Finite Group Databasebeta

← #27413983

Subgroups — #27413983

Order 13568.

988 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810671C211yesmaximal, characteristic, complemented
2#20283278C211yesmaximal, characteristic, complemented
3C2×#16810641C211yesmaximal, characteristic
4C2×#16810639C211yesmaximal, characteristic
5#20283159C211yesmaximal, characteristic
6#20276271C218yesmaximal, complemented
7C2×#16810379C211yesmaximal, characteristic, complemented
8C53×#2905C211yesmaximal, characteristic, complemented
9C22×#11719945C2211yescharacteristic
10C22×#11719917C2211yescharacteristic
11C22×#11719917C2211yescharacteristic
12C2×#13133176C2211yescharacteristic
13#16810671—22no—
14#16810671C2214yescomplemented
15C2×#13133152—22no—
16#16810651—22no—
17#16810649C2214yescomplemented
18#16810646—22no—
19#16810641—22no—
20#16810641C2214yes—
21#16810639—22no—
22#16810639C2214yes—
23C2×#13133145—22no—
24C2×#13133143—22no—
25#16810591—22no—
26#16810589C2214yes—
27#16810586—22no—
28#16810379C2214yescomplemented
29C22×M16×C53C2211yescharacteristic
30C22×M16×C53C2211yescharacteristic
31C2×#117×C53C2211yescharacteristic
32C53×#424—22no—
33C53×#424C2214yescomplemented
34C53×#423—22no—
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×#11719945—21no—
44C2×#11719945—21no—
45C2×#11719945Q812yes—
46C2×#11719945C2312yes—
47C22×Dic106Q811yescharacteristic
48C22×D212Q811yescharacteristic
49C22×C22⋊D53—21no—
50C22×C4×D53—21no—
51C2×#11719917—21no—
52C2×#11719917—21no—
53C2×#11719917—21no—
54C2×#11719917—21no—
55C2×#11719917—21no—
56C2×#11719917—21no—
57C2×#11719917—21no—
58C2×#11719917—21no—
59C2×#11719917—21no—
60C2×#11719917D412yes—
61C2×#11719917C2312yes—
62C2×#11719917C2312yes—
63C22×#88322—21no—
64C22×#88322D412yes—
65#13133176—21no—
66#13133176—21no—
67#13133176D412yes—
68#13133176C2312yes—
69C2×C4⋊Dic53—21no—
70C2×C22⋊Dic53—21no—
71C2×C4×Dic53D412yes—
72#13133152—28no—
73#13133146—28no—
74#13133145—28no—
75#13133143—28no—
76#13133129—28no—
77C23×C4×C53C2311yescharacteristic
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×C53C2312yes—
90C22×C8×C53—21no—
91C22×C8×C53C2312yes—
92#117×C53—21no—
93#117×C53—21no—
94#117×C53D412yes—
95#117×C53C2312yes—
96C2×C4⋊C4×C53—21no—
97C2×C22⋊C4×C53—21no—
98C2×C42×C53D412yes—
99C4⋊C8×C53—28no—
100C24×C53#4111yescharacteristic
101C23×D53—21no—
102C22×C4×C53—22no—
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×C53C2×Q811yescharacteristic
111C22×C4×C53#4111yescharacteristic
112C22×C4×C53#4111yescharacteristic
113C22×C4×C53#4111yescharacteristic
114C22×C4×C53#4111yescharacteristic
115C22×C4×C53#4111yescharacteristic
116C22×C4×C53C2411yescharacteristic
117C2×C22⋊D53—22no—
118C2×C22⋊D53—22no—
119C2×C22⋊D53—41no—
120C2×C22⋊D53—41no—
121C2×C22⋊D53—41no—
122C2×C22⋊D53—41no—
123C22×Dic53—21no—
124C22×Dic53—22no—
125#11719945—22no—
126#11719945—22no—
127#11719945—22no—
128#11719945—22no—
129#11719945—22no—
130#11719945—22no—
131#11719945—22no—
132#11719945—22no—
133#11719945—41no—
134#11719945—41no—
135#11719945—41no—
136#11719945—41no—
137#11719945—41no—
138#11719945—41no—
139#11719945—41no—
140#11719945—41no—
141C2×D212—21no—
142C2×D212—21no—
143C2×D212—21no—
144C2×D212—21no—
145C2×D212—21no—
146C2×D212C2×Q812yes—
147C2×C4×D53—22no—
148C2×C4×D53—22no—
149C2×C4×D53—41no—
150C2×C4×D53—41no—
151C2×C4×D53—41no—
152C2×C4×D53—41no—
153C2×Dic106—21no—
154C2×Dic106—21no—
155C2×Dic106—21no—
156C2×Dic106—21no—
157C2×Dic106—21no—
158C2×Dic106C2×Q812yes—
159M16×C53—22no—
160M16×C53—22no—
161M16×C53—41no—
162M16×C53—41no—
163M16×C53—41no—
164M16×C53—41no—
165M16×C53—41no—
166M16×C53—41no—
167C2×C8×C53—22no—
168C2×C8×C53—22no—
169C2×C8×C53—22no—
170C2×C8×C53—22no—
171C2×C8×C53—41no—
172C2×C8×C53—41no—
173C4⋊C4×C53—22no—
174C4⋊C4×C53—22no—
175C22⋊C4×C53—22no—
176C22⋊C4×C53—22no—
177C42×C53—22no—
178C42×C53C2×D414yes—
179C22⋊Dic53—22no—
180C22⋊Dic53—22no—
181C4⋊Dic53—22no—
182C4⋊Dic53—22no—
183C4×Dic53—22no—
184C4×Dic53C2×D414yes—
185#11719917—22no—
186#11719917—22no—
187#11719917—22no—
188#11719917—22no—
189#11719917—22no—
190#11719917—22no—
191#11719917—41no—
192#11719917—41no—
193C2×#88322—22no—
194C2×#88322—22no—
195C2×#88322—22no—
196C2×#88322—41no—
197C2×#88322—41no—
198C2×#88322C2×D414yes—
199C23×C53—81no—
200C23×C53C22⋊Q811yescharacteristic
201C23×C53C22⋊Q811yescharacteristic
202C23×C53#12311yescharacteristic
203C23×C53#12311yescharacteristic
204C23×C53#12311yescharacteristic
205C23×C53#12311yescharacteristic
206C23×C53C2×#4111yescharacteristic
207C22×D53—22no—
208C22×D53—41no—
209C22×D53—41no—
210C22×D53—82no—
211C2×C4×C53—24no—
212C2×C4×C53—82no—
213C2×C4×C53C22⋊Q811yescharacteristic
214C2×C4×C53C22⋊Q811yescharacteristic
215C2×C4×C53C22⋊Q811yescharacteristic
216C2×C4×C53C22⋊Q811yescharacteristic
217C2×C4×C53C22⋊Q811yescharacteristic
218C2×C4×C53C22⋊Q811yescharacteristic
219C2×C4×C53C22⋊Q811yescharacteristic
220C2×C4×C53C22⋊Q811yescharacteristic
221C2×C4×C53C22⋊Q811yescharacteristic
222C2×C4×C53C22⋊Q811yescharacteristic
223C2×C4×C53#12311yescharacteristic
224C2×C4×C53#12311yescharacteristic
225C2×C4×C53#12311yescharacteristic
226C2×C4×C53#12311yescharacteristic
227C2×C4×C53#12311yescharacteristic
228C2×C4×C53#12311yescharacteristic
229C2×C4×C53#12311yescharacteristic
230C2×C4×C53#12311yescharacteristic
231C2×C4×C53#12711yescharacteristic
232C2×C4×C53#12711yescharacteristic
233C2×C4×C53#12711yescharacteristic
234C2×C4×C53#12711yescharacteristic
235C2×C4×C53C22×D411yescharacteristic
236C2×C4×C53C22×D411yescharacteristic
237C2×C4×C53C22×D411yescharacteristic
238C2×C4×C53C22×Q811yescharacteristic
239C2×C4×C53C2×#4111yescharacteristic
240C2×C4×C53C2×#4111yescharacteristic
241C22⋊D53—42no—
242C22⋊D53—42no—
243C22⋊D53—42no—
244C22⋊D53—42no—
245C2×Dic53—22no—
246C2×Dic53—24no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—82no—
250D212—22no—
251D212—22no—
252D212—41no—
253D212—41no—
254C4×D53—42no—
255C4×D53—42no—
256C4×D53—42no—
257C4×D53—42no—
258Dic106—22no—
259Dic106—22no—
260Dic106—41no—
261Dic106—41no—
262C8×C53—42no—
263C8×C53—42no—
264#88322—24no—
265#88322—42no—
266#23740—531nomaximal, Sylow
267C22×C53—41no—
268C22×C53—41no—
269C22×C53—41no—
270C22×C53—41no—
271C22×C53—41no—
272C22×C53—41no—
273C22×C53—41no—
274C22×C53#39911yescharacteristic
275C22×C53#39911yescharacteristic
276C22×C53#39911yescharacteristic
277C22×C53#39911yescharacteristic
278C22×C53C2×C22⋊Q811yescharacteristic
279C22×C53C2×#12311yescharacteristic
280C22×C53C2×#12311yescharacteristic
281C2×D53—44no—
282C2×D53—42no—
283C2×D53—82no—
284C2×D53—82no—
285C4×C53—44no—
286C4×C53#39911yescharacteristic
287C4×C53#39911yescharacteristic
288C4×C53#39911yescharacteristic
289C4×C53#39911yescharacteristic
290C4×C53C2×C22⋊Q811yescharacteristic
291C4×C53C2×C22⋊Q811yescharacteristic
292C4×C53C2×#12311yescharacteristic
293C4×C53C2×#12711yescharacteristic
294Dic53—44no—
295Dic53—42no—
296C2×#430—531no—
297#2905—531no—
298#2905D5311yescharacteristic, complemented
299C2×#421—531no—
300C2×#421—531no—
301#2879—531no—
302#2098—538no—
303C2×H(3,ℤ/4)—531no—
304C2×C53—21no—
305C2×C53—21no—
306C2×C53—21no—
307C2×C53—21no—
308C2×C53—21no—
309C2×C53—21no—
310C2×C53—21no—
311C2×C53C2×#39911yescharacteristic
312D53—82no—
313C22×#41—531no—
314C22×M16—531no—
315C22×M16—531no—
316C22×M16C2×D5311yescharacteristic
317C22×M16C2×D5311yescharacteristic
318C2×#117—531no—
319C2×#117C2×D5311yescharacteristic
320#430—534no—
321#430—1062no—
322C2×#108—1062no—
323#424—22no—
324#424—534no—
325#424—1062no—
326#424C2×D5314yescomplemented
327#423—22no—
328#423—1062no—
329#421—534no—
330#421—534no—
331#421—1062no—
332#421—1062no—
333C2×#104—1062no—
334C2×#104—1062no—
335#408—534no—
336#408—1062no—
337#407—1062no—
338H(3,ℤ/4)—534no—
339C53#2374011yescharacteristic, complemented, Sylow
340C2×#41—532no—
341C2×#41—532no—
342C2×#41—1061no—
343C2×#41—1061no—
344C2×#41—1061no—
345C2×#41—1061no—
346C2×#41—1061no—
347C2×#41—1061no—
348C2×#41—1061no—
349C2×#41—1061no—
350C2×#41—1061no—
351C2×#41—1061no—
352C22×Q8—531no—
353C22×D4—531no—
354C22×D4—1061no—
355C23×C4—1061no—
356C23×C4C22×D5311yescharacteristic
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—532no—
369C2×M16—532no—
370C2×M16—532no—
371C2×M16—1061no—
372C2×M16—1061no—
373C2×M16—1061no—
374C2×M16—1061no—
375C2×M16—1061no—
376C2×M16—1061no—
377C2×M16—1061no—
378C2×M16—1061no—
379C2×M16—1061no—
380C2×M16C22×D5312yes—
381C22×C8—21no—
382C22×C8—532no—
383C22×C8—1061no—
384C22×C8C22×D5312yes—
385#117—21no—
386#117—21no—
387#117—532no—
388#117—532no—
389#117—1061no—
390#117—1061no—
391#117C22⋊D5312yes—
392#117C22×D5312yes—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—1061no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—1061no—
397C2×C42—532no—
398C2×C42C22⋊D5312yes—
399#108—1068no—
400C4⋊C8—28no—
401C4⋊C8—1068no—
402#104—1068no—
403#104—1068no—
404C22⋊C8—1068no—
405C24—1061no—
406C24#1171994511yescharacteristic
407#41—1062no—
408#41—1062no—
409#41—1062no—
410#41—1062no—
411#41—1062no—
412#41—1062no—
413#41—1062no—
414#41—1062no—
415#41—2121no—
416#41—2121no—
417#41—2121no—
418#41—2121no—
419#41—2121no—
420#41—2121no—
421#41—2121no—
422#41—2121no—
423C2×Q8—532no—
424C2×Q8—1061no—
425C2×Q8—1061no—
426C2×Q8—1061no—
427C2×Q8—1061no—
428C2×Q8—1061no—
429C2×D4—532no—
430C2×D4—1062no—
431C2×D4—1062no—
432C2×D4—1061no—
433C2×D4—1061no—
434C2×D4—1061no—
435C2×D4—1061no—
436C2×D4—1061no—
437C2×D4—2121no—
438C2×D4—2121no—
439C2×D4—2121no—
440C2×D4—2121no—
441C22×C4—22no—
442C22×C4—1062no—
443C22×C4—1061no—
444C22×C4—1062no—
445C22×C4—1062no—
446C22×C4—2121no—
447C22×C4—2121no—
448C22×C4—2121no—
449C22×C4—2121no—
450C22×C4#1171994511yescharacteristic
451C22×C4#1171994511yescharacteristic
452C22×C4#1171994511yescharacteristic
453C22×C4D4×D5311yescharacteristic
454C22×C4D4×D5311yescharacteristic
455C22×C4D4×D5311yescharacteristic
456C22×C4D4×D5311yescharacteristic
457C22×C4Q8×D5311yescharacteristic
458C22×C4Q8×D5311yescharacteristic
459C22×C4#1171994911yescharacteristic
460C22×C4#1171994911yescharacteristic
461C22×C4C2×C22⋊D5311yescharacteristic
462C22×C4C2×C22⋊D5311yescharacteristic
463C22×C4C23×D5311yescharacteristic
464M16—22no—
465M16—22no—
466M16—41no—
467M16—41no—
468M16—41no—
469M16—41no—
470M16—41no—
471M16—41no—
472M16—1062no—
473M16—1062no—
474M16—1062no—
475M16—1062no—
476M16—1062no—
477M16—1062no—
478M16—2121no—
479M16—2121no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—41no—
485C2×C8—41no—
486C2×C8—534no—
487C2×C8—1062no—
488C2×C8—1062no—
489C2×C8—1062no—
490C2×C8—2121no—
491C2×C8—2121no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—1062no—
495C4⋊C4—1062no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—1062no—
499C22⋊C4—1062no—
500C42—22no—
501C42—534no—
502C42—1062no—
503C42C2×C22⋊D5314yes—
504C23—81no—
505C23—1062no—
506C23—4242no—
507C23—2121no—
508C23—2121no—
509C23#1313319511yescharacteristic
510C23#1313319511yescharacteristic
511C23#1313320011yescharacteristic
512C23#1313320011yescharacteristic
513C23#1313320211yescharacteristic
514C23#1313320211yescharacteristic
515C23C2×#1171994511yescharacteristic
516Q8—1062no—
517Q8—1062no—
518Q8—2121no—
519Q8—2121no—
520D4—1062no—
521D4—1062no—
522D4—2122no—
523D4—2122no—
524D4—2122no—
525D4—2122no—
526D4—2121no—
527D4—2121no—
528C2×C4—24no—
529C2×C4—82no—
530C2×C4—1062no—
531C2×C4—1064no—
532C2×C4—4242no—
533C2×C4—2121no—
534C2×C4—2121no—
535C2×C4—2122no—
536C2×C4—2122no—
537C2×C4—2122no—
538C2×C4—2122no—
539C2×C4#1313319311yescharacteristic
540C2×C4#1313319311yescharacteristic
541C2×C4#1313319311yescharacteristic
542C2×C4#1313319311yescharacteristic
543C2×C4#1313319511yescharacteristic
544C2×C4#1313319511yescharacteristic
545C2×C4#1313319511yescharacteristic
546C2×C4#1313319511yescharacteristic
547C2×C4#1313319511yescharacteristic
548C2×C4#1313319511yescharacteristic
549C2×C4#1313320011yescharacteristic
550C2×C4#1313320011yescharacteristic
551C2×C4#1313320211yescharacteristic
552C2×C4#1313320211yescharacteristic
553C2×C4#1313320211yescharacteristic
554C2×C4#1313320211yescharacteristic
555C2×C4#1313320211yescharacteristic
556C2×C4#1313320211yescharacteristic
557C2×C4#1313321311yescharacteristic
558C2×C4#1313321311yescharacteristic
559C2×C4#1313321311yescharacteristic
560C2×C4#1313321311yescharacteristic
561C2×C4C2×D4×D5311yescharacteristic
562C2×C4C2×D4×D5311yescharacteristic
563C2×C4C22×C22⋊D5311yescharacteristic
564C2×C4C2×Q8×D5311yescharacteristic
565C2×C4C2×#1171994511yescharacteristic
566C2×C4C2×#1171994911yescharacteristic
567C8—42no—
568C8—42no—
569C8—1064no—
570C8—2122no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—4242no—
579C22—4242no—
580C22—2124no—
581C22—2122no—
582C22#1681055811yescharacteristic
583C22#1681055811yescharacteristic
584C22#1681055811yescharacteristic
585C22#1681055811yescharacteristic
586C22C2×#1313319511yescharacteristic
587C22C2×#1313320011yescharacteristic
588C22C2×#1313320211yescharacteristic
589C4—44no—
590C4—2124no—
591C4—2122no—
592C4#1681055811yescharacteristic
593C4#1681055811yescharacteristic
594C4#1681055811yescharacteristic
595C4#1681055811yescharacteristic
596C4C2×#1313319311yescharacteristic
597C4C2×#1313319511yescharacteristic
598C4C2×#1313320211yescharacteristic
599C4C2×#1313321311yescharacteristic
600C2—21no—
601C2—21no—
602C2—21no—
603C2—21no—
604C2—21no—
605C2—21no—
606C2—21no—
607C2—4242no—
608C2C2×#1681055811yescharacteristic
subgroupG/H
centerC4C2×#13133213
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2905C2
socleC2×C53C2×#399
prime radicalC22×#11719945C22
hypercenter#2905D53
hypocenterC53#23740
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C23×C4 ◁ #2905

Complements

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

NH ≅ G/Ntype
1. C2×#16810671C2semidirect
2. #20283278C2semidirect
6. #20276271C2semidirect
7. C2×#16810379C2semidirect
8. C53×#2905C2semidirect
14. #16810671C22semidirect
17. #16810649C22semidirect
28. #16810379C22semidirect
33. C53×#424C22semidirect
298. #2905D53semidirect
326. #424C2×D53semidirect
339. C53#23740semidirect

Maximal subgroups

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

subgroupindex
C2×#168106712
#202832782
C2×#168106412
C2×#168106392
#202831592
#202762712
C2×#168103792
C53×#29052
#2374053