Finite Group Databasebeta

← #27390278

Subgroups — #27390278

Order 13568.

1104 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810827C211yesmaximal, characteristic
2C2×#16810710C211yesmaximal, characteristic
3C2×#16810708C211yesmaximal, characteristic
4#20277284C211yesmaximal, characteristic, complemented
5C4×#13133179C211yesmaximal, characteristic
6#20275636C211yesmaximal, characteristic, complemented
7#20275475C211yesmaximal, characteristic, complemented
8#20275349C211yesmaximal, characteristic, complemented
9#20275052C211yesmaximal, characteristic
10#20275048C211yesmaximal, characteristic
11#20274714C211yesmaximal, characteristic
12#20274563C211yesmaximal, characteristic, complemented
13#103×Dic53C211yesmaximal, characteristic, complemented
14#20274534C211yesmaximal, characteristic, complemented
15C53×#1927C211yesmaximal, characteristic, complemented
16C22×#11719913C2211yescharacteristic
17C22×#11719924C2211yescharacteristic
18C2×#13133215C2211yescharacteristic
19C2×Q8×Dic53C2211yescharacteristic
20C2×#13133182C2211yescharacteristic
21C2×C4×D212C2211yescharacteristic
22C2×#13133179C2211yescharacteristic
23C2×#13133179C2211yescharacteristic
24C2×C42×D53—21no—
25#16810827C418yescomplemented
26#16810710—24no—
27#16810708—24no—
28C2×#13133148C2211yescharacteristic
29C2×#13133146C2211yescharacteristic
30C2×#13133146C2211yescharacteristic
31C2×#13133142C2211yescharacteristic
32C2×#13133141C2211yescharacteristic
33C2×#13133140C2211yescharacteristic
34C2×#13133140C2211yescharacteristic
35C2×#13133139C2211yescharacteristic
36C2×#13133137C2211yescharacteristic
37C2×C8×Dic53C2211yescharacteristic
38C2×C8×Dic53C2211yescharacteristic
39#16810529C2211yescharacteristic
40#16810520—21no—
41#16810520—21no—
42#16810492C2211yescharacteristic
43C4⋊C4×Dic53C2211yescharacteristic
44C4⋊C4×Dic53C2211yescharacteristic
45#16810474—21no—
46#16810474—21no—
47C4×#11719921—21no—
48#16810470—21no—
49#16810470—21no—
50C42×Dic53C2211yescharacteristic
51#16810386C2211yescharacteristic
52#16810385C2211yescharacteristic
53#16810385C2211yescharacteristic
54#16810378C2211yescharacteristic
55C2×#106×C53C2211yescharacteristic
56C2×#103×C53C2211yescharacteristic
57C2×#103×C53C2211yescharacteristic
58C53×#391C2211yescharacteristic
59C4×C4⋊C4×C53C2211yescharacteristic
60C53×#340C2211yescharacteristic
61C53×#336C2211yescharacteristic
62C22×Dic106C2311yescharacteristic
63C22×D212C2311yescharacteristic
64C22×C4×D53—21no—
65C22×C4×D53—21no—
66C22×C4×D53—21no—
67C2×#11719913—22no—
68C2×#11719913—22no—
69C2×#11719913C2×C414yes—
70C2×#11719924—22no—
71C2×#11719924—22no—
72C2×#11719924C2×C414yes—
73C22×#88322C2311yescharacteristic
74C22×#88322C2311yescharacteristic
75#13133215—22no—
76#13133215C2×C414yescomplemented
77Q8×Dic53—22no—
78Q8×Dic53—22no—
79#13133182—22no—
80#13133182—22no—
81C4×D212—22no—
82C4×D212—22no—
83#13133179—22no—
84#13133179—22no—
85#13133179C2×C414yes—
86#13133179D414yes—
87C2×C4⋊Dic53—21no—
88C2×C4⋊Dic53C2311yescharacteristic
89C2×C4⋊Dic53C2311yescharacteristic
90C2×#11719919—21no—
91C2×#11719919—21no—
92C2×#11719919—21no—
93C2×#11719919—21no—
94C2×#11719919—21no—
95C2×#11719921—21no—
96C2×#11719921—21no—
97C2×#11719921—21no—
98C2×C4×Dic53—21no—
99C2×C4×Dic53—21no—
100C2×C4×Dic53—21no—
101C2×C4×Dic53—21no—
102C2×C4×Dic53D411yescharacteristic
103C2×C4×Dic53D411yescharacteristic
104C2×C4×Dic53D411yescharacteristic
105C2×C4×Dic53D411yescharacteristic
106C2×C4×Dic53C2311yescharacteristic
107C2×C4×Dic53C2311yescharacteristic
108C42×D53—44no—
109#13133148—22no—
110#13133146—22no—
111#13133146C2×C414yescomplemented
112#13133142—22no—
113#13133141—22no—
114#13133140—22no—
115#13133140C2×C414yescomplemented
116#13133139—22no—
117#13133137—22no—
118C8×Dic53—22no—
119C8×Dic53—22no—
120#13133124—21no—
121#13133124—21no—
122#13133124—21no—
123#13133124—21no—
124C22×Q8×C53C2311yescharacteristic
125C22×C8×C53C2311yescharacteristic
126C22×C8×C53C2311yescharacteristic
127C2×C4⋊C4×C53C2311yescharacteristic
128C2×C4⋊C4×C53C2311yescharacteristic
129C2×C4⋊C4×C53C2311yescharacteristic
130C2×C42×C53—21no—
131C2×C42×C53C2311yescharacteristic
132#106×C53—22no—
133#103×C53—22no—
134#103×C53C2×C414yescomplemented
135#95×C53—21no—
136#95×C53—21no—
137#95×C53—21no—
138C23×D53—21no—
139C2×Q8×C53—21no—
140C2×Q8×C53—21no—
141C2×Q8×C53—21no—
142C2×Q8×C53—21no—
143C2×Q8×C53—21no—
144C2×Q8×C53C22×C412yes—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53—21no—
148C22×C4×C53—21no—
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2411yescharacteristic
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53—21no—
155C22×Dic53C2×D411yescharacteristic
156C22×Dic53C2×D411yescharacteristic
157C22×Dic53C2×D411yescharacteristic
158C22×Dic53C2×D411yescharacteristic
159C2×D212—21no—
160C2×D212—21no—
161C2×D212—21no—
162C2×D212C22×C412yes—
163C2×D212C2×D412yes—
164C2×D212#4112yes—
165C2×C4×D53—24no—
166C2×C4×D53—24no—
167C2×C4×D53—24no—
168C2×C4×D53—42no—
169C2×C4×D53—42no—
170C2×C4×D53—42no—
171C2×C4×D53—42no—
172C2×C4×D53—42no—
173C2×C4×D53—42no—
174C2×Dic106—21no—
175C2×Dic106—21no—
176C2×Dic106—21no—
177C2×Dic106—21no—
178C2×Dic106—21no—
179C2×Dic106C22×C412yes—
180C2×C8×C53—21no—
181C2×C8×C53—21no—
182C2×C8×C53—21no—
183C2×C8×C53—21no—
184C2×C8×C53—21no—
185C2×C8×C53C22×C412yes—
186C4⋊C4×C53—21no—
187C4⋊C4×C53—21no—
188C4⋊C4×C53—21no—
189C4⋊C4×C53—21no—
190C4⋊C4×C53C22×C412yes—
191C4⋊C4×C53C2×D412yes—
192C42×C53—21no—
193C42×C53—21no—
194C42×C53—42no—
195#11719924—28no—
196#11719921—24no—
197#11719921—24no—
198#11719921—24no—
199C4⋊Dic53—21no—
200C4⋊Dic53—21no—
201C4⋊Dic53—21no—
202C4⋊Dic53—42no—
203C4⋊Dic53#4112yes—
204#11719919—24no—
205#11719919—24no—
206#11719919—42no—
207#11719919—42no—
208#11719919—42no—
209C4×Dic53—22no—
210C4×Dic53—22no—
211C4×Dic53—41no—
212C4×Dic53—42no—
213C4×Dic53—42no—
214C4×Dic53—42no—
215C4×Dic53—42no—
216C4×Dic53SD1614yescomplemented
217C4×Dic53C22×C412yes—
218C4×Dic53C2×D412yes—
219C4×Dic53#4112yes—
220C4×Dic53#4112yes—
221C2×#88322—21no—
222C2×#88322—21no—
223C2×#88322—21no—
224C2×#88322—21no—
225C2×#88322—21no—
226C2×#88322C22×C412yes—
227#11719913—28no—
228C23×C53C22×D411yescharacteristic
229C22×D53—22no—
230C22×D53—22no—
231C22×D53—22no—
232C22×D53—41no—
233C22×D53—41no—
234C22×D53—41no—
235C22×D53—41no—
236Q8×C53—22no—
237Q8×C53—22no—
238Q8×C53—41no—
239Q8×C53—41no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—22no—
243C2×C4×C53—22no—
244C2×C4×C53—22no—
245C2×C4×C53—21no—
246C2×C4×C53—21no—
247C2×C4×C53—21no—
248C2×C4×C53—41no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53—41no—
253C2×C4×C53—41no—
254C2×C4×C53—41no—
255C2×C4×C53—41no—
256C2×C4×C53—41no—
257C2×C4×C53C2×SD1612yes—
258C2×C4×C53C23×C411yescharacteristic
259C2×C4×C53C22×D411yescharacteristic
260C2×C4×C53C2×#4111yescharacteristic
261C2×C4×C53C2×#4111yescharacteristic
262C2×C4×C53C2×#4111yescharacteristic
263C2×C4×C53C2×#4111yescharacteristic
264C2×Dic53—21no—
265C2×Dic53—21no—
266C2×Dic53—21no—
267C2×Dic53—21no—
268C2×Dic53—22no—
269C2×Dic53—22no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275C2×Dic53—41no—
276C2×Dic53—41no—
277C2×Dic53C4×D412yes—
278C2×Dic53C4×D412yes—
279C2×Dic53#12112yes—
280C2×Dic53#12112yes—
281C2×Dic53C2×SD1612yes—
282C2×Dic53C2×SD1612yes—
283C2×Dic53#13512yes—
284C2×Dic53#13712yes—
285D212—22no—
286D212—22no—
287D212—22no—
288D212C4×D414yes—
289C4×D53—48no—
290C4×D53—48no—
291C4×D53—48no—
292Dic106—22no—
293Dic106—22no—
294Dic106—41no—
295Dic106—41no—
296C8×C53—22no—
297C8×C53—41no—
298#88322—22no—
299#88322—41no—
300C4×#461—531nomaximal, Sylow
301C22×C53C2×C4×D411yescharacteristic
302C22×C53C2×#12111yescharacteristic
303C22×C53#52511yescharacteristic
304C22×C53C22×SD1611yescharacteristic
305C22×C53C2×#13511yescharacteristic
306C22×C53C2×#13711yescharacteristic
307C22×C53#57511yescharacteristic
308C2×D53—24no—
309C2×D53—42no—
310C2×D53—42no—
311C2×D53—42no—
312C2×D53—42no—
313C2×D53—42no—
314C2×D53—42no—
315C4×C53—22no—
316C4×C53—41no—
317C4×C53—42no—
318C4×C53—42no—
319C4×C53—42no—
320C4×C53—81no—
321C4×C53C2×C4×D411yescharacteristic
322C4×C53C4×#4111yescharacteristic
323C4×C53#52911yescharacteristic
324C4×C53#53211yescharacteristic
325Dic53—22no—
326Dic53—22no—
327Dic53—42no—
328Dic53—42no—
329Dic53—81no—
330Dic53C4×SD1614yescomplemented
331Dic53#46114yescomplemented
332C2×#461—531no—
333C2×C4×SD16—531no—
334C2×C4×SD16—531no—
335C4×C4⋊Q8—531no—
336C42×D4—531no—
337#1981—531no—
338#1948—531no—
339#1927—531no—
340#1927D5311yescharacteristic, complemented
341#1877—531no—
342#1877—531no—
343#1802—531no—
344C4×C4⋊C8—531no—
345C4×#103—531no—
346C4×#103—531no—
347C2×C53C4×#12111yescharacteristic
348C2×C53C2×C4×SD1611yescharacteristic
349C2×C53C4×#13711yescharacteristic
350C2×C53C2×#46111yescharacteristic
351C2×C53#303511yescharacteristic
352C2×C53#310111yescharacteristic
353C2×C53#310711yescharacteristic
354D53—44no—
355C22×SD16—531no—
356C22×SD16—531no—
357C2×C4⋊Q8—531no—
358C2×C4×Q8—531no—
359C2×C4×Q8—531no—
360C2×C4×D4—531no—
361C2×C4×D4—531no—
362C2×C4×D4—531no—
363C22×C42—1061no—
364#461—538no—
365C4×SD16—1064no—
366C4×SD16—1064no—
367C2×#106—531no—
368C2×#106C2×D5311yescharacteristic
369C2×C4⋊C8—531no—
370C2×C4⋊C8—531no—
371C2×#103—531no—
372C2×#103—531no—
373C2×#103—531no—
374C2×#103—531no—
375C2×#103C2×D5311yescharacteristic
376C2×#103C2×D5311yescharacteristic
377C2×#102—531no—
378C2×#102—531no—
379C2×C4×C8—531no—
380C2×C4×C8—531no—
381#391—531no—
382#391C2×D5311yescharacteristic
383#389—1061no—
384#389—1061no—
385#384—531no—
386C4×C4⋊C4—531no—
387C4×C4⋊C4—531no—
388C4×C4⋊C4—1061no—
389C4×C4⋊C4—1061no—
390C4×C4⋊C4C2×D5311yescharacteristic
391C4×C22⋊C4—1061no—
392C4×C22⋊C4—1061no—
393C4×C22⋊C4—1061no—
394C43—531no—
395#340—531no—
396#340—531no—
397#340—531no—
398#340C2×D5311yescharacteristic
399#336—531no—
400#336C2×D5311yescharacteristic
401C53C4×#46111yescharacteristic, complemented, Sylow
402C22×Q8—531no—
403C22×Q8C22×D5311yescharacteristic
404C22×D4—531no—
405C23×C4—1061no—
406C23×C4—1061no—
407C23×C4—1061no—
408C2×SD16—534no—
409C2×SD16—534no—
410C2×SD16—1062no—
411C2×SD16—1062no—
412C2×SD16—1062no—
413C2×SD16—1062no—
414C22×C8—531no—
415C22×C8—531no—
416C22×C8C22×D5311yescharacteristic
417C22×C8C22×D5311yescharacteristic
418C4⋊Q8—534no—
419C4⋊Q8—1062no—
420C4×Q8—1062no—
421C4×Q8—1062no—
422C4×Q8—1062no—
423C4×Q8—1062no—
424C4×D4—534no—
425C4×D4—534no—
426C4×D4—1062no—
427C4×D4—1062no—
428C4×D4—1062no—
429C4×D4—1062no—
430C2×C4⋊C4—531no—
431C2×C4⋊C4—531no—
432C2×C4⋊C4—1061no—
433C2×C4⋊C4—1061no—
434C2×C4⋊C4—1061no—
435C2×C4⋊C4—1061no—
436C2×C4⋊C4—1061no—
437C2×C4⋊C4—1061no—
438C2×C4⋊C4C22×D5311yescharacteristic
439C2×C4⋊C4C22×D5311yescharacteristic
440C2×C4⋊C4C22×D5311yescharacteristic
441C2×C22⋊C4—1061no—
442C2×C22⋊C4—1061no—
443C2×C22⋊C4—1061no—
444C2×C42—21no—
445C2×C42—531no—
446C2×C42—531no—
447C2×C42—531no—
448C2×C42—531no—
449C2×C42—531no—
450C2×C42—531no—
451C2×C42—1061no—
452C2×C42—1061no—
453C2×C42—1061no—
454C2×C42—1061no—
455C2×C42—2124no—
456C2×C42C22×D5311yescharacteristic
457#106—22no—
458#106—1062no—
459C4⋊C8—534no—
460C4⋊C8—1062no—
461#103—22no—
462#103—534no—
463#103—1062no—
464#103—1062no—
465#103—1062no—
466#103C4×D5314yescomplemented
467#102—1062no—
468#102—1062no—
469C4×C8—1062no—
470C4×C8—1062no—
471#95—21no—
472#95—21no—
473#95—21no—
474#95—1061no—
475#95—1061no—
476#95—1061no—
477#95—1061no—
478C24—1061no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—532no—
485C2×Q8—1061no—
486C2×Q8—1061no—
487C2×Q8—1061no—
488C2×Q8—1061no—
489C2×Q8—1061no—
490C2×Q8C2×C4×D5312yes—
491C2×D4—532no—
492C2×D4—532no—
493C2×D4—532no—
494C2×D4—1061no—
495C2×D4—1061no—
496C2×D4—1061no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—531no—
502C22×C4—531no—
503C22×C4—531no—
504C22×C4—531no—
505C22×C4—1064no—
506C22×C4—1064no—
507C22×C4—1064no—
508C22×C4—1061no—
509C22×C4—1061no—
510C22×C4—1061no—
511C22×C4—2122no—
512C22×C4—2122no—
513C22×C4—2122no—
514C22×C4—2122no—
515C22×C4—2122no—
516C22×C4—2122no—
517C22×C4D4×D5311yescharacteristic
518C22×C4D4×D5311yescharacteristic
519C22×C4C23×D5311yescharacteristic
520SD16—1068no—
521SD16—1068no—
522C2×C8—21no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—532no—
528C2×C8—1061no—
529C2×C8—1061no—
530C2×C8—1061no—
531C2×C8—1061no—
532C2×C8—1061no—
533C2×C8C2×C4×D5312yes—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—532no—
539C4⋊C4—1064no—
540C4⋊C4—1064no—
541C4⋊C4—1061no—
542C4⋊C4—1061no—
543C4⋊C4—1061no—
544C4⋊C4—2122no—
545C4⋊C4—2122no—
546C4⋊C4—2122no—
547C4⋊C4—2122no—
548C4⋊C4C2×C4×D5312yes—
549C4⋊C4D4×D5312yes—
550C22⋊C4—1064no—
551C22⋊C4—1064no—
552C22⋊C4—1064no—
553C42—21no—
554C42—21no—
555C42—42no—
556C42—534no—
557C42—532no—
558C42—532no—
559C42—532no—
560C42—532no—
561C42—1062no—
562C42—1062no—
563C42—2121no—
564C42—2122no—
565C42—2122no—
566C42—2122no—
567C42—2122no—
568C23—1062no—
569C23—1062no—
570C23—1062no—
571C23—2121no—
572C23—2121no—
573C23—2121no—
574C23—2121no—
575C23C2×D4×D5311yescharacteristic
576Q8—22no—
577Q8—22no—
578Q8—41no—
579Q8—41no—
580Q8—1062no—
581Q8—1062no—
582Q8—2121no—
583Q8—2121no—
584D4—534no—
585D4—1062no—
586D4—1062no—
587D4—1062no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—532no—
606C2×C4—532no—
607C2×C4—532no—
608C2×C4—532no—
609C2×C4—532no—
610C2×C4—532no—
611C2×C4—532no—
612C2×C4—532no—
613C2×C4—1061no—
614C2×C4—1061no—
615C2×C4—1061no—
616C2×C4—1061no—
617C2×C4—1062no—
618C2×C4—1062no—
619C2×C4—2128no—
620C2×C4—2128no—
621C2×C4—2128no—
622C2×C4—2121no—
623C2×C4—2121no—
624C2×C4—2121no—
625C2×C4—2121no—
626C2×C4—2121no—
627C2×C4—2121no—
628C2×C4—2121no—
629C2×C4SD16×D5312yes—
630C2×C4C22×C4×D5311yescharacteristic
631C2×C4C2×D4×D5311yescharacteristic
632C2×C4#41×D5311yescharacteristic
633C2×C4#41×D5311yescharacteristic
634C2×C4#41×D5311yescharacteristic
635C2×C4#41×D5311yescharacteristic
636C8—22no—
637C8—41no—
638C8—1062no—
639C8—2121no—
640C22—1064no—
641C22—2122no—
642C22—2122no—
643C22—2122no—
644C22—2122no—
645C22—2122no—
646C22—2122no—
647C22C4×D4×D5311yescharacteristic
648C22#121×D5311yescharacteristic
649C22#1681113111yescharacteristic
650C22C2×SD16×D5311yescharacteristic
651C22#135×D5311yescharacteristic
652C22#137×D5311yescharacteristic
653C22#1681142211yescharacteristic
654C4—22no—
655C4—42no—
656C4—41no—
657C4—42no—
658C4—42no—
659C4—81no—
660C4—534no—
661C4—534no—
662C4—1062no—
663C4—1062no—
664C4—4241no—
665C4—2122no—
666C4—2122no—
667C4C4×D4×D5311yescharacteristic
668C4#1681109111yescharacteristic
669C4#1681115611yescharacteristic
670C4#1681117211yescharacteristic
671C2—2124no—
672C2#2027722711yescharacteristic
673C2C4×SD16×D5311yescharacteristic
674C2#2028353111yescharacteristic
675C2#461×D5311yescharacteristic
676C2#2028416911yescharacteristic
677C2#2028472011yescharacteristic
678C2#2028478111yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1927C2
socleC23×C53C22×D4
prime radicalC22×D212C23
cosocleC22×C4×C53C24
hypercenter#1927D53
hypocenterC53C4×#461
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1927

Complements

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

NH ≅ G/Ntype
4. #20277284C2semidirect
6. #20275636C2semidirect
7. #20275475C2semidirect
8. #20275349C2semidirect
12. #20274563C2semidirect
13. #103×Dic53C2semidirect
14. #20274534C2semidirect
15. C53×#1927C2semidirect
25. #16810827C4semidirect
76. #13133215C2×C4semidirect
111. #13133146C2×C4semidirect
115. #13133140C2×C4semidirect
134. #103×C53C2×C4semidirect
216. C4×Dic53SD16semidirect
330. Dic53C4×SD16semidirect
331. Dic53#461semidirect
340. #1927D53semidirect
401. C53C4×#461semidirect
466. #103C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168108272
C2×#168107102
C2×#168107082
#202772842
C4×#131331792
#202756362
#202754752
#202753492
#202750522
#202750482
#202747142
#202745632
#103×Dic532
#202745342
C53×#19272
C4×#46153