Finite Group Databasebeta

← #27386682

Subgroups — #27386682

Order 13568.

1000 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×#16810672C211yesmaximal, characteristic, complemented
2C2×#16810641C211yesmaximal, characteristic
3C2×#16810641C211yesmaximal, characteristic
4C2×#16810605C211yesmaximal, characteristic
5#20283134C211yesmaximal, characteristic, complemented
6#20275385C218yesmaximal, complemented
7C2×#16810393C211yesmaximal, characteristic, complemented
8C53×#1733C211yesmaximal, characteristic, complemented
9C22×#11719945C2211yescharacteristic
10C22×#11719917C2211yescharacteristic
11C22×#11719917C2211yescharacteristic
12C22×#11719917C2211yescharacteristic
13#16810672—22no—
14#16810672C2214yescomplemented
15C2×#13133153—22no—
16#16810641—22no—
17#16810641—22no—
18#16810641—22no—
19#16810641C2214yes—
20C2×#13133145—22no—
21C2×#13133145C2214yes—
22#16810605C414yescomplemented
23#16810605C2214yes—
24C2×#13133135C412yescomplemented
25C2×#13133133C412yescomplemented
26#16810579—22no—
27#16810579—22no—
28C4×#11719917C2214yescomplemented
29#16810393C2214yescomplemented
30C22×M16×C53C2211yescharacteristic
31C2×#117×C53C2211yescharacteristic
32C2×#117×C53C2211yescharacteristic
33C53×#339C2214yescomplemented
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×#11719945—21no—
44C2×#11719945C2×C412yes—
45C2×#11719945C2312yes—
46C22×Dic106C2×C411yescharacteristic
47C22×D212C2×C411yescharacteristic
48C22×C22⋊D53—21no—
49C22×C4×D53—21no—
50C2×#11719917—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×#11719917—21no—
61C2×#11719917C2×C412yes—
62C2×#11719917C2×C412yes—
63C2×#11719917D412yes—
64C2×#11719917D412yes—
65C2×#11719917C2312yes—
66C2×#11719917C2312yes—
67C2×#11719917C2312yes—
68C22×#88322—21no—
69C22×#88322—21no—
70C22×#88322—21no—
71#13133153—28no—
72#13133145—28no—
73#13133145—28no—
74#13133135C2×C418yescomplemented
75#13133133C2×C418yescomplemented
76#13133128—24no—
77C4×#88322—24no—
78C23×C4×C53C2311yescharacteristic
79C2×M16×C53—21no—
80C2×M16×C53—21no—
81C2×M16×C53—21no—
82C2×M16×C53C2×C412yes—
83C2×M16×C53C2×C412yes—
84C2×M16×C53C2312yes—
85C22×C8×C53—21no—
86#117×C53—21no—
87#117×C53—21no—
88#117×C53—21no—
89#117×C53—21no—
90#117×C53—21no—
91#117×C53D412yes—
92#117×C53D412yes—
93#117×C53C2312yes—
94C2×C4⋊C4×C53—21no—
95C2×C4⋊C4×C53—21no—
96C2×C22⋊C4×C53—21no—
97C2×C22⋊C4×C53—21no—
98C2×C42×C53—21no—
99C2×C42×C53C2312yes—
100C24×C53#4111yescharacteristic
101C23×D53—21no—
102C22×C4×C53—22no—
103C22×C4×C53—41no—
104C22×C4×C53C22×C411yescharacteristic
105C22×C4×C53C22×C411yescharacteristic
106C22×C4×C53C2×D411yescharacteristic
107C22×C4×C53C2×D411yescharacteristic
108C22×C4×C53C2×D411yescharacteristic
109C22×C4×C53C2×D411yescharacteristic
110C22×C4×C53#4111yescharacteristic
111C22×C4×C53#4111yescharacteristic
112C22×C4×C53#4111yescharacteristic
113C22×C4×C53#4111yescharacteristic
114C22×C4×C53#4111yescharacteristic
115C22×C4×C53#4111yescharacteristic
116C22×C4×C53#4111yescharacteristic
117C22×C4×C53C2411yescharacteristic
118C2×C22⋊D53—22no—
119C2×C22⋊D53—22no—
120C2×C22⋊D53—41no—
121C2×C22⋊D53—41no—
122C2×C22⋊D53—41no—
123C2×C22⋊D53—41no—
124C22×Dic53—21no—
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×D212C22×C412yes—
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×Dic106C22×C412yes—
159M16×C53—22no—
160M16×C53—22no—
161M16×C53—22no—
162M16×C53C22×C414yes—
163C2×C8×C53—22no—
164C2×C8×C53—22no—
165C2×C8×C53—22no—
166C4⋊C4×C53—22no—
167C4⋊C4×C53—22no—
168C4⋊C4×C53—22no—
169C4⋊C4×C53—41no—
170C22⋊C4×C53—22no—
171C22⋊C4×C53—22no—
172C22⋊C4×C53—22no—
173C22⋊C4×C53—41no—
174C42×C53—22no—
175C42×C53—22no—
176C42×C53—41no—
177C42×C53C2×D414yes—
178#11719917—22no—
179#11719917—22no—
180#11719917—22no—
181#11719917—22no—
182#11719917—22no—
183#11719917—22no—
184#11719917—22no—
185#11719917—22no—
186#11719917—41no—
187#11719917—41no—
188#11719917C22×C414yes—
189#11719917C2×D414yes—
190C2×#88322—22no—
191C2×#88322—22no—
192C2×#88322—22no—
193C2×#88322—22no—
194C2×#88322—22no—
195C2×#88322—22no—
196C2×#88322—22no—
197C2×#88322—41no—
198C2×#88322—41no—
199C23×C53—81no—
200C23×C53#11711yescharacteristic
201C23×C53#11711yescharacteristic
202C23×C53#12311yescharacteristic
203C23×C53#12311yescharacteristic
204C23×C53#12411yescharacteristic
205C23×C53#12411yescharacteristic
206C23×C53C2×#4111yescharacteristic
207C22×D53—22no—
208C22×D53—41no—
209C22×D53—41no—
210C22×D53—82no—
211C2×C4×C53—24no—
212C2×C4×C53—42no—
213C2×C4×C53—82no—
214C2×C4×C53—82no—
215C2×C4×C53#11711yescharacteristic
216C2×C4×C53#11711yescharacteristic
217C2×C4×C53C4×D411yescharacteristic
218C2×C4×C53C4×D411yescharacteristic
219C2×C4×C53C4×D411yescharacteristic
220C2×C4×C53C4×D411yescharacteristic
221C2×C4×C53C4×D411yescharacteristic
222C2×C4×C53C4×D411yescharacteristic
223C2×C4×C53C4×D411yescharacteristic
224C2×C4×C53C4×D411yescharacteristic
225C2×C4×C53#12111yescharacteristic
226C2×C4×C53#12111yescharacteristic
227C2×C4×C53#12111yescharacteristic
228C2×C4×C53#12111yescharacteristic
229C2×C4×C53#12311yescharacteristic
230C2×C4×C53#12311yescharacteristic
231C2×C4×C53#12411yescharacteristic
232C2×C4×C53#12411yescharacteristic
233C2×C4×C53#12611yescharacteristic
234C2×C4×C53#12611yescharacteristic
235C2×C4×C53#12611yescharacteristic
236C2×C4×C53#12611yescharacteristic
237C2×C4×C53C23×C411yescharacteristic
238C2×C4×C53C22×D411yescharacteristic
239C2×C4×C53C22×D411yescharacteristic
240C2×C4×C53C2×#4111yescharacteristic
241C2×C4×C53C2×#4111yescharacteristic
242C2×C4×C53C2×#4111yescharacteristic
243C22⋊D53—42no—
244C22⋊D53—42no—
245C22⋊D53—42no—
246C22⋊D53—42no—
247C2×Dic53—22no—
248C2×Dic53—41no—
249C2×Dic53—41no—
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—24no—
263#88322—24no—
264#88322—24no—
265#88322—42no—
266#21080—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#38811yescharacteristic
275C22×C53#38811yescharacteristic
276C22×C53#38811yescharacteristic
277C22×C53#38811yescharacteristic
278C22×C53C2×#11711yescharacteristic
279C22×C53C2×#12311yescharacteristic
280C22×C53C2×#12411yescharacteristic
281C2×D53—42no—
282C2×D53—82no—
283C2×D53—82no—
284C2×D53—82no—
285C4×C53—44no—
286C4×C53—82no—
287C4×C53#38811yescharacteristic
288C4×C53#38811yescharacteristic
289C4×C53#38811yescharacteristic
290C4×C53#38811yescharacteristic
291C4×C53C2×C4×D411yescharacteristic
292C4×C53C2×C4×D411yescharacteristic
293C4×C53C2×#12111yescharacteristic
294C4×C53C2×#12611yescharacteristic
295Dic53—42no—
296C2×#430—531no—
297C2×#421—531no—
298C2×#421—531no—
299C2×#413—531no—
300#2873—531no—
301#1932—538no—
302C2×#344—531no—
303#1733D5311yescharacteristic, complemented
304C2×C53—21no—
305C2×C53—21no—
306C2×C53—21no—
307C2×C53—21no—
308C2×C53—21no—
309C2×C53—21no—
310C2×C53—21no—
311C2×C53C2×#38811yescharacteristic
312D53—82no—
313C22×#41—531no—
314C22×M16—531no—
315C22×M16—531no—
316C22×M16—531no—
317C22×M16C2×D5311yescharacteristic
318C2×#117C2×D5311yescharacteristic
319C2×#117C2×D5311yescharacteristic
320#430—534no—
321#430—1062no—
322C2×#108—1062no—
323#421—534no—
324#421—1062no—
325#421—1062no—
326#421—1062no—
327C2×#104—534no—
328C2×#104—1062no—
329#413—534no—
330#413—534no—
331C2×#101—532no—
332C2×#100—532no—
333#405—1062no—
334#405—1062no—
335C4×M16—534no—
336#344—534no—
337#339C2×D5314yescomplemented
338C53#2108011yescharacteristic, complemented, Sylow
339C2×#41—532no—
340C2×#41—532no—
341C2×#41—1061no—
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—
351C22×Q8—531no—
352C22×D4—531no—
353C22×D4—1061no—
354C23×C4—1061no—
355C23×C4C22×D5311yescharacteristic
356C2×M16—21no—
357C2×M16—21no—
358C2×M16—21no—
359C2×M16—532no—
360C2×M16—532no—
361C2×M16—532no—
362C2×M16—532no—
363C2×M16—532no—
364C2×M16—532no—
365C2×M16—532no—
366C2×M16—1061no—
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×M16—1061no—
376C2×M16—1061no—
377C2×M16C4×D5312yes—
378C2×M16C4×D5312yes—
379C2×M16C22×D5312yes—
380C22×C8—21no—
381C22×C8—1061no—
382C22×C8—1061no—
383C22×C8—1061no—
384#117—21no—
385#117—21no—
386#117—21no—
387#117—21no—
388#117—21no—
389#117D21212yes—
390#117D21212yes—
391#117C22×D5312yes—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—21no—
396C2×C42—21no—
397C2×C42C22×D5312yes—
398#108—1068no—
399#104—1068no—
400#104—1068no—
401#101—538no—
402#100—538no—
403#97—1064no—
404C4×C8—1064no—
405C24—1061no—
406C24#1171994711yescharacteristic
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—41no—
443C22×C4—1061no—
444C22×C4—1062no—
445C22×C4—1062no—
446C22×C4—2121no—
447C22×C4—2121no—
448C22×C4—2121no—
449C22×C4—2121no—
450C22×C4C2×C4×D5311yescharacteristic
451C22×C4C2×C4×D5311yescharacteristic
452C22×C4C2×D21211yescharacteristic
453C22×C4C2×D21211yescharacteristic
454C22×C4#1171994511yescharacteristic
455C22×C4#1171994511yescharacteristic
456C22×C4D4×D5311yescharacteristic
457C22×C4D4×D5311yescharacteristic
458C22×C4#1171994711yescharacteristic
459C22×C4#1171994711yescharacteristic
460C22×C4#1171994711yescharacteristic
461C22×C4#1171994911yescharacteristic
462C22×C4#1171994911yescharacteristic
463C22×C4C23×D5311yescharacteristic
464M16—22no—
465M16—22no—
466M16—22no—
467M16—534no—
468M16—534no—
469M16—1062no—
470M16—1062no—
471M16—1062no—
472M16—1062no—
473M16—1062no—
474M16—1062no—
475M16—1062no—
476M16—1062no—
477M16—2121no—
478M16—2121no—
479M16C2×C4×D5314yes—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—1062no—
484C2×C8—1062no—
485C2×C8—1062no—
486C2×C8—1062no—
487C2×C8—1062no—
488C2×C8—1062no—
489C2×C8—1062no—
490C2×C8—2121no—
491C2×C8—2121no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—41no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—41no—
500C42—22no—
501C42—22no—
502C42—41no—
503C42C2×D21214yes—
504C23—81no—
505C23—1062no—
506C23—4242no—
507C23—2121no—
508C23—2121no—
509C23#1313317511yescharacteristic
510C23#1313317511yescharacteristic
511C23#1313320111yescharacteristic
512C23#1313320111yescharacteristic
513C23#1313320311yescharacteristic
514C23#1313320311yescharacteristic
515C23C2×#1171994711yescharacteristic
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—42no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—1062no—
533C2×C4—2121no—
534C2×C4—2121no—
535C2×C4—2122no—
536C2×C4—2122no—
537C2×C4—2122no—
538C2×C4—2122no—
539C2×C4#1313317511yescharacteristic
540C2×C4#1313317511yescharacteristic
541C2×C4#1313317711yescharacteristic
542C2×C4#1313317711yescharacteristic
543C2×C4#1313317711yescharacteristic
544C2×C4#1313317711yescharacteristic
545C2×C4C4×D21211yescharacteristic
546C2×C4C4×D21211yescharacteristic
547C2×C4C4×D21211yescharacteristic
548C2×C4C4×D21211yescharacteristic
549C2×C4#1313319211yescharacteristic
550C2×C4#1313319211yescharacteristic
551C2×C4#1313319211yescharacteristic
552C2×C4#1313319211yescharacteristic
553C2×C4#1313320111yescharacteristic
554C2×C4#1313320111yescharacteristic
555C2×C4#1313320311yescharacteristic
556C2×C4#1313320311yescharacteristic
557C2×C4#1313321211yescharacteristic
558C2×C4#1313321211yescharacteristic
559C2×C4#1313321211yescharacteristic
560C2×C4#1313321211yescharacteristic
561C2×C4C22×C4×D5311yescharacteristic
562C2×C4C2×D4×D5311yescharacteristic
563C2×C4C22×D21211yescharacteristic
564C2×C4C2×#1171994711yescharacteristic
565C2×C4C2×#1171994511yescharacteristic
566C2×C4C2×#1171994911yescharacteristic
567C8—24no—
568C8—1064no—
569C8—1064no—
570C8—2122no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—4242no—
579C22—4242no—
580C22—4242no—
581C22—2122no—
582C22#1681051511yescharacteristic
583C22#1681051511yescharacteristic
584C22#1681051511yescharacteristic
585C22#1681051511yescharacteristic
586C22C2×#1313317511yescharacteristic
587C22C2×#1313320111yescharacteristic
588C22C2×#1313320311yescharacteristic
589C4—44no—
590C4—82no—
591C4—2122no—
592C4#1681051511yescharacteristic
593C4#1681051511yescharacteristic
594C4#1681051511yescharacteristic
595C4#1681051511yescharacteristic
596C4C2×#1313317711yescharacteristic
597C4C2×C4×D21211yescharacteristic
598C4C2×#1313319211yescharacteristic
599C4C2×#1313321211yescharacteristic
600C2—21no—
601C2—21no—
602C2—21no—
603C2—21no—
604C2—21no—
605C2—21no—
606C2—21no—
607C2—4242no—
608C2C2×#1681051511yescharacteristic
subgroupG/H
centerC4C2×#13133192
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1733C2
socleC2×C53C2×#388
prime radicalC22×#11719945C22
cosocleC22×C4×C53C24
hypercenter#1733D53
hypocenterC53#21080
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C23×C4 ◁ #1733

Complements

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

NH ≅ G/Ntype
1. C2×#16810672C2semidirect
5. #20283134C2semidirect
6. #20275385C2semidirect
7. C2×#16810393C2semidirect
8. C53×#1733C2semidirect
14. #16810672C22semidirect
22. #16810605C4semidirect
24. C2×#13133135C4semidirect
25. C2×#13133133C4semidirect
28. C4×#11719917C22semidirect
29. #16810393C22semidirect
33. C53×#339C22semidirect
74. #13133135C2×C4semidirect
75. #13133133C2×C4semidirect
303. #1733D53semidirect
337. #339C2×D53semidirect
338. C53#21080semidirect

Maximal subgroups

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

subgroupindex
C2×#168106722
C2×#168106412
C2×#168106052
#202831342
#202753852
C2×#168103932
C53×#17332
#2108053