Finite Group Databasebeta

← #27413024

Subgroups — #27413024

Order 13568.

892 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283278C211yesmaximal, characteristic, complemented
2C2×#16810639C211yesmaximal, characteristic
3#20283235C211yesmaximal, characteristic
4C2×#16810605C211yesmaximal, characteristic
5#20276171C218yesmaximal, complemented
6C2×#16810394C211yesmaximal, characteristic, complemented
7C2×#16810380C211yesmaximal, characteristic, complemented
8C53×#1738C211yesmaximal, characteristic, complemented
9C22×#11719945C2211yescharacteristic
10C22×#11719917C2211yescharacteristic
11C2×#13133176C2211yescharacteristic
12C2×#13133176C2211yescharacteristic
13#16810651—22no—
14#16810649C2214yescomplemented
15#16810646—22no—
16#16810639—22no—
17#16810639C2214yes—
18C2×#13133143—22no—
19#16810633—21no—
20#16810629—21no—
21#16810625—22no—
22#16810620C2214yes—
23#16810605—22no—
24#16810605C2214yes—
25C2×#13133135—21no—
26C2×#13133133—21no—
27#16810394C2214yescomplemented
28#16810380C2214yescomplemented
29C22×M16×C53C2211yescharacteristic
30C22×M16×C53C2211yescharacteristic
31C22×M16×C53C2211yescharacteristic
32C53×#341C2214yescomplemented
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×#11719945—21no—
42C2×#11719945—21no—
43C2×#11719945D412yes—
44C2×#11719945C2312yes—
45C22×Dic106D411yescharacteristic
46C22×D212D411yescharacteristic
47C22×C22⋊D53—21no—
48C22×C4×D53—21no—
49C2×#11719917—21no—
50C2×#11719917—21no—
51C2×#11719917—21no—
52C2×#11719917—21no—
53C2×#11719917—21no—
54C2×#11719917C2312yes—
55C22×#88322—21no—
56#13133176—21no—
57#13133176—21no—
58#13133176—21no—
59#13133176—21no—
60#13133176—21no—
61#13133176D412yes—
62#13133176C2312yes—
63#13133176C2312yes—
64C2×C4⋊Dic53—21no—
65C2×C4⋊Dic53—21no—
66C2×C22⋊Dic53—21no—
67C2×C22⋊Dic53—21no—
68C2×C4×Dic53—21no—
69C2×C4×Dic53D412yes—
70#13133146—28no—
71#13133143—28no—
72#13133142—24no—
73#13133139—24no—
74#13133135—24no—
75#13133133—24no—
76C23×C4×C53C2311yescharacteristic
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×C53—21no—
86C2×M16×C53—21no—
87C2×M16×C53—21no—
88C2×M16×C53—21no—
89C2×M16×C53—21no—
90C2×M16×C53—21no—
91C2×M16×C53—21no—
92C2×M16×C53—21no—
93C2×M16×C53C2312yes—
94C2×M16×C53C2312yes—
95C22×C8×C53—21no—
96C22×C8×C53—21no—
97C22×C8×C53C2312yes—
98C24×C53#4111yescharacteristic
99C23×D53—21no—
100C22×C4×C53C2×D411yescharacteristic
101C22×C4×C53C2×D411yescharacteristic
102C22×C4×C53C2×D411yescharacteristic
103C22×C4×C53C2×D411yescharacteristic
104C22×C4×C53#4111yescharacteristic
105C22×C4×C53#4111yescharacteristic
106C22×C4×C53#4111yescharacteristic
107C22×C4×C53#4111yescharacteristic
108C22×C4×C53#4111yescharacteristic
109C22×C4×C53#4111yescharacteristic
110C22×C4×C53#4111yescharacteristic
111C22×C4×C53#4111yescharacteristic
112C22×C4×C53#4111yescharacteristic
113C22×C4×C53C2411yescharacteristic
114C2×C22⋊D53—22no—
115C2×C22⋊D53—22no—
116C2×C22⋊D53—41no—
117C2×C22⋊D53—41no—
118C2×C22⋊D53—41no—
119C2×C22⋊D53—41no—
120C22×Dic53—21no—
121C22×Dic53—22no—
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×D212C2×D412yes—
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×Dic106C2×D412yes—
157M16×C53—22no—
158M16×C53—22no—
159M16×C53—22no—
160M16×C53—22no—
161M16×C53—41no—
162M16×C53—41no—
163M16×C53—41no—
164M16×C53—41no—
165M16×C53—41no—
166M16×C53—41no—
167M16×C53—41no—
168M16×C53—41no—
169C2×C8×C53—22no—
170C2×C8×C53—22no—
171C2×C8×C53—22no—
172C2×C8×C53—22no—
173C2×C8×C53—22no—
174C2×C8×C53—41no—
175C2×C8×C53—41no—
176C2×C8×C53—41no—
177C2×C8×C53—41no—
178C22⋊Dic53—22no—
179C22⋊Dic53—22no—
180C22⋊Dic53—22no—
181C22⋊Dic53—41no—
182C4⋊Dic53—22no—
183C4⋊Dic53—22no—
184C4⋊Dic53—22no—
185C4⋊Dic53—41no—
186C4×Dic53—22no—
187C4×Dic53—22no—
188C4×Dic53—41no—
189C4×Dic53C2×D414yes—
190#11719917—22no—
191#11719917—22no—
192#11719917—41no—
193#11719917—41no—
194C2×#88322—22no—
195C2×#88322—41no—
196C2×#88322—41no—
197C23×C53—81no—
198C23×C53#12311yescharacteristic
199C23×C53#12311yescharacteristic
200C23×C53#12311yescharacteristic
201C23×C53#12311yescharacteristic
202C23×C53#12611yescharacteristic
203C23×C53#12611yescharacteristic
204C23×C53C2×#4111yescharacteristic
205C22×D53—22no—
206C22×D53—41no—
207C22×D53—41no—
208C22×D53—82no—
209C2×C4×C53#12111yescharacteristic
210C2×C4×C53#12111yescharacteristic
211C2×C4×C53#12111yescharacteristic
212C2×C4×C53#12111yescharacteristic
213C2×C4×C53#12311yescharacteristic
214C2×C4×C53#12311yescharacteristic
215C2×C4×C53#12311yescharacteristic
216C2×C4×C53#12311yescharacteristic
217C2×C4×C53#12311yescharacteristic
218C2×C4×C53#12311yescharacteristic
219C2×C4×C53#12311yescharacteristic
220C2×C4×C53#12311yescharacteristic
221C2×C4×C53#12411yescharacteristic
222C2×C4×C53#12411yescharacteristic
223C2×C4×C53#12411yescharacteristic
224C2×C4×C53#12411yescharacteristic
225C2×C4×C53#12611yescharacteristic
226C2×C4×C53#12611yescharacteristic
227C2×C4×C53#12611yescharacteristic
228C2×C4×C53#12611yescharacteristic
229C2×C4×C53#12611yescharacteristic
230C2×C4×C53#12611yescharacteristic
231C2×C4×C53C22×D411yescharacteristic
232C2×C4×C53C22×D411yescharacteristic
233C2×C4×C53C2×#4111yescharacteristic
234C2×C4×C53C2×#4111yescharacteristic
235C2×C4×C53C2×#4111yescharacteristic
236C2×C4×C53C2×#4111yescharacteristic
237C22⋊D53—42no—
238C22⋊D53—42no—
239C22⋊D53—42no—
240C22⋊D53—42no—
241C2×Dic53—22no—
242C2×Dic53—24no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—42no—
246C2×Dic53—82no—
247C2×Dic53—82no—
248D212—22no—
249D212—22no—
250D212—41no—
251D212—41no—
252C4×D53—42no—
253C4×D53—42no—
254C4×D53—42no—
255C4×D53—42no—
256Dic106—22no—
257Dic106—22no—
258Dic106—41no—
259Dic106—41no—
260C8×C53—42no—
261C8×C53—42no—
262C8×C53—42no—
263#88322—42no—
264#23649—531nomaximal, Sylow
265C22×C53—41no—
266C22×C53—41no—
267C22×C53—41no—
268C22×C53—41no—
269C22×C53—41no—
270C22×C53—41no—
271C22×C53—41no—
272C22×C53#39711yescharacteristic
273C22×C53#39711yescharacteristic
274C22×C53#39711yescharacteristic
275C22×C53#39711yescharacteristic
276C22×C53C2×#12311yescharacteristic
277C22×C53C2×#12311yescharacteristic
278C22×C53C2×#12611yescharacteristic
279C2×D53—42no—
280C2×D53—82no—
281C2×D53—82no—
282C2×D53—82no—
283C4×C53#39711yescharacteristic
284C4×C53#39711yescharacteristic
285C4×C53#39711yescharacteristic
286C4×C53#39711yescharacteristic
287C4×C53C2×#12111yescharacteristic
288C4×C53C2×#12311yescharacteristic
289C4×C53C2×#12411yescharacteristic
290C4×C53C2×#12611yescharacteristic
291Dic53—44no—
292Dic53—42no—
293Dic53—82no—
294#2905—531no—
295C2×#421—531no—
296#2896—531no—
297C2×#413—531no—
298#2080—538no—
299C2×#344—531no—
300#1738D5311yescharacteristic, complemented
301C2×H(3,ℤ/4)—531no—
302C2×C53—21no—
303C2×C53—21no—
304C2×C53—21no—
305C2×C53—21no—
306C2×C53—21no—
307C2×C53—21no—
308C2×C53—21no—
309C2×C53C2×#39711yescharacteristic
310D53—82no—
311C22×#41—531no—
312C22×M16—531no—
313C22×M16C2×D5311yescharacteristic
314C22×M16C2×D5311yescharacteristic
315C22×M16C2×D5311yescharacteristic
316C2×#117—531no—
317C2×#117—531no—
318#424—534no—
319#424—1062no—
320#423—1062no—
321#421—534no—
322#421—1062no—
323C2×#104—1062no—
324#419—1061no—
325#418—1061no—
326#417—1062no—
327#416—534no—
328#413—534no—
329#413—1062no—
330C2×#101—1061no—
331C2×#100—1061no—
332#344—534no—
333#341C2×D5314yescomplemented
334H(3,ℤ/4)—534no—
335C53#2364911yescharacteristic, complemented, Sylow
336C2×#41—532no—
337C2×#41—532no—
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—
347C2×#41—1061no—
348C22×Q8—531no—
349C22×D4—531no—
350C22×D4—1061no—
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—21no—
364C2×M16—21no—
365C2×M16—21no—
366C2×M16—21no—
367C2×M16—21no—
368C2×M16—21no—
369C2×M16—532no—
370C2×M16—1061no—
371C2×M16—1061no—
372C2×M16—1061no—
373C2×M16—1061no—
374C2×M16—1061no—
375C2×M16C22×D5312yes—
376C2×M16C22×D5312yes—
377C22×C8—21no—
378C22×C8—21no—
379C22×C8—1061no—
380C22×C8C22×D5312yes—
381#117—532no—
382#117—532no—
383#117—532no—
384#117—1061no—
385#117—1061no—
386#117—1061no—
387#117—1061no—
388#117—1061no—
389C2×C4⋊C4—1061no—
390C2×C4⋊C4—1061no—
391C2×C22⋊C4—1061no—
392C2×C22⋊C4—1061no—
393C2×C42—532no—
394C2×C42—1061no—
395C4⋊C8—1068no—
396#104—1068no—
397#103—1064no—
398#102—1064no—
399#101—1064no—
400#100—1064no—
401C24—1061no—
402C24#1171994911yescharacteristic
403#41—1062no—
404#41—1062no—
405#41—1062no—
406#41—1062no—
407#41—1062no—
408#41—1062no—
409#41—1062no—
410#41—1062no—
411#41—2121no—
412#41—2121no—
413#41—2121no—
414#41—2121no—
415#41—2121no—
416#41—2121no—
417#41—2121no—
418#41—2121no—
419C2×Q8—532no—
420C2×Q8—1061no—
421C2×Q8—1061no—
422C2×Q8—1061no—
423C2×Q8—1061no—
424C2×Q8—1061no—
425C2×D4—532no—
426C2×D4—1062no—
427C2×D4—1062no—
428C2×D4—1061no—
429C2×D4—1061no—
430C2×D4—1061no—
431C2×D4—1061no—
432C2×D4—1061no—
433C2×D4—2121no—
434C2×D4—2121no—
435C2×D4—2121no—
436C2×D4—2121no—
437C22×C4—1062no—
438C22×C4—1061no—
439C22×C4—1062no—
440C22×C4—1062no—
441C22×C4—2121no—
442C22×C4—2121no—
443C22×C4—2121no—
444C22×C4—2121no—
445C22×C4—2121no—
446C22×C4#1171994511yescharacteristic
447C22×C4#1171994511yescharacteristic
448C22×C4#1171994511yescharacteristic
449C22×C4#1171994511yescharacteristic
450C22×C4#1171994511yescharacteristic
451C22×C4#1171994511yescharacteristic
452C22×C4D4×D5311yescharacteristic
453C22×C4D4×D5311yescharacteristic
454C22×C4D4×D5311yescharacteristic
455C22×C4D4×D5311yescharacteristic
456C22×C4#1171994711yescharacteristic
457C22×C4#1171994711yescharacteristic
458C22×C4#1171994911yescharacteristic
459C22×C4C23×D5311yescharacteristic
460M16—22no—
461M16—22no—
462M16—22no—
463M16—22no—
464M16—41no—
465M16—41no—
466M16—41no—
467M16—41no—
468M16—41no—
469M16—41no—
470M16—41no—
471M16—41no—
472M16—1062no—
473M16—1062no—
474M16—2121no—
475M16—2121no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—41no—
482C2×C8—41no—
483C2×C8—41no—
484C2×C8—41no—
485C2×C8—1062no—
486C2×C8—2121no—
487C2×C8—2121no—
488C4⋊C4—1062no—
489C4⋊C4—1062no—
490C4⋊C4—1062no—
491C4⋊C4—2121no—
492C22⋊C4—1062no—
493C22⋊C4—1062no—
494C22⋊C4—1062no—
495C22⋊C4—2121no—
496C42—534no—
497C42—1062no—
498C42—1062no—
499C42—2121no—
500C23—81no—
501C23—1062no—
502C23—4242no—
503C23—2121no—
504C23—2121no—
505C23#1313320211yescharacteristic
506C23#1313320211yescharacteristic
507C23#1313320211yescharacteristic
508C23#1313320211yescharacteristic
509C23#1313321211yescharacteristic
510C23#1313321211yescharacteristic
511C23C2×#1171994911yescharacteristic
512Q8—1062no—
513Q8—1062no—
514Q8—2121no—
515Q8—2121no—
516D4—1062no—
517D4—1062no—
518D4—2122no—
519D4—2122no—
520D4—2122no—
521D4—2122no—
522D4—2121no—
523D4—2121no—
524C2×C4—1064no—
525C2×C4—1062no—
526C2×C4—4242no—
527C2×C4—4242no—
528C2×C4—2122no—
529C2×C4—2121no—
530C2×C4—2121no—
531C2×C4—2122no—
532C2×C4—2122no—
533C2×C4—2122no—
534C2×C4—2122no—
535C2×C4#1313319011yescharacteristic
536C2×C4#1313319011yescharacteristic
537C2×C4#1313319011yescharacteristic
538C2×C4#1313319011yescharacteristic
539C2×C4#1313319911yescharacteristic
540C2×C4#1313319911yescharacteristic
541C2×C4#1313319911yescharacteristic
542C2×C4#1313319911yescharacteristic
543C2×C4#1313320211yescharacteristic
544C2×C4#1313320211yescharacteristic
545C2×C4#1313320211yescharacteristic
546C2×C4#1313320211yescharacteristic
547C2×C4#1313320311yescharacteristic
548C2×C4#1313320311yescharacteristic
549C2×C4#1313320311yescharacteristic
550C2×C4#1313320311yescharacteristic
551C2×C4#1313321111yescharacteristic
552C2×C4#1313321111yescharacteristic
553C2×C4#1313321111yescharacteristic
554C2×C4#1313321111yescharacteristic
555C2×C4#1313321211yescharacteristic
556C2×C4#1313321211yescharacteristic
557C2×C4C2×D4×D5311yescharacteristic
558C2×C4C2×D4×D5311yescharacteristic
559C2×C4C2×#1171994711yescharacteristic
560C2×C4C2×#1171994511yescharacteristic
561C2×C4C2×#1171994511yescharacteristic
562C2×C4C2×#1171994511yescharacteristic
563C8—42no—
564C8—42no—
565C8—42no—
566C8—2122no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—4242no—
575C22—4242no—
576C22—4242no—
577C22—2122no—
578C22#1681055211yescharacteristic
579C22#1681055211yescharacteristic
580C22#1681055211yescharacteristic
581C22#1681055211yescharacteristic
582C22C2×#1313320211yescharacteristic
583C22C2×#1313320211yescharacteristic
584C22C2×#1313321211yescharacteristic
585C4—4242no—
586C4—2124no—
587C4—2122no—
588C4#1681055211yescharacteristic
589C4#1681055211yescharacteristic
590C4#1681055211yescharacteristic
591C4#1681055211yescharacteristic
592C4C2×#1313319011yescharacteristic
593C4C2×#1313319911yescharacteristic
594C4C2×#1313320311yescharacteristic
595C4C2×#1313321111yescharacteristic
596C2—21no—
597C2—21no—
598C2—21no—
599C2—21no—
600C2—21no—
601C2—21no—
602C2—21no—
603C2—4242no—
604C2C2×#1681055211yescharacteristic
subgroupG/H
centerC4C2×#13133203
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1738C2
socleC2×C53C2×#397
prime radicalC22×#11719945C22
hypercenter#1738D53
hypocenterC53#23649
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C23×C4 ◁ #1738

Complements

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

NH ≅ G/Ntype
1. #20283278C2semidirect
5. #20276171C2semidirect
6. C2×#16810394C2semidirect
7. C2×#16810380C2semidirect
8. C53×#1738C2semidirect
14. #16810649C22semidirect
27. #16810394C22semidirect
28. #16810380C22semidirect
32. C53×#341C22semidirect
300. #1738D53semidirect
333. #341C2×D53semidirect
335. C53#23649semidirect

Maximal subgroups

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

subgroupindex
#202832782
C2×#168106392
#202832352
C2×#168106052
#202761712
C2×#168103942
C2×#168103802
C53×#17382
#2364953