Finite Group Databasebeta

← C2×#20275717 #27410881

Subgroups — C2×#20275717 #27410881

Order 13568.

2526 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11719924C211yesmaximal, characteristic, complemented
2C22×#13133193C211yesmaximal, characteristic, complemented
3C22×#13133192C211yesmaximal, characteristic
4C22×#13133139C211yesmaximal, characteristic, complemented
5C22×#13133129C211yesmaximal, characteristic, complemented
6#20275717C218yesmaximal, direct product factor
7C2×#16810385C211yesmaximal, characteristic, complemented
8C22×#103×C53C211yesmaximal, characteristic, complemented
9C23×D212—21no—
10C23×D212C2211yescharacteristic
11C23×C4×D53C2211yescharacteristic
12C22×#11719924—24no—
13C22×#11719924—24no—
14C22×#11719924—24no—
15C22×#11719924C2214yescomplemented
16C23×#88322C2211yescharacteristic
17C2×#13133193—22no—
18C2×#13133193—24no—
19C2×#13133193—24no—
20C2×#13133193C2214yescomplemented
21C2×#13133192—24no—
22C2×#13133192—22no—
23C2×#13133192—24no—
24C2×#13133192C2214yes—
25C22×C4⋊Dic53C2211yescharacteristic
26C22×#11719919—21no—
27C22×#11719921—21no—
28C22×#11719921—21no—
29C2×#13133139—24no—
30C2×#13133139C2214yescomplemented
31C2×#13133129—24no—
32C2×#13133129C2214yescomplemented
33#16810385C2214yescomplemented
34C23×Q8×C53C2211yescharacteristic
35C23×C8×C53C2211yescharacteristic
36C22×C4⋊C4×C53C2211yescharacteristic
37C2×#103×C53—24no—
38C2×#103×C53C2214yescomplemented
39C24×D53—21no—
40C24×D53D411yescharacteristic
41C22×D212—22no—
42C22×D212—22no—
43C22×D212—22no—
44C22×D212—22no—
45C22×D212—28no—
46C22×D212—24no—
47C22×D212—21no—
48C22×D212—22no—
49C22×D212—44no—
50C22×D212—44no—
51C22×D212D414yes—
52C22×D212C2312yes—
53C23×Dic53—21no—
54C23×Dic53D411yescharacteristic
55C22×C4×D53—22no—
56C22×C4×D53—22no—
57C22×C4×D53—22no—
58C22×C4×D53—22no—
59C22×C4×D53—22no—
60C22×C4×D53—22no—
61C22×C4×D53D412yes—
62C22×C4×D53C2312yes—
63C2×#11719924—232no—
64C2×#11719924—28no—
65C2×#11719924—28no—
66C2×#11719924—28no—
67C22×#88322—22no—
68C22×#88322—22no—
69C22×#88322D414yescomplemented
70C22×#88322C2312yes—
71#13133193—216no—
72#13133193—28no—
73#13133193—28no—
74#13133192—216no—
75#13133192—28no—
76#13133192—28no—
77C2×C4⋊Dic53—22no—
78C2×C4⋊Dic53—21no—
79C2×C4⋊Dic53D414yescomplemented
80C2×C4⋊Dic53C2312yes—
81C2×#11719919—28no—
82C2×#11719919—24no—
83C2×#11719921—28no—
84C2×#11719921—28no—
85C2×#11719921—24no—
86C2×#11719921—24no—
87#13133139—28no—
88#13133129—28no—
89C22×Q8×C53—22no—
90C22×Q8×C53—21no—
91C22×Q8×C53—22no—
92C22×Q8×C53—22no—
93C22×Q8×C53—22no—
94C22×Q8×C53—22no—
95C22×Q8×C53D414yescomplemented
96C22×Q8×C53C2312yes—
97C23×C4×C53—21no—
98C23×C4×C53—21no—
99C23×C4×C53C2311yescharacteristic
100C22×C8×C53—22no—
101C22×C8×C53—22no—
102C22×C8×C53—22no—
103C22×C8×C53C2312yes—
104C2×C4⋊C4×C53—22no—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53D414yes—
107C2×C4⋊C4×C53C2312yes—
108#103×C53—28no—
109C24×C53C2×D411yescharacteristic
110C23×D53—22no—
111C23×D53—22no—
112C23×D53—22no—
113C23×D53—22no—
114C23×D53—22no—
115C23×D53—22no—
116C23×D53—24no—
117C23×D53—22no—
118C23×D53—42no—
119C23×D53—42no—
120C23×D53—42no—
121C23×D53—42no—
122C23×D53—42no—
123C23×D53—42no—
124C23×D53SD1614yescomplemented
125C23×D53C2×D412yes—
126C2×Q8×C53—24no—
127C2×Q8×C53—22no—
128C2×Q8×C53—28no—
129C2×Q8×C53—28no—
130C2×Q8×C53—22no—
131C2×Q8×C53—22no—
132C2×Q8×C53—22no—
133C2×Q8×C53—22no—
134C2×Q8×C53—42no—
135C2×Q8×C53—42no—
136C2×Q8×C53—44no—
137C2×Q8×C53—44no—
138C2×Q8×C53C2×D414yescomplemented
139C22×C4×C53—24no—
140C22×C4×C53—24no—
141C22×C4×C53—22no—
142C22×C4×C53—22no—
143C22×C4×C53—42no—
144C22×C4×C53—42no—
145C22×C4×C53—42no—
146C22×C4×C53—42no—
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2×D412yes—
149C22×C4×C53C2×D412yes—
150C22×C4×C53C2×D412yes—
151C22×C4×C53C2×D412yes—
152C22×C4×C53C2×D412yes—
153C22×C4×C53#4112yes—
154C22×C4×C53C2411yescharacteristic
155C22×Dic53—22no—
156C22×Dic53—24no—
157C22×Dic53—22no—
158C22×Dic53—22no—
159C22×Dic53—22no—
160C22×Dic53—42no—
161C22×Dic53—42no—
162C22×Dic53C2×D412yes—
163C2×D212—28no—
164C2×D212—28no—
165C2×D212—22no—
166C2×D212—22no—
167C2×D212—22no—
168C2×D212—22no—
169C2×D212—216no—
170C2×D212—24no—
171C2×D212—22no—
172C2×D212—48no—
173C2×D212—432no—
174C2×D212—48no—
175C2×D212—42no—
176C2×D212—44no—
177C2×D212—42no—
178C2×D212—44no—
179C2×D212C2×D414yes—
180C2×C4×D53—24no—
181C2×C4×D53—24no—
182C2×C4×D53—24no—
183C2×C4×D53—22no—
184C2×C4×D53—22no—
185C2×C4×D53—22no—
186C2×C4×D53—22no—
187C2×C4×D53—22no—
188C2×C4×D53—22no—
189C2×C4×D53—44no—
190C2×C4×D53—44no—
191C2×C4×D53—44no—
192C2×C4×D53—44no—
193C2×C8×C53—22no—
194C2×C8×C53—22no—
195C2×C8×C53—22no—
196C2×C8×C53—44no—
197C4⋊C4×C53—24no—
198C4⋊C4×C53—22no—
199C4⋊C4×C53C2×D414yes—
200#11719924—232no—
201#11719921—216no—
202#11719921—216no—
203C4⋊Dic53—24no—
204C4⋊Dic53—22no—
205C4⋊Dic53C2×D414yescomplemented
206#11719919—216no—
207C2×#88322—28no—
208C2×#88322—22no—
209C2×#88322—22no—
210C2×#88322C2×D414yescomplemented
211C23×C53#12012yes—
212C23×C53#12012yes—
213C23×C53#12112yes—
214C23×C53C2×SD1612yes—
215C23×C53#13512yes—
216C23×C53AGL(1,ℤ/8)12yes—
217C23×C53#13712yes—
218C23×C53C22×D411yescharacteristic
219C22×D53—22no—
220C22×D53—22no—
221C22×D53—22no—
222C22×D53—22no—
223C22×D53—22no—
224C22×D53—22no—
225C22×D53—28no—
226C22×D53—28no—
227C22×D53—28no—
228C22×D53—24no—
229C22×D53—44no—
230C22×D53—44no—
231C22×D53—44no—
232C22×D53—44no—
233C22×D53—42no—
234C22×D53—42no—
235C22×D53—42no—
236C22×D53—42no—
237C22×D53—42no—
238C22×D53—42no—
239C22×D53—48no—
240C22×D53—48no—
241C22×D53—48no—
242C22×D53—84no—
243C22×D53—84no—
244C22×D53—84no—
245C22×D53—84no—
246C22×D53C2×SD1614yescomplemented
247Q8×C53—24no—
248Q8×C53—24no—
249Q8×C53—42no—
250Q8×C53—42no—
251Q8×C53—48no—
252C2×C4×C53—24no—
253C2×C4×C53—24no—
254C2×C4×C53—42no—
255C2×C4×C53—42no—
256C2×C4×C53—42no—
257C2×C4×C53—42no—
258C2×C4×C53—48no—
259C2×C4×C53—48no—
260C2×C4×C53#12012yes—
261C2×C4×C53#12014yes—
262C2×C4×C53#12012yes—
263C2×C4×C53#12114yes—
264C2×C4×C53#12112yes—
265C2×C4×C53#12114yes—
266C2×C4×C53#12714yes—
267C2×C4×C53C22×D411yescharacteristic
268C2×C4×C53C22×D411yescharacteristic
269C2×C4×C53C22×D411yescharacteristic
270C2×C4×C53C22×D411yescharacteristic
271C2×C4×C53C22×D411yescharacteristic
272C2×C4×C53C2×#4111yescharacteristic
273C2×Dic53—22no—
274C2×Dic53—22no—
275C2×Dic53—24no—
276C2×Dic53—22no—
277C2×Dic53—42no—
278C2×Dic53—42no—
279C2×Dic53—44no—
280C2×Dic53—48no—
281D212—24no—
282D212—24no—
283D212—432no—
284D212—42no—
285D212—42no—
286D212—48no—
287C4×D53—42no—
288C4×D53—42no—
289C4×D53—42no—
290C4×D53—42no—
291C4×D53—48no—
292C8×C53—42no—
293#88322—24no—
294C2×#2000—531nomaximal, Sylow
295C22×C53#39214yes—
296C22×C53#44814yes—
297C22×C53#44914yes—
298C22×C53#45014yes—
299C22×C53#45214yes—
300C22×C53#46514yes—
301C22×C53#46814yes—
302C22×C53C2×#12011yescharacteristic
303C22×C53C2×#12011yescharacteristic
304C22×C53C2×#12111yescharacteristic
305C22×C53C22×SD1611yescharacteristic
306C22×C53C2×#13511yescharacteristic
307C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
308C22×C53C2×#13711yescharacteristic
309C2×D53—24no—
310C2×D53—24no—
311C2×D53—24no—
312C2×D53—42no—
313C2×D53—42no—
314C2×D53—42no—
315C2×D53—416no—
316C2×D53—42no—
317C2×D53—44no—
318C2×D53—44no—
319C2×D53—44no—
320C2×D53—82no—
321C2×D53—82no—
322C2×D53—816no—
323C2×D53—82no—
324C2×D53—82no—
325C4×C53—44no—
326C4×C53—44no—
327C4×C53#39214yes—
328C4×C53C2×#12011yescharacteristic
329C4×C53C2×#12111yescharacteristic
330C4×C53C2×#12111yescharacteristic
331C4×C53C2×#12711yescharacteristic
332Dic53—42no—
333Dic53—44no—
334C23×SD16—531no—
335C22×C22⋊Q8—531no—
336C22×#121—531no—
337C22×#103D5311yescharacteristic, complemented
338C22×#102—531no—
339C22×C22⋊C8—531no—
340#2000—538no—
341C2×#340—531no—
342C2×C53#200018yescomplemented
343C2×C53C2×#39211yescharacteristic
344C2×C53C2×#45011yescharacteristic
345C2×C53C2×#44811yescharacteristic
346C2×C53C2×#44911yescharacteristic
347C2×C53C2×#45211yescharacteristic
348C2×C53C2×#46511yescharacteristic
349C2×C53C2×#46811yescharacteristic
350D53—44no—
351D53—84no—
352C23×Q8C2×D5311yescharacteristic
353C23×D4—531no—
354C23×D4—1061no—
355C24×C4—531no—
356C22×SD16—534no—
357C22×SD16—1064no—
358C22×SD16—1064no—
359C22×SD16—1064no—
360C23×C8—531no—
361C23×C8C2×D5311yescharacteristic
362C2×C22⋊Q8—534no—
363C2×C22⋊Q8—1064no—
364C2×C22⋊Q8—1062no—
365C2×C22⋊Q8—1064no—
366C2×#121—534no—
367C2×#121—1062no—
368C2×#121—1064no—
369C2×#121—1064no—
370C22×C4⋊C4—531no—
371C22×C4⋊C4—1061no—
372C22×C4⋊C4C2×D5311yescharacteristic
373C22×C22⋊C4—1061no—
374C22×C22⋊C4—1061no—
375C2×#103—24no—
376C2×#103C2×D5314yescomplemented
377C2×#102—534no—
378C2×#102—1064no—
379C2×C22⋊C8—534no—
380C2×C22⋊C8—1064no—
381#340—534no—
382C53C2×#200011yescharacteristic, complemented, Sylow
383C25—531no—
384C25—1061no—
385C22×Q8—22no—
386C22×Q8—21no—
387C22×Q8—22no—
388C22×Q8—22no—
389C22×Q8—22no—
390C22×Q8—22no—
391C22×Q8D21214yescomplemented
392C22×Q8C22×D5312yes—
393C22×D4—534no—
394C22×D4—532no—
395C22×D4—1064no—
396C22×D4—1068no—
397C22×D4—1062no—
398C22×D4—1062no—
399C22×D4—1062no—
400C22×D4—1062no—
401C22×D4—1061no—
402C22×D4—1062no—
403C22×D4—2124no—
404C22×D4—2124no—
405C23×C4—21no—
406C23×C4—21no—
407C23×C4—531no—
408C23×C4—532no—
409C23×C4—532no—
410C23×C4—1061no—
411C23×C4—1062no—
412C23×C4—1062no—
413C23×C4—1062no—
414C23×C4—1062no—
415C23×C4—1062no—
416C23×C4—1062no—
417C23×C4C22×D5311yescharacteristic
418C2×SD16—1068no—
419C2×SD16—1068no—
420C2×SD16—10632no—
421C2×SD16—1068no—
422C22×C8—22no—
423C22×C8—22no—
424C22×C8—22no—
425C22×C8—534no—
426C22×C8—532no—
427C22×C8—1062no—
428C22×C8—1062no—
429C22×C8C22×D5312yes—
430C22⋊Q8—1068no—
431C22⋊Q8—10616no—
432C22⋊Q8—1068no—
433#121—1068no—
434#121—10616no—
435#121—1068no—
436C2×C4⋊C4—21no—
437C2×C4⋊C4—22no—
438C2×C4⋊C4—534no—
439C2×C4⋊C4—532no—
440C2×C4⋊C4—1064no—
441C2×C4⋊C4—1068no—
442C2×C4⋊C4—1062no—
443C2×C4⋊C4—1061no—
444C2×C4⋊C4C22⋊D5314yes—
445C2×C4⋊C4C22×D5312yes—
446C2×C22⋊C4—1064no—
447C2×C22⋊C4—1064no—
448C2×C22⋊C4—1068no—
449C2×C22⋊C4—1068no—
450#103—28no—
451#102—1068no—
452C22⋊C8—1068no—
453C24—534no—
454C24—532no—
455C24—1062no—
456C24—1062no—
457C24—1062no—
458C24—1062no—
459C24—1062no—
460C24—1062no—
461C24—1064no—
462C24—1062no—
463C24—2122no—
464C24—2122no—
465C24—2122no—
466C24—2122no—
467C24—2122no—
468C24—2122no—
469C24D4×D5311yescharacteristic
470C2×Q8—24no—
471C2×Q8—22no—
472C2×Q8—28no—
473C2×Q8—28no—
474C2×Q8—22no—
475C2×Q8—22no—
476C2×Q8—22no—
477C2×Q8—22no—
478C2×Q8—42no—
479C2×Q8—44no—
480C2×Q8—42no—
481C2×Q8—44no—
482C2×Q8C2×D21214yescomplemented
483C2×D4—534no—
484C2×D4—10616no—
485C2×D4—1064no—
486C2×D4—1068no—
487C2×D4—1068no—
488C2×D4—1062no—
489C2×D4—1062no—
490C2×D4—1062no—
491C2×D4—1062no—
492C2×D4—1062no—
493C2×D4—2128no—
494C2×D4—21232no—
495C2×D4—2128no—
496C2×D4—2122no—
497C2×D4—2122no—
498C2×D4—2124no—
499C2×D4—2124no—
500C22×C4—24no—
501C22×C4—24no—
502C22×C4—22no—
503C22×C4—22no—
504C22×C4—42no—
505C22×C4—42no—
506C22×C4—42no—
507C22×C4—42no—
508C22×C4—532no—
509C22×C4—1062no—
510C22×C4—1064no—
511C22×C4—1062no—
512C22×C4—1062no—
513C22×C4—1062no—
514C22×C4—1064no—
515C22×C4—1064no—
516C22×C4—1064no—
517C22×C4—1062no—
518C22×C4—1062no—
519C22×C4—1062no—
520C22×C4—1062no—
521C22×C4—1062no—
522C22×C4—1062no—
523C22×C4—2122no—
524C22×C4—2122no—
525C22×C4—2124no—
526C22×C4—2124no—
527C22×C4—2124no—
528C22×C4—2124no—
529C22×C4C2×D21212yes—
530C22×C4#1171994512yes—
531C22×C4D4×D5311yescharacteristic
532C22×C4D4×D5312yes—
533C22×C4D4×D5312yes—
534C22×C4D4×D5312yes—
535C22×C4C2×C22⋊D5312yes—
536C22×C4C23×D5311yescharacteristic
537SD16—10632no—
538C2×C8—22no—
539C2×C8—22no—
540C2×C8—22no—
541C2×C8—44no—
542C2×C8—534no—
543C2×C8—1068no—
544C2×C8—1062no—
545C2×C8—1062no—
546C4⋊C4—24no—
547C4⋊C4—22no—
548C4⋊C4—534no—
549C4⋊C4—10616no—
550C4⋊C4—1064no—
551C4⋊C4—1062no—
552C4⋊C4C2×C22⋊D5314yes—
553C22⋊C4—10616no—
554C22⋊C4—10616no—
555C23—534no—
556C23—1068no—
557C23—1068no—
558C23—1062no—
559C23—1062no—
560C23—1068no—
561C23—1062no—
562C23—1062no—
563C23—1062no—
564C23—1062no—
565C23—1064no—
566C23—4244no—
567C23—4244no—
568C23—4244no—
569C23—4244no—
570C23—2128no—
571C23—2128no—
572C23—2128no—
573C23—2122no—
574C23—2122no—
575C23—2122no—
576C23—2122no—
577C23—2122no—
578C23—2122no—
579C23—2124no—
580C23—2124no—
581C23—2124no—
582C23—2124no—
583C23#1313318512yes—
584C23#1313318712yes—
585C23#1313319012yes—
586C23SD16×D5312yes—
587C23#1313324012yes—
588C23#1313324412yes—
589C23#1313324912yes—
590C23C2×D4×D5311yescharacteristic
591Q8—24no—
592Q8—24no—
593Q8—48no—
594Q8—42no—
595Q8—42no—
596D4—1064no—
597D4—1064no—
598D4—21232no—
599D4—2128no—
600D4—2122no—
601D4—2122no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—48no—
605C2×C4—48no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—1062no—
611C2×C4—1064no—
612C2×C4—1062no—
613C2×C4—1062no—
614C2×C4—2128no—
615C2×C4—2124no—
616C2×C4—2122no—
617C2×C4—2122no—
618C2×C4—2128no—
619C2×C4—2122no—
620C2×C4—2122no—
621C2×C4—2122no—
622C2×C4—2122no—
623C2×C4#1313318512yes—
624C2×C4#1313318712yes—
625C2×C4#1313318714yes—
626C2×C4#1313319012yes—
627C2×C4#1313319014yes—
628C2×C4#1313319114yes—
629C2×C4#1313321314yes—
630C2×C4C2×D4×D5311yescharacteristic
631C2×C4C2×D4×D5311yescharacteristic
632C2×C4C2×D4×D5311yescharacteristic
633C2×C4C22×C22⋊D5311yescharacteristic
634C2×C4C22×D21211yescharacteristic
635C2×C4C2×#1171994511yescharacteristic
636C8—42no—
637C8—1064no—
638C22—1064no—
639C22—1064no—
640C22—1064no—
641C22—4242no—
642C22—42416no—
643C22—4242no—
644C22—4242no—
645C22—4242no—
646C22—2124no—
647C22—2124no—
648C22—2124no—
649C22—21216no—
650C22—2122no—
651C22—2122no—
652C22—2122no—
653C22—2122no—
654C22#1681053214yes—
655C22#1681076314yes—
656C22#1681077114yes—
657C22#1681077214yes—
658C22#1681078414yes—
659C22#1681085514yes—
660C22#1681087214yes—
661C22C2×#1313318511yescharacteristic
662C22C2×#1313318711yescharacteristic
663C22C2×#1313319011yescharacteristic
664C22C2×SD16×D5311yescharacteristic
665C22C2×#1313324011yescharacteristic
666C22C2×#1313324411yescharacteristic
667C22C2×#1313324911yescharacteristic
668C4—44no—
669C4—44no—
670C4—2124no—
671C4—2122no—
672C4#1681053214yes—
673C4C2×#1313318711yescharacteristic
674C4C2×#1313319011yescharacteristic
675C4C2×#1313319111yescharacteristic
676C4C2×#1313321311yescharacteristic
677C2—4244no—
678C2—2124no—
679C2#2027571718yesdirect product factor
680C2C2×#1681053211yescharacteristic
681C2C2×#1681077211yescharacteristic
682C2C2×#1681076311yescharacteristic
683C2C2×#1681077111yescharacteristic
684C2C2×#1681078411yescharacteristic
685C2C2×#1681085511yescharacteristic
686C2C2×#1681087211yescharacteristic
subgroupG/H
centerC24D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC22×#103×C53C2
socleC24×C53C2×D4
prime radicalC22×#13133192C2
hypercenterC22×#103D53
hypocenterC53C2×#2000
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C24 ◁ C23×C4 ◁ C22×#103

Complements

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

NH ≅ G/Ntype
1. C23×#11719924C2semidirect
2. C22×#13133193C2semidirect
4. C22×#13133139C2semidirect
5. C22×#13133129C2semidirect
6. #20275717C2direct (N × H)
7. C2×#16810385C2semidirect
8. C22×#103×C53C2semidirect
15. C22×#11719924C22semidirect
20. C2×#13133193C22semidirect
30. C2×#13133139C22semidirect
32. C2×#13133129C22semidirect
33. #16810385C22semidirect
38. C2×#103×C53C22semidirect
69. C22×#88322D4semidirect
79. C2×C4⋊Dic53D4semidirect
95. C22×Q8×C53D4semidirect
124. C23×D53SD16semidirect
138. C2×Q8×C53C2×D4semidirect
205. C4⋊Dic53C2×D4semidirect
210. C2×#88322C2×D4semidirect
246. C22×D53C2×SD16semidirect
337. C22×#103D53semidirect
342. C2×C53#2000semidirect
376. C2×#103C2×D53semidirect
382. C53C2×#2000semidirect
391. C22×Q8D212semidirect
482. C2×Q8C2×D212semidirect
679. C2#20275717direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#117199242
C22×#131331932
C22×#131331922
C22×#131331392
C22×#131331292
#202757172
C2×#168103852
C22×#103×C532
C2×#200053