C2312yes—
69#13133201—28no—
70#13133197—24no—
71#13133197C2×C418yescomplemented
72#13133192—24no—
73#13133192C2×C418yescomplemented
74#13133187—28no—
75C4×D212—216no—
76C2×C4⋊Dic53—22no—
77C2×C4⋊Dic53—21no—
78C2×C4⋊Dic53—22no—
79C2×C4⋊Dic53D414yescomplemented
80C2×C4⋊Dic53C2312yes—
81C2×#11719919—28no—
82C2×#11719919—24no—
83C4⋊C4×D53—216no—
84C4⋊C4×D53—28no—
85C4⋊C4×D53—28no—
86C2×#11719921—22no—
87C2×#11719921—22no—
88#13133142—24no—
89#13133140—24no—
90#13133139—24no—
91#13133137—24no—
92#13133129—28no—
93C23×C4×C53—21no—
94C23×C4×C53C2311yescharacteristic
95C2×M16×C53—22no—
96C2×M16×C53—22no—
97C2×M16×C53—22no—
98C22×C8×C53C2312yes—
99#117×C53—22no—
100#117×C53C2×C414yescomplemented
101C2×C4⋊C4×C53—22no—
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53D414yescomplemented
104C2×C4⋊C4×C53C2312yes—
105C2×C4⋊C4×C53C2312yes—
106C2×C22⋊C4×C53—21no—
107C2×C42×C53—21no—
108C24×C53C2×D411yescharacteristic
109C23×D53—22no—
110C23×D53—22no—
111C23×D53—24no—
112C23×D53—24no—
113C23×D53—21no—
114C23×D53C22⋊C414yes—
115C23×D53C2×D412yes—
116C22×C4×C53—24no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—44no—
120C22×C4×C53C22×C412yes—
121C22×C4×C53C2×D411yescharacteristic
122C22×C4×C53C2×D412yes—
123C22×C4×C53C2×D412yes—
124C22×C4×C53C2×D412yes—
125C22×C4×C53#4112yes—
126C22×C4×C53#4112yes—
127C22×C4×C53C2411yescharacteristic
128C2×C22⋊D53—24no—
129C2×C22⋊D53—42no—
130C2×C22⋊D53—42no—
131C22×Dic53—22no—
132C22×Dic53—22no—
133C22×Dic53—24no—
134C22×Dic53—21no—
135C22×Dic53—22no—
136C22×Dic53—44no—
137C22×Dic53C22⋊C414yes—
138C22×Dic53C2×D412yes—
139#11719945—28no—
140#11719945—28no—
141#11719945—44no—
142#11719945—44no—
143C2×D212—21no—
144C2×D212—21no—
145C2×D212—21no—
146C2×D212—21no—
147C2×D212—28no—
148C2×D212—21no—
149C2×D212—44no—
150C2×D212—44no—
151C2×D212C22×C412yescomplemented
152C2×C4×D53—28no—
153C2×C4×D53—28no—
154C2×C4×D53—24no—
155C2×C4×D53—28no—
156C2×C4×D53—28no—
157C2×C4×D53—24no—
158C2×C4×D53—22no—
159C2×C4×D53—22no—
160C2×C4×D53—22no—
161C2×C4×D53—22no—
162C2×C4×D53—216no—
163C2×C4×D53—24no—
164C2×C4×D53—432no—
165C2×C4×D53—48no—
166C2×C4×D53—48no—
167C2×C4×D53—42no—
168C2×C4×D53—42no—
169C2×C4×D53C22⋊C418yes—
170C2×C4×D53C22×C414yes—
171C2×C4×D53C2×D414yes—
172C2×Dic106—21no—
173C2×Dic106—21no—
174C2×Dic106—21no—
175C2×Dic106—21no—
176C2×Dic106—21no—
177C2×Dic106C22×C412yescomplemented
178M16×C53—44no—
179C2×C8×C53—22no—
180C2×C8×C53—22no—
181C2×C8×C53—22no—
182C4⋊C4×C53—24no—
183C4⋊C4×C53—22no—
184C4⋊C4×C53—22no—
185C4⋊C4×C53C22×C414yescomplemented
186C4⋊C4×C53C2×D414yes—
187C22⋊C4×C53—24no—
188C42×C53—24no—
189#11719921—28no—
190#11719921—28no—
191C4⋊Dic53—24no—
192C4⋊Dic53—22no—
193C4⋊Dic53—28no—
194C4⋊Dic53C2×D414yes—
195#11719919—216no—
196#11719917—44no—
197C2×#88322—22no—
198C2×#88322—22no—
199C2×#88322—22no—
200C23×C53—22no—
201C23×C53—22no—
202C23×C53C2×C22⋊C412yes—
203C23×C53#12012yes—
204C23×C53#12312yes—
205C23×C53C22×D411yescharacteristic
206C22×D53—28no—
207C22×D53—216no—
208C22×D53—28no—
209C22×D53—22no—
210C22×D53—22no—
211C22×D53—22no—
212C22×D53—22no—
213C22×D53—28no—
214C22×D53—216no—
215C22×D53—22no—
216C22×D53—41no—
217C22×D53—41no—
218C22×D53—42no—
219C22×D53—42no—
220C22×D53C2×C22⋊C414yes—
221C22×D53AGL(1,ℤ/8)14yes—
222C22×D53#13714yes—
223C2×C4×C53—24no—
224C2×C4×C53—24no—
225C2×C4×C53—48no—
226C2×C4×C53—44no—
227C2×C4×C53—42no—
228C2×C4×C53—42no—
229C2×C4×C53C2×C22⋊C412yes—
230C2×C4×C53C4×D414yes—
231C2×C4×C53C4×D414yes—
232C2×C4×C53#12012yes—
233C2×C4×C53#12114yes—
234C2×C4×C53#12114yes—
235C2×C4×C53#12312yes—
236C2×C4×C53C23×C411yescharacteristic
237C2×C4×C53C22×D411yescharacteristic
238C2×C4×C53C22×D411yescharacteristic
239C2×C4×C53C22×D411yescharacteristic
240C2×C4×C53C2×#4111yescharacteristic
241C2×C4×C53C2×#4111yescharacteristic
242C22⋊D53—48no—
243C2×Dic53—22no—
244C2×Dic53—22no—
245C2×Dic53—28no—
246C2×Dic53—22no—
247C2×Dic53—24no—
248C2×Dic53—48no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—44no—
252C2×Dic53C2×C22⋊C414yes—
253D212—22no—
254D212—22no—
255D212—416no—
256D212—41no—
257D212—41no—
258C4×D53—24no—
259C4×D53—24no—
260C4×D53—24no—
261C4×D53—24no—
262C4×D53—216no—
263C4×D53—432no—
264C4×D53—48no—
265Dic106—22no—
266Dic106—22no—
267Dic106—41no—
268Dic106—41no—
269C8×C53—42no—
270#88322—42no—
271#20909—531nomaximal, Sylow
272C22×C53—24no—
273C22×C53—24no—
274C22×C53—22no—
275C22×C53—22no—
276C22×C53#38614yes—
277C22×C53C22×C22⋊C411yescharacteristic
278C22×C53C2×#12011yescharacteristic
279C22×C53C2×#12311yescharacteristic
280C22×C53C2×AGL(1,ℤ/8)12yes—
281C22×C53C2×#13712yes—
282C2×D53—232no—
283C2×D53—24no—
284C2×D53—24no—
285C2×D53—24no—
286C2×D53—24no—
287C2×D53—24no—
288C2×D53—24no—
289C2×D53—44no—
290C2×D53—42no—
291C2×D53—82no—
292C2×D53—81no—
293C2×D53—81no—
294C2×D53#41718yescomplemented
295C4×C53—44no—
296C4×C53—44no—
297C4×C53#38614yes—
298C4×C53C2×C4×D411yescharacteristic
299C4×C53C2×C4×D411yescharacteristic
300C4×C53C2×#12111yescharacteristic
301C4×C53C2×#12111yescharacteristic
302Dic53—24no—
303Dic53—44no—
304Dic53—42no—
305C2×#525—531no—
306C23×C4⋊C4—531no—
307C2×#417—531no—
308C2×#417—531no—
309C2×#407—531no—
310#1874—534no—
311#1872—534no—
312#1736—531no—
313#1736D5311yescharacteristic, complemented
314C2×C53—24no—
315C2×C53C2×#38611yescharacteristic
316C2×C53C2×#41712yes—
317C2×C53#300311yescharacteristic
318C2×C53#300411yescharacteristic
319C2×C53#309212yes—
320D53—28no—
321D53—82no—
322C22×#41—531no—
323C24×C4—531no—
324C24×C4—1061no—
325C22×M16—531no—
326C22×M16C2×D5311yescharacteristic
327#525—538no—
328#525—1064no—
329C2×#123—1061no—
330C2×C22⋊Q8—532no—
331C2×#121—532no—
332C2×#120—1061no—
333C2×C4×D4—1062no—
334C2×#117C2×D5311yescharacteristic
335C22×C4⋊C4—531no—
336C22×C4⋊C4—534no—
337C22×C4⋊C4—1064no—
338C22×C4⋊C4—1062no—
339C22×C4⋊C4—1061no—
340C22×C4⋊C4—1064no—
341C22×C4⋊C4C2×D5311yescharacteristic
342#417—1064no—
343#417—1064no—
344C2×#103—532no—
345C2×#103—532no—
346C2×#102—532no—
347C2×#102—532no—
348#407—1064no—
349C2×C22⋊C8—534no—
350#340—534no—
351#340C2×D5314yescomplemented
352C53#2090911yescharacteristic, complemented, Sylow
353C25—531no—
354C2×#41—534no—
355C2×#41—1062no—
356C2×#41—1062no—
357C2×#41—1062no—
358C2×#41—1062no—
359C2×#41—1062no—
360C22×Q8—531no—
361C22×D4—531no—
362C22×D4—1062no—
363C22×D4—1061no—
364C23×C4—21no—
365C23×C4—531no—
366C23×C4—534no—
367C23×C4—532no—
368C23×C4—534no—
369C23×C4—532no—
370C23×C4—1064no—
371C23×C4—1068no—
372C23×C4—1062no—
373C23×C4—1062no—
374C23×C4—1062no—
375C23×C4—1062no—
376C23×C4—1061no—
377C23×C4—1061no—
378C23×C4—2128no—
379C23×C4C22×D5311yescharacteristic
380C2×M16—22no—
381C2×M16—22no—
382C2×M16—22no—
383C2×M16—1062no—
384C2×M16—1062no—
385C2×M16—1062no—
386C22×C8—532no—
387C22×C8C22×D5312yes—
388#123—1068no—
389C22⋊Q8—538no—
390C22⋊Q8—1064no—
391#121—538no—
392#121—1064no—
393#120—1068no—
394C4×D4—10616no—
395#117—22no—
396#117C4×D5314yescomplemented
397C2×C4⋊C4—21no—
398C2×C4⋊C4—22no—
399C2×C4⋊C4—534no—
400C2×C4⋊C4—532no—
401C2×C4⋊C4—1068no—
402C2×C4⋊C4—10616no—
403C2×C4⋊C4—1064no—
404C2×C4⋊C4—1062no—
405C2×C4⋊C4—1068no—
406C2×C4⋊C4—1062no—
407C2×C4⋊C4—1061no—
408C2×C4⋊C4—1068no—
409C2×C4⋊C4D21214yescomplemented
410C2×C4⋊C4C22×D5312yes—
411C2×C4⋊C4C22×D5312yes—
412C2×C22⋊C4—21no—
413C2×C22⋊C4—1062no—
414C2×C22⋊C4—1062no—
415C2×C42—21no—
416#103—1064no—
417#103—1064no—
418#102—1064no—
419#102—1064no—
420C22⋊C8—1068no—
421C24—532no—
422C24—534no—
423C24—1064no—
424C24—1064no—
425C24—1062no—
426C24—1062no—
427C24—1061no—
428C24D4×D5311yescharacteristic
429#41—1068no—
430#41—1068no—
431#41—2124no—
432#41—2124no—
433C2×Q8—532no—
434C2×Q8—1061no—
435C2×Q8—1061no—
436C2×Q8—1061no—
437C2×Q8—1061no—
438C2×Q8—1061no—
439C2×D4—532no—
440C2×D4—1068no—
441C2×D4—1064no—
442C2×D4—1061no—
443C2×D4—1061no—
444C2×D4—1061no—
445C2×D4—1061no—
446C2×D4—1061no—
447C2×D4—2124no—
448C2×D4—2124no—
449C2×D4—2122no—
450C2×D4—2122no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—44no—
455C22×C4—534no—
456C22×C4—532no—
457C22×C4—538no—
458C22×C4—534no—
459C22×C4—534no—
460C22×C4—10616no—
461C22×C4—1062no—
462C22×C4—1062no—
463C22×C4—1068no—
464C22×C4—1068no—
465C22×C4—1064no—
466C22×C4—1068no—
467C22×C4—1068no—
468C22×C4—1064no—
469C22×C4—1062no—
470C22×C4—1062no—
471C22×C4—1062no—
472C22×C4—1062no—
473C22×C4—1064no—
474C22×C4—1061no—
475C22×C4—1062no—
476C22×C4—1064no—
477C22×C4—2128no—
478C22×C4—21232no—
479C22×C4—2128no—
480C22×C4—2124no—
481C22×C4—2122no—
482C22×C4—2122no—
483C22×C4C2×C4×D5312yes—
484C22×C4C2×D21212yes—
485C22×C4#1171994512yes—
486C22×C4D4×D5311yescharacteristic
487C22×C4D4×D5312yes—
488C22×C4D4×D5312yes—
489C22×C4#1171994912yes—
490C22×C4C23×D5311yescharacteristic
491M16—44no—
492M16—2124no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—1062no—
497C2×C8—1062no—
498C2×C8—1062no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—534no—
503C4⋊C4—1068no—
504C4⋊C4—10616no—
505C4⋊C4—1064no—
506C4⋊C4—1062no—
507C4⋊C4C2×C4×D5314yescomplemented
508C4⋊C4C2×D21214yes—
509C22⋊C4—24no—
510C22⋊C4—1068no—
511C22⋊C4—1068no—
512C42—24no—
513C23—22no—
514C23—22no—
515C23—534no—
516C23—534no—
517C23—534no—
518C23—10616no—
519C23—1068no—
520C23—1068no—
521C23—10616no—
522C23—1068no—
523C23—1062no—
524C23—1062no—
525C23—1062no—
526C23—1062no—
527C23—1062no—
528C23—2121no—
529C23—2121no—
530C23—2122no—
531C23—2122no—
532C23C22⋊C4×D5312yes—
533C23#1313318712yes—
534C23#1313320212yes—
535C23C2×D4×D5311yescharacteristic
536Q8—1062no—
537Q8—1062no—
538Q8—2121no—
539Q8—2121no—
540D4—1062no—
541D4—1062no—
542D4—21216no—
543D4—2128no—
544D4—2121no—
545D4—2121no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—48no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—534no—
553C2×C4—1068no—
554C2×C4—1062no—
555C2×C4—1062no—
556C2×C4—10616no—
557C2×C4—1064no—
558C2×C4—1064no—
559C2×C4—1064no—
560C2×C4—1064no—
561C2×C4—1062no—
562C2×C4—1064no—
563C2×C4—21232no—
564C2×C4—2128no—
565C2×C4—2121no—
566C2×C4—2121no—
567C2×C4—2124no—
568C2×C4—2128no—
569C2×C4C22⋊C4×D5312yes—
570C2×C4#1313317914yes—
571C2×C4C4×D21214yes—
572C2×C4#1313318712yes—
573C2×C4#1313319014yes—
574C2×C4#1313319214yes—
575C2×C4#1313320212yes—
576C2×C4C22×C4×D5311yescharacteristic
577C2×C4C2×D4×D5311yescharacteristic
578C2×C4C2×D4×D5311yescharacteristic
579C2×C4C22×D21211yescharacteristic
580C2×C4C2×#1171994511yescharacteristic
581C2×C4C2×#1171994911yescharacteristic
582C8—42no—
583C8—2122no—
584C22—22no—
585C22—22no—
586C22—24no—
587C22—24no—
588C22—538no—
589C22—10632no—
590C22—1064no—
591C22—1064no—
592C22—1064no—
593C22—1064no—
594C22—1064no—
595C22—1064no—
596C22—4241no—
597C22—4241no—
598C22—4242no—
599C22—2124no—
600C22—2122no—
601C22#1681050114yes—
602C22C2×C22⋊C4×D5311yescharacteristic
603C22C2×#1313318711yescharacteristic
604C22C2×#1313320211yescharacteristic
605C22AGL(1,ℤ/8)×D5312yes—
606C22#137×D5312yes—
607C4—44no—
608C4—44no—
609C4—1064no—
610C4—2124no—
611C4—2122no—
612C4#1681050114yes—
613C4C2×#1313317911yescharacteristic
614C4C2×C4×D21211yescharacteristic
615C4C2×#1313319011yescharacteristic
616C4C2×#1313319211yescharacteristic
617C2—24no—
618C2—1068no—
619C2—4242no—
620C2C2×#1681050111yescharacteristic
621C2#417×D5312yes—
622C2#2028394411yescharacteristic
623C2#2028395111yescharacteristic
624C2#2028464112yes—
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1736C2
socleC23×C53C22×D4
prime radicalC2×#16811135C2
cosocleC22×C4×C53C24
hypercenter#1736D53
hypocenterC53#20909
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
2. C22×C4⋊C4×D53C2semidirect
3. C2×#16810625C2semidirect
4. C2×#16810622C2semidirect
5. C2×#16810586C2semidirect
6. #20275031C2semidirect
7. #20275021C2semidirect
8. #20274436C2semidirect
9. C53×#1736C2semidirect
15. #16811135C4semidirect
29. C2×#13133142C22semidirect
30. C2×#13133140C22semidirect
31. C2×#13133139C22semidirect
32. C2×#13133137C22semidirect
35. #16810385C22semidirect
39. C53×#340C22semidirect
46. C2×#11719945C2×C4semidirect
71. #13133197C2×C4semidirect
73. #13133192C2×C4semidirect
79. C2×C4⋊Dic53D4semidirect
100. #117×C53C2×C4semidirect
103. C2×C4⋊C4×C53D4semidirect
151. C2×D212C22×C4semidirect
177. C2×Dic106C22×C4semidirect
185. C4⋊C4×C53C22×C4semidirect
294. C2×D53#417semidirect
313. #1736D53semidirect
351. #340C2×D53semidirect
352. C53#20909semidirect
396. #117C4×D53semidirect
409. C2×C4⋊C4D212semidirect
507. C4⋊C4C2×C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168111352
C22×C4⋊C4×D532
C2×#168106252
C2×#168106222
C2×#168105862
#202750312
#202750212
#202744362
C53×#17362
#2090953