Finite Group Databasebeta

← C2×#20275157 #27385844

Subgroups — C2×#20275157 #27385844

Order 13568.

2514 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811135C211yesmaximal, characteristic
2C22×C42×D53C211yesmaximal, characteristic, complemented
3C22×#13133145C211yesmaximal, characteristic, complemented
4C22×#13133143C211yesmaximal, characteristic, complemented
5C2×#16810586C211yesmaximal, characteristic, complemented
6#20275157C218yesmaximal, direct product factor
7C2×#16810382C211yesmaximal, characteristic, complemented
8C2×C53×#339C211yesmaximal, characteristic, complemented
9C22×#11719945C2211yescharacteristic
10C23×C4×D53—21no—
11C23×C4×D53C2211yescharacteristic
12C22×#11719917C2211yescharacteristic
13#16811135—22no—
14#16811135—22no—
15#16811135C414yescomplemented
16#16811135C2214yes—
17C2×#13133201—21no—
18C2×#13133197C412yescomplemented
19C2×#13133192C412yescomplemented
20C2×#13133187—21no—
21C2×C4×D212—22no—
22C22×C4×Dic53—21no—
23C22×C4×Dic53C2211yescharacteristic
24C2×C42×D53—22no—
25C2×C42×D53—24no—
26C2×C42×D53—24no—
27C2×C42×D53C2214yescomplemented
28C2×#13133145—24no—
29C2×#13133145C2214yescomplemented
30C2×#13133143—24no—
31C2×#13133143C2214yescomplemented
32#16810586—22no—
33#16810586C2214yescomplemented
34C2×#13133129—22no—
35#16810382C2214yescomplemented
36C22×M16×C53C2211yescharacteristic
37C2×#117×C53C2211yescharacteristic
38C22×C42×C53C2211yescharacteristic
39C53×#339C2214yescomplemented
40C24×D53D411yescharacteristic
41C2×#11719945—21no—
42C2×#11719945—21no—
43C2×#11719945—22no—
44C2×#11719945—22no—
45C2×#11719945—22no—
46C2×#11719945—22no—
47C2×#11719945C2×C412yescomplemented
48C2×#11719945C2312yes—
49C22×Dic106C2×C411yescharacteristic, complemented
50C22×D212—22no—
51C22×D212C2×C411yescharacteristic, complemented
52C22×C22⋊D53—21no—
53C23×Dic53—21no—
54C23×Dic53D411yescharacteristic
55C22×C4×D53—28no—
56C22×C4×D53—216no—
57C22×C4×D53—24no—
58C22×C4×D53—22no—
59C22×C4×D53—22no—
60C22×C4×D53—22no—
61C22×C4×D53—22no—
62C22×C4×D53—21no—
63C22×C4×D53C2×C412yes—
64C22×C4×D53C2×C412yes—
65C22×C4×D53D412yes—
66C22×C4×D53D412yes—
67C22×C4×D53D412yes—
68C22×C4×D53C2312yes—
69C2×#11719917—21no—
70C2×#11719917—22no—
71C2×#11719917—22no—
72C2×#11719917C2312yes—
73C22×#88322—21no—
74#13133201—24no—
75#13133201—24no—
76#13133197—24no—
77#13133197C2×C418yescomplemented
78#13133192—24no—
79#13133192C2×C418yescomplemented
80#13133187—24no—
81#13133187—24no—
82C4×D212—28no—
83C4×D212—28no—
84C2×C4⋊Dic53—22no—
85C2×#11719921—22no—
86C2×#11719921—22no—
87C2×C4×Dic53—28no—
88C2×C4×Dic53—22no—
89C2×C4×Dic53—21no—
90C2×C4×Dic53—24no—
91C2×C4×Dic53D414yescomplemented
92C2×C4×Dic53C2312yes—
93C42×D53—216no—
94C42×D53—28no—
95C42×D53—28no—
96#13133145—28no—
97#13133143—28no—
98#13133129—28no—
99C23×C4×C53—21no—
100C23×C4×C53C2311yescharacteristic
101C2×M16×C53—21no—
102C2×M16×C53—22no—
103C2×M16×C53—22no—
104C2×M16×C53C2312yes—
105C22×C8×C53—21no—
106#117×C53—21no—
107#117×C53—21no—
108#117×C53C2×C412yescomplemented
109#117×C53C2312yes—
110C2×C4⋊C4×C53C2×C412yescomplemented
111C2×C22⋊C4×C53—21no—
112C2×C42×C53—22no—
113C2×C42×C53—21no—
114C2×C42×C53—21no—
115C2×C42×C53D414yescomplemented
116C2×C42×C53C2312yes—
117C24×C53C2×D411yescharacteristic
118C23×D53—22no—
119C23×D53—22no—
120C23×D53—28no—
121C23×D53—21no—
122C23×D53C22⋊C412yes—
123C23×D53C22⋊C412yes—
124C23×D53C2×D412yes—
125C22×C4×C53—28no—
126C22×C4×C53—24no—
127C22×C4×C53—22no—
128C22×C4×C53—22no—
129C22×C4×C53C22×C411yescharacteristic
130C22×C4×C53C22×C411yescharacteristic
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D412yes—
135C22×C4×C53C2×D412yes—
136C22×C4×C53#4112yes—
137C22×C4×C53#4112yes—
138C22×C4×C53C2411yescharacteristic
139C2×C22⋊D53—22no—
140C2×C22⋊D53—22no—
141C2×C22⋊D53—42no—
142C2×C22⋊D53—42no—
143C22×Dic53—28no—
144C22×Dic53—22no—
145C22×Dic53—24no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—21no—
149C22×Dic53C22⋊C412yes—
150C22×Dic53C22⋊C412yes—
151C22×Dic53C2×D412yes—
152#11719945—24no—
153#11719945—24no—
154#11719945—22no—
155#11719945—22no—
156#11719945—22no—
157#11719945—22no—
158#11719945—42no—
159#11719945—42no—
160#11719945—42no—
161#11719945—42no—
162C2×D212—22no—
163C2×D212—22no—
164C2×D212—28no—
165C2×D212—21no—
166C2×D212—48no—
167C2×D212C22×C412yescomplemented
168C2×C4×D53—216no—
169C2×C4×D53—264no—
170C2×C4×D53—232no—
171C2×C4×D53—24no—
172C2×C4×D53—24no—
173C2×C4×D53—24no—
174C2×C4×D53—24no—
175C2×C4×D53—24no—
176C2×C4×D53—24no—
177C2×C4×D53—24no—
178C2×C4×D53—24no—
179C2×C4×D53—24no—
180C2×C4×D53—24no—
181C2×C4×D53—22no—
182C2×C4×D53—22no—
183C2×C4×D53—22no—
184C2×C4×D53—22no—
185C2×C4×D53—22no—
186C2×C4×D53—22no—
187C2×C4×D53—42no—
188C2×C4×D53—42no—
189C2×C4×D53C22⋊C414yes—
190C2×C4×D53C22⋊C414yes—
191C2×C4×D53C22×C414yes—
192C2×C4×D53C2×D414yes—
193C2×Dic106—22no—
194C2×Dic106—22no—
195C2×Dic106—21no—
196C2×Dic106C22×C412yescomplemented
197M16×C53—22no—
198M16×C53—22no—
199M16×C53—42no—
200C2×C8×C53—22no—
201C2×C8×C53—42no—
202C4⋊C4×C53—22no—
203C4⋊C4×C53C22×C414yescomplemented
204C22⋊C4×C53—22no—
205C22⋊C4×C53—22no—
206C42×C53—22no—
207C42×C53—24no—
208C42×C53—22no—
209C42×C53—22no—
210C42×C53C2×D414yescomplemented
211#11719921—28no—
212#11719921—28no—
213C4⋊Dic53—28no—
214C4×Dic53—216no—
215C4×Dic53—24no—
216C4×Dic53—22no—
217C4×Dic53C2×D414yescomplemented
218#11719917—22no—
219#11719917—22no—
220#11719917—42no—
221C2×#88322—22no—
222C2×#88322—42no—
223C23×C53—24no—
224C23×C53C2×C22⋊C411yescharacteristic
225C23×C53C2×C22⋊C411yescharacteristic
226C23×C53#12012yes—
227C23×C53#12312yes—
228C23×C53C22×D411yescharacteristic
229C22×D53—216no—
230C22×D53—216no—
231C22×D53—22no—
232C22×D53—22no—
233C22×D53—24no—
234C22×D53—24no—
235C22×D53—24no—
236C22×D53—28no—
237C22×D53—28no—
238C22×D53—22no—
239C22×D53—42no—
240C22×D53—44no—
241C22×D53#10418yescomplemented
242C22×D53C2×C22⋊C414yes—
243C2×C4×C53—28no—
244C2×C4×C53—216no—
245C2×C4×C53—24no—
246C2×C4×C53—24no—
247C2×C4×C53—44no—
248C2×C4×C53C2×C22⋊C411yescharacteristic
249C2×C4×C53C2×C22⋊C411yescharacteristic
250C2×C4×C53C4×D412yes—
251C2×C4×C53C4×D412yes—
252C2×C4×C53C4×D412yes—
253C2×C4×C53C4×D412yes—
254C2×C4×C53#12012yes—
255C2×C4×C53#12112yes—
256C2×C4×C53#12112yes—
257C2×C4×C53#12112yes—
258C2×C4×C53#12112yes—
259C2×C4×C53#12312yes—
260C2×C4×C53C23×C411yescharacteristic
261C2×C4×C53C22×D411yescharacteristic
262C2×C4×C53C22×D411yescharacteristic
263C2×C4×C53C22×D411yescharacteristic
264C2×C4×C53C2×#4111yescharacteristic
265C2×C4×C53C2×#4111yescharacteristic
266C22⋊D53—42no—
267C22⋊D53—42no—
268C22⋊D53—44no—
269C2×Dic53—28no—
270C2×Dic53—216no—
271C2×Dic53—24no—
272C2×Dic53—22no—
273C2×Dic53—22no—
274C2×Dic53—24no—
275C2×Dic53—24no—
276C2×Dic53—22no—
277C2×Dic53—42no—
278C2×Dic53C2×C22⋊C414yes—
279D212—22no—
280D212—22no—
281D212—416no—
282D212—42no—
283C4×D53—264no—
284C4×D53—24no—
285C4×D53—24no—
286C4×D53—24no—
287C4×D53—24no—
288C4×D53—28no—
289C4×D53—28no—
290C4×D53—42no—
291C4×D53—42no—
292C4×D53—44no—
293Dic106—22no—
294Dic106—22no—
295Dic106—42no—
296C8×C53—42no—
297#88322—42no—
298C2×#1896—531nomaximal, Sylow
299C22×C53—22no—
300C22×C53—22no—
301C22×C53—24no—
302C22×C53—24no—
303C22×C53#38612yes—
304C22×C53#38612yes—
305C22×C53C2×#10414yes—
306C22×C53C22×C22⋊C411yescharacteristic
307C22×C53C2×#12011yescharacteristic
308C22×C53C2×#12311yescharacteristic
309C2×D53—28no—
310C2×D53—28no—
311C2×D53—216no—
312C2×D53—24no—
313C2×D53—216no—
314C2×D53—24no—
315C2×D53—44no—
316C2×D53—42no—
317C2×D53—84no—
318C2×D53C2×#10418yescomplemented
319C4×C53—28no—
320C4×C53—44no—
321C4×C53#38612yes—
322C4×C53#38612yes—
323C4×C53C2×C4×D411yescharacteristic
324C4×C53C2×C4×D411yescharacteristic
325C4×C53C2×#12111yescharacteristic
326C4×C53C2×#12111yescharacteristic
327Dic53—28no—
328Dic53—24no—
329Dic53—42no—
330C2×#525—531no—
331C23×C42—531no—
332C22×#104—531no—
333C22×#104—531no—
334C2×#407—531no—
335#1896—538no—
336C2×#339—531no—
337C2×#339D5311yescharacteristic, complemented
338C2×C53—21no—
339C2×C53—21no—
340C2×C53—22no—
341C2×C53#189614yescomplemented
342C2×C53C2×#38611yescharacteristic
343C2×C53C22×#10412yes—
344D53—28no—
345D53—82no—
346C22×#41—531no—
347C24×C4—531no—
348C24×C4—1061no—
349C22×M16—531no—
350C22×M16C2×D5311yescharacteristic
351#525—534no—
352#525—534no—
353#525—1062no—
354#525—1062no—
355C2×#123—1061no—
356C2×C22⋊Q8—532no—
357C2×#121—532no—
358C2×#120—1061no—
359C2×C4×D4—1062no—
360C2×#117C2×D5311yescharacteristic
361C22×C42—531no—
362C22×C42—534no—
363C22×C42—1064no—
364C22×C42—1062no—
365C22×C42—1061no—
366C22×C42—1064no—
367C22×C42C2×D5311yescharacteristic
368C2×#104—534no—
369C2×#104—534no—
370C2×#104—1064no—
371C2×#104—1064no—
372#407—534no—
373#407—1062no—
374C2×C22⋊C8—1062no—
375#339—534no—
376#339C2×D5314yescomplemented
377C53C2×#189611yescharacteristic, complemented, Sylow
378C25—531no—
379C2×#41—532no—
380C2×#41—532no—
381C2×#41—1061no—
382C2×#41—1061no—
383C2×#41—1062no—
384C2×#41—1062no—
385C2×#41—1062no—
386C2×#41—1062no—
387C22×Q8—531no—
388C22×D4—531no—
389C22×D4—1062no—
390C22×D4—1061no—
391C23×C4—21no—
392C23×C4—531no—
393C23×C4—532no—
394C23×C4—532no—
395C23×C4—532no—
396C23×C4—532no—
397C23×C4—532no—
398C23×C4—532no—
399C23×C4—1064no—
400C23×C4—10616no—
401C23×C4—1068no—
402C23×C4—1061no—
403C23×C4—1062no—
404C23×C4—1062no—
405C23×C4—1062no—
406C23×C4—1062no—
407C23×C4—1061no—
408C23×C4C22×D5311yescharacteristic
409C2×M16—21no—
410C2×M16—22no—
411C2×M16—22no—
412C2×M16—532no—
413C2×M16—1061no—
414C2×M16—1062no—
415C2×M16—1062no—
416C2×M16C22×D5312yes—
417C22×C8—21no—
418C22×C8—1061no—
419#123—1064no—
420#123—1064no—
421C22⋊Q8—538no—
422C22⋊Q8—1064no—
423#121—538no—
424#121—1064no—
425#120—1064no—
426#120—1064no—
427C4×D4—1068no—
428C4×D4—1068no—
429#117—21no—
430#117—21no—
431#117C4×D5312yescomplemented
432#117C22×D5312yes—
433C2×C4⋊C4—1062no—
434C2×C4⋊C4C4×D5312yescomplemented
435C2×C22⋊C4—21no—
436C2×C22⋊C4—1062no—
437C2×C22⋊C4—1062no—
438C2×C42—21no—
439C2×C42—22no—
440C2×C42—21no—
441C2×C42—534no—
442C2×C42—532no—
443C2×C42—1068no—
444C2×C42—10616no—
445C2×C42—1064no—
446C2×C42—1068no—
447C2×C42—1062no—
448C2×C42—1061no—
449C2×C42—1068no—
450C2×C42D21214yescomplemented
451C2×C42C22×D5312yes—
452#104—1068no—
453#104—1068no—
454C22⋊C8—1068no—
455C24—532no—
456C24—532no—
457C24—532no—
458C24—1068no—
459C24—1062no—
460C24—1062no—
461C24—1061no—
462C24D4×D5311yescharacteristic
463#41—1064no—
464#41—1064no—
465#41—1062no—
466#41—1062no—
467#41—1062no—
468#41—1062no—
469#41—2122no—
470#41—2122no—
471#41—2122no—
472#41—2122no—
473C2×Q8—532no—
474C2×Q8—1061no—
475C2×Q8—1062no—
476C2×Q8—1062no—
477C2×D4—532no—
478C2×D4—1068no—
479C2×D4—1062no—
480C2×D4—1062no—
481C2×D4—1061no—
482C2×D4—1062no—
483C2×D4—1062no—
484C2×D4—2128no—
485C2×D4—2122no—
486C2×D4—2122no—
487C22×C4—28no—
488C22×C4—24no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—532no—
492C22×C4—532no—
493C22×C4—532no—
494C22×C4—534no—
495C22×C4—534no—
496C22×C4—534no—
497C22×C4—534no—
498C22×C4—10616no—
499C22×C4—10632no—
500C22×C4—10664no—
501C22×C4—1068no—
502C22×C4—1064no—
503C22×C4—1062no—
504C22×C4—1062no—
505C22×C4—1062no—
506C22×C4—1064no—
507C22×C4—1064no—
508C22×C4—1064no—
509C22×C4—1064no—
510C22×C4—1064no—
511C22×C4—1064no—
512C22×C4—1064no—
513C22×C4—1064no—
514C22×C4—1064no—
515C22×C4—1064no—
516C22×C4—1062no—
517C22×C4—1062no—
518C22×C4—1062no—
519C22×C4—1062no—
520C22×C4—1061no—
521C22×C4—1062no—
522C22×C4—1062no—
523C22×C4—2122no—
524C22×C4—2122no—
525C22×C4C2×C4×D5311yescharacteristic
526C22×C4C2×C4×D5311yescharacteristic
527C22×C4C2×D21212yes—
528C22×C4#1171994512yes—
529C22×C4D4×D5311yescharacteristic
530C22×C4D4×D5311yescharacteristic
531C22×C4D4×D5312yes—
532C22×C4D4×D5311yescharacteristic
533C22×C4#1171994912yes—
534C22×C4C23×D5311yescharacteristic
535M16—22no—
536M16—22no—
537M16—42no—
538M16—1062no—
539M16—1062no—
540M16—2122no—
541C2×C8—22no—
542C2×C8—42no—
543C2×C8—1062no—
544C2×C8—2122no—
545C4⋊C4—22no—
546C4⋊C4—1068no—
547C4⋊C4C2×C4×D5314yescomplemented
548C22⋊C4—22no—
549C22⋊C4—22no—
550C22⋊C4—1068no—
551C22⋊C4—1068no—
552C42—22no—
553C42—24no—
554C42—22no—
555C42—22no—
556C42—534no—
557C42—10616no—
558C42—1064no—
559C42—1062no—
560C42C2×D21214yescomplemented
561C23—24no—
562C23—534no—
563C23—538no—
564C23—1068no—
565C23—1068no—
566C23—10616no—
567C23—10616no—
568C23—1064no—
569C23—1064no—
570C23—1062no—
571C23—1062no—
572C23—1064no—
573C23—1062no—
574C23—2122no—
575C23—2124no—
576C23C22⋊C4×D5311yescharacteristic
577C23C22⋊C4×D5311yescharacteristic
578C23#1313318712yes—
579C23#1313320212yes—
580C23C2×D4×D5311yescharacteristic
581Q8—1062no—
582Q8—1062no—
583Q8—2122no—
584D4—1062no—
585D4—1062no—
586D4—21216no—
587D4—2124no—
588D4—2122no—
589D4—2122no—
590D4—2122no—
591C2×C4—28no—
592C2×C4—216no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—44no—
596C2×C4—534no—
597C2×C4—10664no—
598C2×C4—1068no—
599C2×C4—10616no—
600C2×C4—1064no—
601C2×C4—1064no—
602C2×C4—1064no—
603C2×C4—1062no—
604C2×C4—1062no—
605C2×C4—1068no—
606C2×C4—1068no—
607C2×C4—1064no—
608C2×C4—1064no—
609C2×C4—1064no—
610C2×C4—1064no—
611C2×C4—1062no—
612C2×C4—2122no—
613C2×C4—2124no—
614C2×C4—2122no—
615C2×C4—2122no—
616C2×C4C22⋊C4×D5311yescharacteristic
617C2×C4C22⋊C4×D5311yescharacteristic
618C2×C4#1313317912yes—
619C2×C4#1313317912yes—
620C2×C4C4×D21212yes—
621C2×C4C4×D21212yes—
622C2×C4#1313318712yes—
623C2×C4#1313319012yes—
624C2×C4#1313319012yes—
625C2×C4#1313319212yes—
626C2×C4#1313319212yes—
627C2×C4#1313320212yes—
628C2×C4C22×C4×D5311yescharacteristic
629C2×C4C2×D4×D5311yescharacteristic
630C2×C4C2×D4×D5311yescharacteristic
631C2×C4C22×D21211yescharacteristic
632C2×C4C2×#1171994511yescharacteristic
633C2×C4C2×#1171994911yescharacteristic
634C8—42no—
635C8—2122no—
636C22—22no—
637C22—22no—
638C22—24no—
639C22—24no—
640C22—538no—
641C22—1068no—
642C22—10616no—
643C22—1068no—
644C22—1064no—
645C22—1064no—
646C22—10616no—
647C22—4244no—
648C22—2124no—
649C22—2122no—
650C22#1681050112yes—
651C22#1681050112yes—
652C22#104×D5314yes—
653C22C2×C22⋊C4×D5311yescharacteristic
654C22C2×#1313318711yescharacteristic
655C22C2×#1313320211yescharacteristic
656C4—28no—
657C4—44no—
658C4—1068no—
659C4—1064no—
660C4—2122no—
661C4#1681050112yes—
662C4#1681050112yes—
663C4C2×#1313317911yescharacteristic
664C4C2×C4×D21211yescharacteristic
665C4C2×#1313319011yescharacteristic
666C4C2×#1313319211yescharacteristic
667C2—21no—
668C2—22no—
669C2—21no—
670C2—1068no—
671C2—4242no—
672C2#2027515714yesdirect product factor
673C2C2×#1681050111yescharacteristic
674C2C2×#104×D5312yes—
subgroupG/H
centerC22×C4D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#339C2
socleC23×C53C2×C22⋊C4
prime radicalC2×#16811135C2
cosocleC22×C4×C53C24
hypercenterC2×#339D53
hypocenterC53C2×#1896
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C2×#339

Complements

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

NH ≅ G/Ntype
2. C22×C42×D53C2semidirect
3. C22×#13133145C2semidirect
4. C22×#13133143C2semidirect
5. C2×#16810586C2semidirect
6. #20275157C2direct (N × H)
7. C2×#16810382C2semidirect
8. C2×C53×#339C2semidirect
15. #16811135C4semidirect
18. C2×#13133197C4semidirect
19. C2×#13133192C4semidirect
27. C2×C42×D53C22semidirect
29. C2×#13133145C22semidirect
31. C2×#13133143C22semidirect
33. #16810586C22semidirect
35. #16810382C22semidirect
39. C53×#339C22semidirect
47. C2×#11719945C2×C4semidirect
49. C22×Dic106C2×C4semidirect
51. C22×D212C2×C4semidirect
77. #13133197C2×C4semidirect
79. #13133192C2×C4semidirect
91. C2×C4×Dic53D4semidirect
108. #117×C53C2×C4semidirect
110. C2×C4⋊C4×C53C2×C4semidirect
115. C2×C42×C53D4semidirect
167. C2×D212C22×C4semidirect
196. C2×Dic106C22×C4semidirect
203. C4⋊C4×C53C22×C4semidirect
210. C42×C53C2×D4semidirect
217. C4×Dic53C2×D4semidirect
241. C22×D53#104semidirect
318. C2×D53C2×#104semidirect
337. C2×#339D53semidirect
341. C2×C53#1896semidirect
376. #339C2×D53semidirect
377. C53C2×#1896semidirect
431. #117C4×D53semidirect
434. C2×C4⋊C4C4×D53semidirect
450. C2×C42D212semidirect
547. C4⋊C4C2×C4×D53semidirect
560. C42C2×D212semidirect
672. C2#20275157direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168111352
C22×C42×D532
C22×#131331452
C22×#131331432
C2×#168105862
#202751572
C2×#168103822
C2×C53×#3392
C2×#189653