Finite Group Databasebeta

← C2×#21425282 #28616150

Subgroups — C2×#21425282 #28616150

Order 15104.

2006 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14273658C211yesmaximal, characteristic, complemented
2C22×#14273648C211yesmaximal, characteristic, complemented
3C22×#14273648C211yesmaximal, characteristic, complemented
4C22×C22⋊(C59⋊C8)C211yesmaximal, characteristic, complemented
5C2×#16837262C211yesmaximal, characteristic, complemented
6#21425282C218yesmaximal, direct product factor
7C2×#16837164C211yesmaximal, characteristic, complemented
8C22×#121×C59C211yesmaximal, characteristic
9C23×C59⋊C8C2211yescharacteristic
10C23×C59⋊C8C2211yescharacteristic
11C22×C4⋊Dic59C2211yescharacteristic
12C22×C4⋊Dic59C2211yescharacteristic
13C22×C22⋊Dic59—21no—
14C2×#14273658—24no—
15C2×#14273658C2214yescomplemented
16C2×#14273648—24no—
17C2×#14273648—24no—
18C2×#14273648C2214yescomplemented
19C2×#14273648C2214yescomplemented
20C2×C22⋊(C59⋊C8)—24no—
21C2×C22⋊(C59⋊C8)C2214yescomplemented
22#16837262—22no—
23#16837262C2214yescomplemented
24C2×C59⋊#95—21no—
25#16837164C2214yescomplemented
26C23×D4×C59—21no—
27C23×D4×C59C2211yescharacteristic
28C24×C4×C59C2211yescharacteristic
29C2×#121×C59—24no—
30C2×#121×C59—22no—
31C2×#121×C59—24no—
32C2×#121×C59C2214yes—
33C22×C4⋊C4×C59C2211yescharacteristic
34C22×C22⋊C4×C59—21no—
35C23×Dic59—21no—
36C23×Dic59—21no—
37C22×C59⋊C8—22no—
38C22×C59⋊C8—22no—
39C22×C59⋊C8—22no—
40C22×C59⋊C8—22no—
41C22×C59⋊C8—22no—
42C22×C59⋊C8D414yescomplemented
43C22×C59⋊C8C2312yes—
44C22×C59⋊C8C2312yes—
45C2×C4⋊Dic59—22no—
46C2×C4⋊Dic59—22no—
47C2×C4⋊Dic59—22no—
48C2×C4⋊Dic59—22no—
49C2×C4⋊Dic59—21no—
50C2×C4⋊Dic59—21no—
51C2×C4⋊Dic59C2312yes—
52C2×C4⋊Dic59C2312yes—
53C2×C22⋊Dic59—24no—
54C2×C22⋊Dic59—44no—
55#14273658—28no—
56#14273648—28no—
57#14273648—28no—
58C22⋊(C59⋊C8)—28no—
59C59⋊#95—24no—
60C25×C59—21no—
61C25×C59D411yescharacteristic
62C22×D4×C59—28no—
63C22×D4×C59—24no—
64C22×D4×C59—22no—
65C22×D4×C59—21no—
66C22×D4×C59—22no—
67C22×D4×C59—22no—
68C22×D4×C59—22no—
69C22×D4×C59—22no—
70C22×D4×C59—44no—
71C22×D4×C59—44no—
72C22×D4×C59D414yes—
73C22×D4×C59C2312yes—
74C23×C4×C59—21no—
75C23×C4×C59—22no—
76C23×C4×C59—22no—
77C23×C4×C59—22no—
78C23×C4×C59—22no—
79C23×C4×C59—22no—
80C23×C4×C59—22no—
81C23×C4×C59D411yescharacteristic
82C23×C4×C59D412yes—
83C23×C4×C59C2312yes—
84C23×C4×C59C2311yescharacteristic
85#121×C59—216no—
86#121×C59—28no—
87#121×C59—28no—
88C2×C4⋊C4×C59—21no—
89C2×C4⋊C4×C59—22no—
90C2×C4⋊C4×C59D414yes—
91C2×C4⋊C4×C59C2312yes—
92C2×C22⋊C4×C59—28no—
93C2×C22⋊C4×C59—24no—
94C24×C59—22no—
95C24×C59—22no—
96C24×C59—22no—
97C24×C59—22no—
98C24×C59—22no—
99C24×C59—22no—
100C24×C59—24no—
101C24×C59—22no—
102C24×C59—42no—
103C24×C59—42no—
104C24×C59—42no—
105C24×C59—42no—
106C24×C59—42no—
107C24×C59—42no—
108C24×C59SD1614yes—
109C24×C59C2×D412yes—
110C24×C59C2×D411yescharacteristic
111C2×D4×C59—216no—
112C2×D4×C59—24no—
113C2×D4×C59—28no—
114C2×D4×C59—28no—
115C2×D4×C59—22no—
116C2×D4×C59—22no—
117C2×D4×C59—22no—
118C2×D4×C59—22no—
119C2×D4×C59—22no—
120C2×D4×C59—48no—
121C2×D4×C59—432no—
122C2×D4×C59—48no—
123C2×D4×C59—44no—
124C2×D4×C59—42no—
125C2×D4×C59—42no—
126C2×D4×C59—44no—
127C2×D4×C59C2×D414yes—
128C22×C4×C59—22no—
129C22×C4×C59—24no—
130C22×C4×C59—22no—
131C22×C4×C59—22no—
132C22×C4×C59—22no—
133C22×C4×C59—24no—
134C22×C4×C59—24no—
135C22×C4×C59—24no—
136C22×C4×C59—22no—
137C22×C4×C59—22no—
138C22×C4×C59—22no—
139C22×C4×C59—22no—
140C22×C4×C59—22no—
141C22×C4×C59—22no—
142C22×C4×C59—42no—
143C22×C4×C59—42no—
144C22×C4×C59—44no—
145C22×C4×C59—44no—
146C22×C4×C59—44no—
147C22×C4×C59—44no—
148C22×C4×C59C2×D412yes—
149C22×C4×C59C2×D411yescharacteristic
150C22×C4×C59C2×D412yes—
151C22×C4×C59C2×D412yes—
152C22×C4×C59C2×D412yes—
153C22×C4×C59#4112yes—
154C22×C4×C59#4112yes—
155C22×C4×C59#4112yes—
156C22×C4×C59C2411yescharacteristic
157C22×Dic59—22no—
158C22×Dic59—22no—
159C22×Dic59—42no—
160C22×Dic59—42no—
161C22×Dic59—42no—
162C22×Dic59—42no—
163C22×Dic59—42no—
164C22×Dic59—42no—
165C4⋊C4×C59—24no—
166C4⋊C4×C59—22no—
167C4⋊C4×C59C2×D414yes—
168C22⋊C4×C59—216no—
169C22⋊Dic59—48no—
170C4⋊Dic59—22no—
171C4⋊Dic59—22no—
172C4⋊Dic59—22no—
173C4⋊Dic59—22no—
174C4⋊Dic59—42no—
175C4⋊Dic59—42no—
176C2×C59⋊C8—28no—
177C2×C59⋊C8—22no—
178C2×C59⋊C8—22no—
179C2×C59⋊C8—22no—
180C2×C59⋊C8—22no—
181C2×C59⋊C8—22no—
182C2×C59⋊C8—44no—
183C2×C59⋊C8C2×D414yescomplemented
184C23×C59—22no—
185C23×C59—22no—
186C23×C59—22no—
187C23×C59—22no—
188C23×C59—22no—
189C23×C59—22no—
190C23×C59—28no—
191C23×C59—28no—
192C23×C59—28no—
193C23×C59—24no—
194C23×C59—44no—
195C23×C59—44no—
196C23×C59—44no—
197C23×C59—44no—
198C23×C59—42no—
199C23×C59—42no—
200C23×C59—42no—
201C23×C59—42no—
202C23×C59—42no—
203C23×C59—42no—
204C23×C59—48no—
205C23×C59—48no—
206C23×C59—48no—
207C23×C59—84no—
208C23×C59—84no—
209C23×C59—84no—
210C23×C59—84no—
211C23×C59#12012yes—
212C23×C59#12112yes—
213C23×C59#12312yes—
214C23×C59C2×SD1614yes—
215C23×C59C2×SD1612yes—
216C23×C59#13512yes—
217C23×C59AGL(1,ℤ/8)12yes—
218C23×C59AGL(1,ℤ/8)12yes—
219C23×C59C22×D411yescharacteristic
220D4×C59—24no—
221D4×C59—24no—
222D4×C59—432no—
223D4×C59—42no—
224D4×C59—42no—
225D4×C59—48no—
226C2×C4×C59—22no—
227C2×C4×C59—24no—
228C2×C4×C59—22no—
229C2×C4×C59—22no—
230C2×C4×C59—42no—
231C2×C4×C59—42no—
232C2×C4×C59—44no—
233C2×C4×C59—48no—
234C2×C4×C59—42no—
235C2×C4×C59—42no—
236C2×C4×C59—42no—
237C2×C4×C59—42no—
238C2×C4×C59—48no—
239C2×C4×C59#12012yes—
240C2×C4×C59#12114yes—
241C2×C4×C59#12114yes—
242C2×C4×C59#12112yes—
243C2×C4×C59#12314yes—
244C2×C4×C59#12312yes—
245C2×C4×C59#12414yes—
246C2×C4×C59C22×D411yescharacteristic
247C2×C4×C59C22×D411yescharacteristic
248C2×C4×C59C22×D411yescharacteristic
249C2×C4×C59C2×#4111yescharacteristic
250C2×C4×C59C2×#4111yescharacteristic
251C2×C4×C59C2×#4111yescharacteristic
252C2×Dic59—42no—
253C2×Dic59—42no—
254C2×Dic59—42no—
255C2×Dic59—42no—
256C2×Dic59—42no—
257C2×Dic59—42no—
258C2×Dic59—84no—
259C2×Dic59—84no—
260C59⋊C8—24no—
261C59⋊C8—42no—
262C2×#2034—591nomaximal, Sylow
263C22×C59—24no—
264C22×C59—24no—
265C22×C59—24no—
266C22×C59—42no—
267C22×C59—42no—
268C22×C59—42no—
269C22×C59—42no—
270C22×C59—416no—
271C22×C59—44no—
272C22×C59—44no—
273C22×C59—44no—
274C22×C59—82no—
275C22×C59—82no—
276C22×C59—82no—
277C22×C59—82no—
278C22×C59—816no—
279C22×C59#39414yes—
280C22×C59#44914yes—
281C22×C59#45014yes—
282C22×C59#46514yes—
283C22×C59#46914yes—
284C22×C59#48114yes—
285C22×C59#48214yes—
286C22×C59C2×#12011yescharacteristic
287C22×C59C2×#12111yescharacteristic
288C22×C59C2×#12311yescharacteristic
289C22×C59C22×SD1611yescharacteristic
290C22×C59C2×#13511yescharacteristic
291C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
292C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
293C4×C59—42no—
294C4×C59—44no—
295C4×C59#39414yes—
296C4×C59C2×#12111yescharacteristic
297C4×C59C2×#12111yescharacteristic
298C4×C59C2×#12311yescharacteristic
299C4×C59C2×#12411yescharacteristic
300Dic59—82no—
301Dic59—82no—
302C22×#121D5911yescharacteristic
303C22×#106—591no—
304C22×#102—591no—
305C22×#102—591no—
306C22×C22⋊C8—591no—
307C2×#380—591no—
308#2034—598no—
309C2×#340—591no—
310C2×C59—44no—
311C2×C59—84no—
312C2×C59#203418yescomplemented
313C2×C59C2×#39411yescharacteristic
314C2×C59C2×#45011yescharacteristic
315C2×C59C2×#44911yescharacteristic
316C2×C59C2×#46511yescharacteristic
317C2×C59C2×#46911yescharacteristic
318C2×C59C2×#48211yescharacteristic
319C2×C59C2×#48111yescharacteristic
320C23×D4—21no—
321C23×D4C2×D5911yescharacteristic
322C24×C4C2×D5911yescharacteristic
323C23×C8—591no—
324C23×C8—591no—
325C2×#121—24no—
326C2×#121—22no—
327C2×#121—24no—
328C2×#121C2×D5914yes—
329C22×C4⋊C4—591no—
330C22×C4⋊C4—591no—
331C22×C4⋊C4C2×D5911yescharacteristic
332C22×C22⋊C4—21no—
333C22×C22⋊C4—1181no—
334C2×#106—594no—
335C2×#106—1184no—
336C2×#102—594no—
337C2×#102—594no—
338C2×#102—1184no—
339C2×#102—1184no—
340C2×C22⋊C8—594no—
341C2×C22⋊C8—1184no—
342#380—594no—
343#380—1182no—
344C2×#95—1181no—
345#340—594no—
346C59C2×#203411yescharacteristic, complemented, Sylow
347C25—21no—
348C25C22⋊D5911yescharacteristic
349C22×D4—28no—
350C22×D4—24no—
351C22×D4—22no—
352C22×D4—22no—
353C22×D4—22no—
354C22×D4—22no—
355C22×D4—21no—
356C22×D4—22no—
357C22×D4—44no—
358C22×D4—44no—
359C22×D4C22⋊D5914yes—
360C22×D4C22×D5912yes—
361C23×C4—21no—
362C23×C4—22no—
363C23×C4—22no—
364C23×C4—22no—
365C23×C4—22no—
366C23×C4—22no—
367C23×C4—22no—
368C23×C4—1181no—
369C23×C4—1181no—
370C23×C4C22⋊D5912yes—
371C23×C4C22⋊D5911yescharacteristic
372C23×C4C22×D5912yes—
373C23×C4C22×D5911yescharacteristic
374C22×C8—594no—
375C22×C8—592no—
376C22×C8—592no—
377C22×C8—1182no—
378C22×C8—1182no—
379C22×C8—1182no—
380C22×C8—1182no—
381C22×C8—1182no—
382#121—216no—
383#121—28no—
384#121—28no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—22no—
387C2×C4⋊C4—592no—
388C2×C4⋊C4—592no—
389C2×C4⋊C4—1182no—
390C2×C4⋊C4—1182no—
391C2×C4⋊C4—1182no—
392C2×C4⋊C4—1182no—
393C2×C4⋊C4—1181no—
394C2×C4⋊C4—1181no—
395C2×C4⋊C4C22⋊D5914yes—
396C2×C4⋊C4C22×D5912yes—
397C2×C22⋊C4—28no—
398C2×C22⋊C4—24no—
399C2×C22⋊C4—1184no—
400C2×C22⋊C4—2364no—
401#106—1188no—
402#102—1188no—
403#102—1188no—
404C22⋊C8—1188no—
405#95—1184no—
406C24—22no—
407C24—22no—
408C24—22no—
409C24—22no—
410C24—22no—
411C24—22no—
412C24—24no—
413C24—22no—
414C24—42no—
415C24—42no—
416C24—42no—
417C24—42no—
418C24—42no—
419C24—42no—
420C24#1174567014yes—
421C24C2×C22⋊D5912yes—
422C24C2×C22⋊D5911yescharacteristic
423C2×D4—216no—
424C2×D4—24no—
425C2×D4—28no—
426C2×D4—28no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—48no—
433C2×D4—432no—
434C2×D4—48no—
435C2×D4—44no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—44no—
439C2×D4C2×C22⋊D5914yes—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—1182no—
461C22×C4—1182no—
462C22×C4—2362no—
463C22×C4—2362no—
464C22×C4—2362no—
465C22×C4—2362no—
466C22×C4—2362no—
467C22×C4—2362no—
468C22×C4D4×D5912yes—
469C22×C4#1174568712yes—
470C22×C4#1174568712yes—
471C22×C4#1174568712yes—
472C22×C4C2×C22⋊D5912yes—
473C22×C4C2×C22⋊D5911yescharacteristic
474C22×C4C2×C22⋊D5912yes—
475C22×C4C2×C22⋊D5912yes—
476C22×C4C23×D5911yescharacteristic
477C2×C8—594no—
478C2×C8—1188no—
479C2×C8—1182no—
480C2×C8—1182no—
481C2×C8—1182no—
482C2×C8—1182no—
483C2×C8—1182no—
484C2×C8—2364no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—1182no—
488C4⋊C4—1182no—
489C4⋊C4—1182no—
490C4⋊C4—1182no—
491C4⋊C4—2362no—
492C4⋊C4—2362no—
493C4⋊C4C2×C22⋊D5914yes—
494C22⋊C4—216no—
495C22⋊C4—2368no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—28no—
503C23—28no—
504C23—28no—
505C23—24no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—48no—
517C23—48no—
518C23—48no—
519C23—84no—
520C23—84no—
521C23—84no—
522C23—84no—
523C23#1427369612yes—
524C23#1427369912yes—
525C23#1427370812yes—
526C23C2×#1174567014yes—
527C23C2×#1174567012yes—
528C23#1427374712yes—
529C23#1427375212yes—
530C23#1427375312yes—
531C23C22×C22⋊D5911yescharacteristic
532D4—24no—
533D4—24no—
534D4—432no—
535D4—42no—
536D4—42no—
537D4—48no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—48no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—48no—
551C2×C4—4724no—
552C2×C4—4724no—
553C2×C4—2362no—
554C2×C4—2362no—
555C2×C4—2362no—
556C2×C4—2362no—
557C2×C4—2362no—
558C2×C4—2362no—
559C2×C4#1427369612yes—
560C2×C4#1427369814yes—
561C2×C4#1427369914yes—
562C2×C4#1427369912yes—
563C2×C4#1427370814yes—
564C2×C4#1427370812yes—
565C2×C4#1427371414yes—
566C2×C4C2×D4×D5911yescharacteristic
567C2×C4C22×C22⋊D5911yescharacteristic
568C2×C4C22×C22⋊D5911yescharacteristic
569C2×C4C2×#1174568711yescharacteristic
570C2×C4C2×#1174568711yescharacteristic
571C2×C4C2×#1174568711yescharacteristic
572C8—1184no—
573C8—2362no—
574C22—24no—
575C22—24no—
576C22—24no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—416no—
581C22—42no—
582C22—44no—
583C22—44no—
584C22—44no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22—816no—
589C22—82no—
590C22#1683731614yes—
591C22#1683754714yes—
592C22#1683755314yes—
593C22#1683762914yes—
594C22#1683765314yes—
595C22#1683771814yes—
596C22#1683772314yes—
597C22C2×#1427369611yescharacteristic
598C22C2×#1427369911yescharacteristic
599C22C2×#1427370811yescharacteristic
600C22C22×#1174567011yescharacteristic
601C22C2×#1427374711yescharacteristic
602C22C2×#1427375211yescharacteristic
603C22C2×#1427375311yescharacteristic
604C4—42no—
605C4—44no—
606C4—4722no—
607C4—4722no—
608C4#1683731614yes—
609C4C2×#1427369811yescharacteristic
610C4C2×#1427369911yescharacteristic
611C4C2×#1427370811yescharacteristic
612C4C2×#1427371411yescharacteristic
613C2—44no—
614C2—84no—
615C2#2142528218yesdirect product factor
616C2C2×#1683731611yescharacteristic
617C2C2×#1683755311yescharacteristic
618C2C2×#1683754711yescharacteristic
619C2C2×#1683762911yescharacteristic
620C2C2×#1683765311yescharacteristic
621C2C2×#1683772311yescharacteristic
622C2C2×#1683771811yescharacteristic
subgroupG/H
centerC24C2×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
Fitting, prime radicalC22×#121×C59C2
socleC24×C59C2×D4
hypercenterC22×#121D59
hypocenterC59C2×#2034
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C24 ◁ C24×C4 ◁ C22×#121

Complements

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

NH ≅ G/Ntype
1. C22×#14273658C2semidirect
2. C22×#14273648C2semidirect
3. C22×#14273648C2semidirect
4. C22×C22⋊(C59⋊C8)C2semidirect
5. C2×#16837262C2semidirect
6. #21425282C2direct (N × H)
7. C2×#16837164C2semidirect
15. C2×#14273658C22semidirect
18. C2×#14273648C22semidirect
19. C2×#14273648C22semidirect
21. C2×C22⋊(C59⋊C8)C22semidirect
23. #16837262C22semidirect
25. #16837164C22semidirect
42. C22×C59⋊C8D4semidirect
183. C2×C59⋊C8C2×D4semidirect
312. C2×C59#2034semidirect
346. C59C2×#2034semidirect
615. C2#21425282direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#142736582
C22×#142736482
C22×C22⋊(C59⋊C8)2
C2×#168372622
#214252822
C2×#168371642
C22×#121×C592
C2×#203459