Finite Group Databasebeta

← #27291272

Subgroups — #27291272

Order 13568.

866 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810650C211yesmaximal, characteristic, complemented
2C2×#16810641C211yesmaximal, characteristic
3C2×#16810617C211yesmaximal, characteristic
4C2×#16810605C211yesmaximal, characteristic
5C2×#16810393C211yesmaximal, characteristic, complemented
6C2×#16810387C211yesmaximal, characteristic, complemented
7C2×C53×H(3,ℤ/4)C211yesmaximal, characteristic, complemented
8C22×#11719945C2211yescharacteristic
9C22×#11719917C2211yescharacteristic
10C22×#11719917C2211yescharacteristic
11C23×#88322C2211yescharacteristic
12#16810650—22no—
13#16810650—22no—
14C2×C4⋊#88322—22no—
15#16810641—22no—
16#16810641C414yescomplemented
17C2×#13133145—22no—
18#16810617—22no—
19#16810617—22no—
20C2×#13133140—21no—
21C2×#13133137—21no—
22#16810605—22no—
23#16810605—22no—
24C2×#13133135—21no—
25C2×#13133133—21no—
26#16810393—22no—
27#16810387—22no—
28C22×M16×C53C2211yescharacteristic
29C2×#117×C53C2211yescharacteristic
30C2×#117×C53C2211yescharacteristic
31C53×H(3,ℤ/4)—22no—
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×#11719945—21no—
42C2×#11719945C2×C412yes—
43C2×#11719945Q812yes—
44C22×Dic106D411yescharacteristic
45C22×D212D411yescharacteristic
46C22×C22⋊D53—21no—
47C22×C4×D53—21no—
48C2×#11719917—21no—
49C2×#11719917—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×#11719917C2×C412yes—
59C2×#11719917D412yes—
60C22×#88322—21no—
61C22×#88322—21no—
62C22×#88322—21no—
63C22×#88322—21no—
64C22×#88322—21no—
65C22×#88322—21no—
66C22×#88322—21no—
67C22×#88322—21no—
68C22×#88322D412yes—
69C4⋊#88322—28no—
70#13133145—28no—
71#13133140—24no—
72#13133137—24no—
73#13133135—24no—
74#13133133—24no—
75C23×C4×C53C2311yescharacteristic
76C2×M16×C53—21no—
77C2×M16×C53—21no—
78C2×M16×C53—21no—
79C2×M16×C53—21no—
80C2×M16×C53—21no—
81C2×M16×C53D412yes—
82C22×C8×C53—21no—
83#117×C53—21no—
84#117×C53—21no—
85#117×C53—21no—
86#117×C53—21no—
87#117×C53—21no—
88#117×C53C2×C412yes—
89#117×C53D412yes—
90#117×C53D412yes—
91C2×C4⋊C4×C53—21no—
92C2×C4⋊C4×C53—21no—
93C2×C22⋊C4×C53—21no—
94C2×C22⋊C4×C53—21no—
95C2×C42×C53—21no—
96C2×C42×C53D412yes—
97C24×C53#4111yescharacteristic
98C23×D53—21no—
99C22×C4×C53—22no—
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—
122#11719945—22no—
123#11719945—22no—
124#11719945—22no—
125#11719945—22no—
126#11719945—22no—
127#11719945—22no—
128#11719945—22no—
129#11719945—22no—
130#11719945—41no—
131#11719945—41no—
132#11719945—41no—
133#11719945—41no—
134#11719945—41no—
135#11719945—41no—
136#11719945—41no—
137#11719945—41no—
138C2×D212—21no—
139C2×D212—21no—
140C2×D212—21no—
141C2×D212—21no—
142C2×D212—21no—
143C2×D212C4⋊C412yes—
144C2×C4×D53—22no—
145C2×C4×D53—22no—
146C2×C4×D53—41no—
147C2×C4×D53—41no—
148C2×C4×D53—41no—
149C2×C4×D53—41no—
150C2×Dic106—21no—
151C2×Dic106—21no—
152C2×Dic106—21no—
153C2×Dic106—21no—
154C2×Dic106—21no—
155C2×Dic106C4⋊C412yes—
156M16×C53—22no—
157M16×C53—22no—
158M16×C53—41no—
159M16×C53—41no—
160C2×C8×C53—22no—
161C2×C8×C53—41no—
162C2×C8×C53—41no—
163C4⋊C4×C53—22no—
164C4⋊C4×C53—22no—
165C4⋊C4×C53—22no—
166C4⋊C4×C53—41no—
167C22⋊C4×C53—22no—
168C22⋊C4×C53—22no—
169C22⋊C4×C53—22no—
170C22⋊C4×C53—41no—
171C42×C53—22no—
172C42×C53—22no—
173C42×C53—41no—
174C42×C53C22⋊C414yes—
175#11719917—22no—
176#11719917—22no—
177#11719917—22no—
178#11719917—22no—
179#11719917—41no—
180#11719917—41no—
181#11719917—41no—
182#11719917—41no—
183C2×#88322—22no—
184C2×#88322—22no—
185C2×#88322—22no—
186C2×#88322—22no—
187C2×#88322—22no—
188C2×#88322—41no—
189C2×#88322—41no—
190C2×#88322—41no—
191C2×#88322—41no—
192C2×#88322—41no—
193C2×#88322—41no—
194C2×#88322—41no—
195C2×#88322—41no—
196C23×C53—42no—
197C23×C53C4×D411yescharacteristic
198C23×C53#12111yescharacteristic
199C23×C53C22⋊Q811yescharacteristic
200C23×C53#12311yescharacteristic
201C23×C53#12311yescharacteristic
202C23×C53#12311yescharacteristic
203C23×C53#12611yescharacteristic
204C22×D53—22no—
205C22×D53—41no—
206C22×D53—41no—
207C22×D53—82no—
208C2×C4×C53—24no—
209C2×C4×C53—44no—
210C2×C4×C53—42no—
211C2×C4×C53—82no—
212C2×C4×C53C2×C22⋊C411yescharacteristic
213C2×C4×C53C2×C4⋊C411yescharacteristic
214C2×C4×C53#11711yescharacteristic
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#12411yescharacteristic
236C2×C4×C53#12411yescharacteristic
237C2×C4×C53#12411yescharacteristic
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—
247D212—22no—
248D212—22no—
249D212—41no—
250D212—41no—
251C4×D53—42no—
252C4×D53—42no—
253C4×D53—42no—
254C4×D53—42no—
255Dic106—22no—
256Dic106—22no—
257Dic106—41no—
258Dic106—41no—
259C8×C53—42no—
260#88322—42no—
261#88322—42no—
262#88322—42no—
263#11086—531nomaximal, Sylow
264C22×C53—22no—
265C22×C53—22no—
266C22×C53—22no—
267C22×C53—41no—
268C22×C53—41no—
269C22×C53—41no—
270C22×C53—41no—
271C22×C53#38511yescharacteristic
272C22×C53#38611yescharacteristic
273C22×C53#38811yescharacteristic
274C22×C53#39411yescharacteristic
275C22×C53#39611yescharacteristic
276C22×C53#39711yescharacteristic
277C22×C53#39911yescharacteristic
278C2×D53—42no—
279C2×D53—82no—
280C2×D53—82no—
281C2×D53—82no—
282C4×C53—44no—
283C4×C53—82no—
284C4×C53#38011yescharacteristic
285C4×C53#38511yescharacteristic
286C4×C53#38811yescharacteristic
287C4×C53#39011yescharacteristic
288C4×C53#39211yescharacteristic
289C4×C53#39311yescharacteristic
290C4×C53#39411yescharacteristic
291C4×C53#39611yescharacteristic
292Dic53—42no—
293C2×#424—531no—
294C2×#421—531no—
295C2×#416—531no—
296C2×#413—531no—
297C2×#344—531no—
298C2×#341—531no—
299C2×H(3,ℤ/4)D5311yescharacteristic, complemented
300C2×C53—21no—
301C2×C53—21no—
302C2×C53—21no—
303C2×C53—21no—
304C2×C53—21no—
305C2×C53—21no—
306C2×C53#143911yescharacteristic
307C2×C53#208012yes—
308D53—82no—
309C22×#41—531no—
310C22×M16—531no—
311C22×M16—531no—
312C22×M16C2×D5311yescharacteristic
313C23×C8—531no—
314C2×#117C2×D5311yescharacteristic
315C2×#117C2×D5311yescharacteristic
316#424—1062no—
317#424—1062no—
318C2×C4⋊C8—1062no—
319#421—534no—
320#421—1062no—
321C2×#104—1062no—
322#416—1062no—
323#416—1062no—
324C2×#103—1061no—
325C2×#102—1061no—
326#413—1062no—
327#413—1062no—
328C2×#101—1061no—
329C2×#100—1061no—
330#344—1062no—
331#341—1062no—
332H(3,ℤ/4)—22no—
333C53#1108611yescharacteristic, complemented, Sylow
334C2×#41—532no—
335C2×#41—532no—
336C2×#41—1061no—
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—
346C22×Q8—531no—
347C22×D4—531no—
348C22×D4—1061no—
349C23×C4—1061no—
350C23×C4C22×D5311yescharacteristic
351C2×M16—21no—
352C2×M16—21no—
353C2×M16—21no—
354C2×M16—21no—
355C2×M16—21no—
356C2×M16—532no—
357C2×M16—532no—
358C2×M16—1061no—
359C2×M16—1061no—
360C2×M16—1061no—
361C2×M16—1061no—
362C2×M16—1061no—
363C2×M16—1061no—
364C2×M16—1061no—
365C2×M16—1061no—
366C2×M16—1061no—
367C2×M16—1061no—
368C2×M16D21212yes—
369C22×C8—21no—
370C22×C8—532no—
371C22×C8—1061no—
372C22×C8—1061no—
373C22×C8—1061no—
374C22×C8—1061no—
375C22×C8—1061no—
376C22×C8—1061no—
377C22×C8—1061no—
378C22×C8—1061no—
379#117—21no—
380#117—21no—
381#117—21no—
382#117—21no—
383#117—21no—
384#117C4×D5312yes—
385#117D21212yes—
386#117C22⋊D5312yes—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C42—21no—
392C2×C42D21212yes—
393C4⋊C8—1068no—
394#104—1068no—
395#103—1064no—
396#102—1064no—
397#101—1064no—
398#100—1064no—
399C24—1061no—
400C24#1171994711yescharacteristic
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—41no—
437C22×C4—1061no—
438C22×C4—1062no—
439C22×C4—1062no—
440C22×C4—2121no—
441C22×C4—2121no—
442C22×C4—2121no—
443C22×C4—2121no—
444C22×C4C2×C4×D5311yescharacteristic
445C22×C4C2×D21211yescharacteristic
446C22×C4C2×D21211yescharacteristic
447C22×C4C2×D21211yescharacteristic
448C22×C4#1171994511yescharacteristic
449C22×C4#1171994511yescharacteristic
450C22×C4D4×D5311yescharacteristic
451C22×C4D4×D5311yescharacteristic
452C22×C4D4×D5311yescharacteristic
453C22×C4#1171994711yescharacteristic
454C22×C4#1171994711yescharacteristic
455C22×C4Q8×D5311yescharacteristic
456C22×C4#1171994911yescharacteristic
457C22×C4C2×C22⋊D5311yescharacteristic
458M16—22no—
459M16—22no—
460M16—41no—
461M16—41no—
462M16—1062no—
463M16—1062no—
464M16—1062no—
465M16—1062no—
466M16—2121no—
467M16—2121no—
468M16—2121no—
469M16—2121no—
470C2×C8—22no—
471C2×C8—41no—
472C2×C8—41no—
473C2×C8—1062no—
474C2×C8—1062no—
475C2×C8—1062no—
476C2×C8—1062no—
477C2×C8—1062no—
478C2×C8—2121no—
479C2×C8—2121no—
480C2×C8—2121no—
481C2×C8—2121no—
482C2×C8—2121no—
483C2×C8—2121no—
484C2×C8—2121no—
485C2×C8—2121no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—41no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—41no—
494C42—22no—
495C42—22no—
496C42—41no—
497C42#1171992114yes—
498C23—42no—
499C23—1062no—
500C23—4242no—
501C23—2121no—
502C23—2121no—
503C23#1313317711yescharacteristic
504C23#1313318911yescharacteristic
505C23#1313319711yescharacteristic
506C23#1313319911yescharacteristic
507C23#1313320111yescharacteristic
508C23#1313320111yescharacteristic
509C23#1313321211yescharacteristic
510Q8—1062no—
511Q8—1062no—
512Q8—2121no—
513Q8—2121no—
514D4—1062no—
515D4—1062no—
516D4—2122no—
517D4—2122no—
518D4—2122no—
519D4—2122no—
520D4—2121no—
521D4—2121no—
522C2×C4—24no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—82no—
526C2×C4—1062no—
527C2×C4—2121no—
528C2×C4—2121no—
529C2×C4—2122no—
530C2×C4—2122no—
531C2×C4—2122no—
532C2×C4—2122no—
533C2×C4C2×#1171992111yescharacteristic
534C2×C4C4⋊C4×D5311yescharacteristic
535C2×C4#1313317511yescharacteristic
536C2×C4#1313317711yescharacteristic
537C2×C4C4×D21211yescharacteristic
538C2×C4C4×D21211yescharacteristic
539C2×C4#1313318711yescharacteristic
540C2×C4#1313318711yescharacteristic
541C2×C4#1313318711yescharacteristic
542C2×C4#1313318811yescharacteristic
543C2×C4#1313318911yescharacteristic
544C2×C4#1313319111yescharacteristic
545C2×C4#1313319111yescharacteristic
546C2×C4#1313319211yescharacteristic
547C2×C4#1313319211yescharacteristic
548C2×C4#1313319211yescharacteristic
549C2×C4#1313319311yescharacteristic
550C2×C4#1313319511yescharacteristic
551C2×C4#1313319511yescharacteristic
552C2×C4#1313319711yescharacteristic
553C2×C4#1313319711yescharacteristic
554C2×C4#1313319911yescharacteristic
555C2×C4#1313320111yescharacteristic
556C2×C4#1313320311yescharacteristic
557C2×C4#1313320311yescharacteristic
558C2×C4#1313320511yescharacteristic
559C2×C4#1313321211yescharacteristic
560C2×C4#1313321411yescharacteristic
561C8—42no—
562C8—2122no—
563C8—2122no—
564C8—2122no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—4242no—
573C22—4242no—
574C22—4242no—
575C22—2122no—
576C22#1681049711yescharacteristic
577C22#1681050011yescharacteristic
578C22#1681051511yescharacteristic
579C22#1681053911yescharacteristic
580C22#1681054511yescharacteristic
581C22#1681055011yescharacteristic
582C22#1681056011yescharacteristic
583C4—44no—
584C4—82no—
585C4—2122no—
586C4#1681048011yescharacteristic
587C4#1681049711yescharacteristic
588C4#1681051511yescharacteristic
589C4#1681052511yescharacteristic
590C4#1681053311yescharacteristic
591C4#1681053611yescharacteristic
592C4#1681053911yescharacteristic
593C4#1681054511yescharacteristic
594C2—21no—
595C2—21no—
596C2—21no—
597C2—21no—
598C2—21no—
599C2—21no—
600C2—4242no—
601C2#2027296111yescharacteristic
602C2#2027616712yes—
subgroupG/H
centerC2×C4#13133205
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC2×C53×H(3,ℤ/4)C2
socleC22×C53#397
prime radicalC22×#11719945C22
cosocleC23×C4×C53C23
hypercenterC2×H(3,ℤ/4)D53
hypocenterC53#11086
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
1. C2×#16810650C2semidirect
5. C2×#16810393C2semidirect
6. C2×#16810387C2semidirect
7. C2×C53×H(3,ℤ/4)C2semidirect
16. #16810641C4semidirect
299. C2×H(3,ℤ/4)D53semidirect
333. C53#11086semidirect

Maximal subgroups

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

subgroupindex
C2×#168106502
C2×#168106412
C2×#168106172
C2×#168106052
C2×#168103932
C2×#168103872
C2×C53×H(3,ℤ/4)2
#1108653