Finite Group Databasebeta

← #27291654

Subgroups — #27291654

Order 13568.

894 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810671C211yesmaximal, characteristic, complemented
2C2×#16810639C211yesmaximal, characteristic
3C22×#13133145C211yesmaximal, characteristic
4C2×#16810605C211yesmaximal, characteristic
5C2×#16810395C211yesmaximal, characteristic, complemented
6C2×#16810383C211yesmaximal, characteristic, complemented
7C53×#1728C211yesmaximal, characteristic, complemented
8C22×#11719945C2211yescharacteristic
9C22×#11719917C2211yescharacteristic
10C22×#11719917C2211yescharacteristic
11C2×#13133176C2211yescharacteristic
12#16810671—22no—
13#16810671—22no—
14C2×#13133152—22no—
15#16810639—22no—
16#16810639—22no—
17C2×#13133145—22no—
18C2×#13133145—22no—
19C2×#13133145—22no—
20C2×#13133143—22no—
21#16810605—22no—
22#16810605—22no—
23C2×#13133135C412yescomplemented
24C2×#13133133C412yescomplemented
25#16810395—22no—
26#16810383—22no—
27C22×M16×C53C2211yescharacteristic
28C22×M16×C53C2211yescharacteristic
29C22×C42×C53C2211yescharacteristic
30C53×#336—22no—
31C2×#11719945—21no—
32C2×#11719945—21no—
33C2×#11719945—21no—
34C2×#11719945—21no—
35C2×#11719945—21no—
36C2×#11719945—21no—
37C2×#11719945—21no—
38C2×#11719945—21no—
39C2×#11719945—21no—
40C2×#11719945—21no—
41C2×#11719945D412yes—
42C2×#11719945Q812yes—
43C22×Dic106C2×C411yescharacteristic
44C22×D212C2×C411yescharacteristic
45C22×C22⋊D53—21no—
46C22×C4×D53—21no—
47C2×#11719917—21no—
48C2×#11719917—21no—
49C2×#11719917—21no—
50C2×#11719917—21no—
51C2×#11719917—21no—
52C2×#11719917—21no—
53C2×#11719917—21no—
54C2×#11719917—21no—
55C2×#11719917C2×C412yes—
56C2×#11719917D412yes—
57C2×#11719917D412yes—
58C2×#11719917D412yes—
59C22×#88322—21no—
60C22×#88322—21no—
61#13133176—21no—
62#13133176—21no—
63#13133176—21no—
64#13133176D412yes—
65C2×C4⋊Dic53—21no—
66C2×C22⋊Dic53—21no—
67C2×C4×Dic53—21no—
68#13133152—28no—
69#13133145—28no—
70#13133143—28no—
71#13133135—24no—
72#13133133—24no—
73C23×C4×C53—21no—
74C23×C4×C53C2311yescharacteristic
75C2×M16×C53—21no—
76C2×M16×C53—21no—
77C2×M16×C53—21no—
78C2×M16×C53—21no—
79C2×M16×C53—21no—
80C2×M16×C53—21no—
81C2×M16×C53—21no—
82C2×M16×C53—21no—
83C2×M16×C53—21no—
84C2×M16×C53—21no—
85C2×M16×C53C2×C412yes—
86C2×M16×C53D412yes—
87C22×C8×C53—21no—
88C22×C8×C53—21no—
89C2×C42×C53—21no—
90C2×C42×C53—21no—
91C2×C42×C53—21no—
92C2×C42×C53—21no—
93C2×C42×C53—21no—
94C2×C42×C53D412yes—
95C24×C53#4111yescharacteristic
96C23×D53—21no—
97C22×C4×C53—22no—
98C22×C4×C53—44no—
99C22×C4×C53—41no—
100C22×C4×C53—41no—
101C22×C4×C53C22×C411yescharacteristic
102C22×C4×C53C2×D411yescharacteristic
103C22×C4×C53C2×D411yescharacteristic
104C22×C4×C53C2×D411yescharacteristic
105C22×C4×C53C2×D411yescharacteristic
106C22×C4×C53C2×D411yescharacteristic
107C22×C4×C53C2×D411yescharacteristic
108C22×C4×C53C2×D411yescharacteristic
109C22×C4×C53C2×Q811yescharacteristic
110C22×C4×C53#4111yescharacteristic
111C22×C4×C53#4111yescharacteristic
112C22×C4×C53#4111yescharacteristic
113C22×C4×C53#4111yescharacteristic
114C22×C4×C53#4111yescharacteristic
115C2×C22⋊D53—22no—
116C2×C22⋊D53—22no—
117C2×C22⋊D53—41no—
118C2×C22⋊D53—41no—
119C2×C22⋊D53—41no—
120C2×C22⋊D53—41no—
121C22×Dic53—21no—
122C22×Dic53—41no—
123#11719945—22no—
124#11719945—22no—
125#11719945—22no—
126#11719945—22no—
127#11719945—22no—
128#11719945—22no—
129#11719945—22no—
130#11719945—22no—
131#11719945—41no—
132#11719945—41no—
133#11719945—41no—
134#11719945—41no—
135#11719945—41no—
136#11719945—41no—
137#11719945—41no—
138#11719945—41no—
139C2×D212—21no—
140C2×D212—21no—
141C2×D212—21no—
142C2×D212—21no—
143C2×D212—21no—
144C2×D212C4⋊C412yes—
145C2×C4×D53—22no—
146C2×C4×D53—22no—
147C2×C4×D53—41no—
148C2×C4×D53—41no—
149C2×C4×D53—41no—
150C2×C4×D53—41no—
151C2×Dic106—21no—
152C2×Dic106—21no—
153C2×Dic106—21no—
154C2×Dic106—21no—
155C2×Dic106—21no—
156C2×Dic106C4⋊C412yes—
157M16×C53—22no—
158M16×C53—22no—
159M16×C53—22no—
160M16×C53—22no—
161M16×C53—41no—
162M16×C53—41no—
163M16×C53—41no—
164M16×C53—41no—
165C2×C8×C53—22no—
166C2×C8×C53—22no—
167C2×C8×C53—41no—
168C2×C8×C53—41no—
169C2×C8×C53—41no—
170C2×C8×C53—41no—
171C42×C53—22no—
172C42×C53—22no—
173C42×C53—41no—
174C42×C53—41no—
175C22⋊Dic53—22no—
176C22⋊Dic53—41no—
177C4⋊Dic53—22no—
178C4⋊Dic53—41no—
179C4×Dic53—22no—
180C4×Dic53—41no—
181#11719917—22no—
182#11719917—22no—
183#11719917—22no—
184#11719917—22no—
185#11719917—22no—
186#11719917—41no—
187#11719917—41no—
188#11719917C22⋊C414yes—
189C2×#88322—22no—
190C2×#88322—22no—
191C2×#88322—22no—
192C2×#88322—22no—
193C2×#88322—41no—
194C2×#88322—41no—
195C23×C53—42no—
196C23×C53#11711yescharacteristic
197C23×C53#12111yescharacteristic
198C23×C53C22⋊Q811yescharacteristic
199C23×C53#12311yescharacteristic
200C23×C53#12311yescharacteristic
201C23×C53#12411yescharacteristic
202C23×C53#12411yescharacteristic
203C22×D53—22no—
204C22×D53—41no—
205C22×D53—41no—
206C22×D53—82no—
207C2×C4×C53—24no—
208C2×C4×C53—42no—
209C2×C4×C53—44no—
210C2×C4×C53—44no—
211C2×C4×C53—42no—
212C2×C4×C53C2×C22⋊C411yescharacteristic
213C2×C4×C53C2×C4⋊C411yescharacteristic
214C2×C4×C53C4×D411yescharacteristic
215C2×C4×C53C4×D411yescharacteristic
216C2×C4×C53C4×D411yescharacteristic
217C2×C4×C53C4×D411yescharacteristic
218C2×C4×C53#12011yescharacteristic
219C2×C4×C53#12011yescharacteristic
220C2×C4×C53#12011yescharacteristic
221C2×C4×C53#12111yescharacteristic
222C2×C4×C53#12111yescharacteristic
223C2×C4×C53#12111yescharacteristic
224C2×C4×C53#12111yescharacteristic
225C2×C4×C53#12111yescharacteristic
226C2×C4×C53#12111yescharacteristic
227C2×C4×C53#12111yescharacteristic
228C2×C4×C53C22⋊Q811yescharacteristic
229C2×C4×C53C22⋊Q811yescharacteristic
230C2×C4×C53C22⋊Q811yescharacteristic
231C2×C4×C53C22⋊Q811yescharacteristic
232C2×C4×C53C22⋊Q811yescharacteristic
233C2×C4×C53#12311yescharacteristic
234C2×C4×C53#12311yescharacteristic
235C2×C4×C53#12311yescharacteristic
236C2×C4×C53#12411yescharacteristic
237C2×C4×C53#12611yescharacteristic
238C2×C4×C53#12611yescharacteristic
239C2×C4×C53#12711yescharacteristic
240C22⋊D53—42no—
241C22⋊D53—42no—
242C22⋊D53—42no—
243C22⋊D53—42no—
244C2×Dic53—22no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—42no—
248C2×Dic53—82no—
249D212—22no—
250D212—22no—
251D212—41no—
252D212—41no—
253C4×D53—42no—
254C4×D53—42no—
255C4×D53—42no—
256C4×D53—42no—
257Dic106—22no—
258Dic106—22no—
259Dic106—41no—
260Dic106—41no—
261C8×C53—42no—
262C8×C53—42no—
263#88322—24no—
264#88322—42no—
265#11146—531nomaximal, Sylow
266C22×C53—22no—
267C22×C53—22no—
268C22×C53—22no—
269C22×C53—41no—
270C22×C53—41no—
271C22×C53—41no—
272C22×C53—41no—
273C22×C53#38011yescharacteristic
274C22×C53#38811yescharacteristic
275C22×C53#38811yescharacteristic
276C22×C53#39311yescharacteristic
277C22×C53#39411yescharacteristic
278C22×C53#39411yescharacteristic
279C22×C53#39911yescharacteristic
280C2×D53—42no—
281C2×D53—82no—
282C2×D53—82no—
283C2×D53—82no—
284C4×C53—44no—
285C4×C53#38511yescharacteristic
286C4×C53#38511yescharacteristic
287C4×C53#38611yescharacteristic
288C4×C53#39011yescharacteristic
289C4×C53#39211yescharacteristic
290C4×C53#39611yescharacteristic
291C4×C53#39611yescharacteristic
292C4×C53#39711yescharacteristic
293Dic53—42no—
294Dic53—82no—
295C2×#430—531no—
296C2×#421—531no—
297C22×#104—531no—
298C2×#413—531no—
299C2×#344—531no—
300C2×#339—531no—
301#1728D5311yescharacteristic, complemented
302C2×C53—21no—
303C2×C53—21no—
304C2×C53—21no—
305C2×C53—21no—
306C2×C53—21no—
307C2×C53—21no—
308C2×C53#143911yescharacteristic
309C2×C53#193212yes—
310D53—82no—
311C22×#41—531no—
312C22×M16—531no—
313C22×M16—531no—
314C22×M16C2×D5311yescharacteristic
315C22×M16C2×D5311yescharacteristic
316C2×#117—531no—
317C22×C42C2×D5311yescharacteristic
318#430—1062no—
319#430—1062no—
320C2×#108—1062no—
321#421—1062no—
322#421—1062no—
323C2×#104—1062no—
324C2×#104—1062no—
325C2×#104—1062no—
326C2×#104—1062no—
327#413—1062no—
328#413—1062no—
329C2×#101—532no—
330C2×#100—532no—
331#344—1062no—
332#339—1062no—
333#336—22no—
334C53#1114611yescharacteristic, complemented, Sylow
335C2×#41—532no—
336C2×#41—532no—
337C2×#41—1061no—
338C2×#41—1061no—
339C2×#41—1061no—
340C2×#41—1061no—
341C2×#41—1061no—
342C2×#41—1061no—
343C2×#41—1061no—
344C2×#41—1061no—
345C2×#41—1061no—
346C2×#41—1061no—
347C22×Q8—531no—
348C22×D4—531no—
349C22×D4—1061no—
350C23×C4—21no—
351C23×C4—1061no—
352C23×C4C22×D5311yescharacteristic
353C2×M16—21no—
354C2×M16—21no—
355C2×M16—21no—
356C2×M16—21no—
357C2×M16—21no—
358C2×M16—21no—
359C2×M16—21no—
360C2×M16—21no—
361C2×M16—21no—
362C2×M16—21no—
363C2×M16—532no—
364C2×M16—532no—
365C2×M16—532no—
366C2×M16—532no—
367C2×M16—1061no—
368C2×M16—1061no—
369C2×M16—1061no—
370C2×M16—1061no—
371C2×M16—1061no—
372C2×M16—1061no—
373C2×M16—1061no—
374C2×M16—1061no—
375C2×M16C4×D5312yes—
376C2×M16D21212yes—
377C22×C8—21no—
378C22×C8—21no—
379C22×C8—1061no—
380C22×C8—1061no—
381#117—532no—
382#117—1061no—
383#117—1061no—
384#117—1061no—
385C2×C4⋊C4—1061no—
386C2×C22⋊C4—1061no—
387C2×C42—21no—
388C2×C42—21no—
389C2×C42—21no—
390C2×C42—21no—
391C2×C42—21no—
392C2×C42—1061no—
393C2×C42C22⋊D5312yes—
394#108—1068no—
395#104—1068no—
396#104—1068no—
397#101—1064no—
398#100—1064no—
399C24—1061no—
400C24#1171994511yescharacteristic
401#41—1062no—
402#41—1062no—
403#41—1062no—
404#41—1062no—
405#41—1062no—
406#41—1062no—
407#41—1062no—
408#41—1062no—
409#41—2121no—
410#41—2121no—
411#41—2121no—
412#41—2121no—
413#41—2121no—
414#41—2121no—
415#41—2121no—
416#41—2121no—
417C2×Q8—532no—
418C2×Q8—1061no—
419C2×Q8—1061no—
420C2×Q8—1061no—
421C2×Q8—1061no—
422C2×Q8—1061no—
423C2×D4—532no—
424C2×D4—1062no—
425C2×D4—1062no—
426C2×D4—1061no—
427C2×D4—1061no—
428C2×D4—1061no—
429C2×D4—1061no—
430C2×D4—1061no—
431C2×D4—2121no—
432C2×D4—2121no—
433C2×D4—2121no—
434C2×D4—2121no—
435C22×C4—22no—
436C22×C4—44no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—1061no—
440C22×C4—1062no—
441C22×C4—1062no—
442C22×C4—2121no—
443C22×C4—2121no—
444C22×C4—2121no—
445C22×C4—2121no—
446C22×C4—2121no—
447C22×C4C2×C4×D5311yescharacteristic
448C22×C4C2×D21211yescharacteristic
449C22×C4#1171994511yescharacteristic
450C22×C4#1171994511yescharacteristic
451C22×C4#1171994511yescharacteristic
452C22×C4D4×D5311yescharacteristic
453C22×C4D4×D5311yescharacteristic
454C22×C4D4×D5311yescharacteristic
455C22×C4D4×D5311yescharacteristic
456C22×C4D4×D5311yescharacteristic
457C22×C4#1171994711yescharacteristic
458C22×C4Q8×D5311yescharacteristic
459C22×C4#1171994911yescharacteristic
460C22×C4C2×C22⋊D5311yescharacteristic
461M16—22no—
462M16—22no—
463M16—22no—
464M16—22no—
465M16—41no—
466M16—41no—
467M16—41no—
468M16—41no—
469M16—534no—
470M16—1062no—
471M16—1062no—
472M16—1062no—
473M16—1062no—
474M16—1062no—
475M16—2121no—
476M16—2121no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—41no—
480C2×C8—41no—
481C2×C8—41no—
482C2×C8—41no—
483C2×C8—1062no—
484C2×C8—1062no—
485C2×C8—1062no—
486C2×C8—1062no—
487C2×C8—2121no—
488C2×C8—2121no—
489C4⋊C4—1062no—
490C4⋊C4—2121no—
491C22⋊C4—1062no—
492C22⋊C4—2121no—
493C42—22no—
494C42—22no—
495C42—41no—
496C42—41no—
497C42—1062no—
498C42—2121no—
499C23—42no—
500C23—1062no—
501C23—4242no—
502C23—2121no—
503C23—2121no—
504C23#1313317411yescharacteristic
505C23#1313319011yescharacteristic
506C23#1313319511yescharacteristic
507C23#1313320011yescharacteristic
508C23#1313320211yescharacteristic
509C23#1313320311yescharacteristic
510C23#1313320511yescharacteristic
511Q8—1062no—
512Q8—1062no—
513Q8—2121no—
514Q8—2121no—
515D4—1062no—
516D4—1062no—
517D4—2122no—
518D4—2122no—
519D4—2122no—
520D4—2122no—
521D4—2121no—
522D4—2121no—
523C2×C4—24no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—1062no—
529C2×C4—4242no—
530C2×C4—2122no—
531C2×C4—2121no—
532C2×C4—2121no—
533C2×C4—2122no—
534C2×C4—2122no—
535C2×C4—2122no—
536C2×C4—2122no—
537C2×C4C22⋊C4×D5311yescharacteristic
538C2×C4C4⋊C4×D5311yescharacteristic
539C2×C4#1313317711yescharacteristic
540C2×C4C4×C22⋊D5311yescharacteristic
541C2×C4#1313317911yescharacteristic
542C2×C4C4×D21211yescharacteristic
543C2×C4#1313318511yescharacteristic
544C2×C4#1313318711yescharacteristic
545C2×C4#1313318711yescharacteristic
546C2×C4#1313318911yescharacteristic
547C2×C4#1313319011yescharacteristic
548C2×C4#1313319011yescharacteristic
549C2×C4#1313319011yescharacteristic
550C2×C4#1313319011yescharacteristic
551C2×C4#1313319111yescharacteristic
552C2×C4#1313319211yescharacteristic
553C2×C4#1313319311yescharacteristic
554C2×C4#1313319511yescharacteristic
555C2×C4#1313319511yescharacteristic
556C2×C4#1313319511yescharacteristic
557C2×C4#1313319711yescharacteristic
558C2×C4#1313319911yescharacteristic
559C2×C4#1313320211yescharacteristic
560C2×C4#1313320211yescharacteristic
561C2×C4#1313320311yescharacteristic
562C2×C4#1313321111yescharacteristic
563C2×C4#1313321211yescharacteristic
564C2×C4#1313321311yescharacteristic
565C8—42no—
566C8—42no—
567C8—1064no—
568C8—2122no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—4242no—
577C22—4242no—
578C22—4242no—
579C22—2122no—
580C22#1681048211yescharacteristic
581C22#1681051211yescharacteristic
582C22#1681051411yescharacteristic
583C22#1681053611yescharacteristic
584C22#1681053811yescharacteristic
585C22#1681054111yescharacteristic
586C22#1681055811yescharacteristic
587C4—44no—
588C4—4242no—
589C4—2122no—
590C4#1681049411yescharacteristic
591C4#1681049711yescharacteristic
592C4#1681050111yescharacteristic
593C4#1681052311yescharacteristic
594C4#1681053211yescharacteristic
595C4#1681054511yescharacteristic
596C4#1681054711yescharacteristic
597C4#1681055211yescharacteristic
598C2—21no—
599C2—21no—
600C2—21no—
601C2—21no—
602C2—21no—
603C2—21no—
604C2—4242no—
605C2#2027295911yescharacteristic
606C2#2027538412yes—
subgroupG/H
centerC2×C4#13133191
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1728C2
socleC22×C53#388
prime radicalC22×#11719945C22
cosocleC23×C4×C53C23
hypercenter#1728D53
hypocenterC53#11146
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1728

Complements

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

NH ≅ G/Ntype
1. C2×#16810671C2semidirect
5. C2×#16810395C2semidirect
6. C2×#16810383C2semidirect
7. C53×#1728C2semidirect
23. C2×#13133135C4semidirect
24. C2×#13133133C4semidirect
301. #1728D53semidirect
334. C53#11146semidirect

Maximal subgroups

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

subgroupindex
C2×#168106712
C2×#168106392
C22×#131331452
C2×#168106052
C2×#168103952
C2×#168103832
C53×#17282
#1114653