Finite Group Databasebeta

← C2×#21425286 #28616154

Subgroups — C2×#21425286 #28616154

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×#14273701C211yesmaximal, characteristic
2C22×#14273659C211yesmaximal, characteristic, complemented
3C22×#14273649C211yesmaximal, characteristic, complemented
4C22×#14273649C211yesmaximal, characteristic, complemented
5C2×#16837262C211yesmaximal, characteristic, complemented
6#21425286C218yesmaximal, direct product factor
7C2×#16837165C211yesmaximal, characteristic, complemented
8C22×C22⋊C8×C59C211yesmaximal, characteristic, complemented
9C23×D236C2211yescharacteristic
10C23×C22⋊D59—21no—
11C2×#14273701—24no—
12C2×#14273701—24no—
13C2×#14273701—22no—
14C2×#14273701C2214yes—
15C22×C4⋊Dic59C2211yescharacteristic
16C22×C4⋊Dic59C2211yescharacteristic
17C22×C4⋊Dic59C2211yescharacteristic
18C22×C22⋊Dic59—21no—
19C22×(C2×D59)⋊C4—21no—
20C2×#14273659—24no—
21C2×#14273659C2214yescomplemented
22C2×#14273649—24no—
23C2×#14273649—24no—
24C2×#14273649C2214yescomplemented
25C2×#14273649C2214yescomplemented
26#16837262—22no—
27#16837262C2214yescomplemented
28C2×C59⋊#95—21no—
29#16837165C2214yescomplemented
30C24×C4×C59C2211yescharacteristic
31C23×C8×C59C2211yescharacteristic
32C23×C8×C59C2211yescharacteristic
33C2×C22⋊C8×C59—24no—
34C2×C22⋊C8×C59C2214yescomplemented
35C24×D59—21no—
36C22×D236—22no—
37C22×D236—22no—
38C22×D236—22no—
39C22×D236—22no—
40C22×D236—21no—
41C22×D236—22no—
42C22×D236D414yes—
43C22×D236C2312yes—
44C22×C22⋊D59—28no—
45C22×C22⋊D59—24no—
46C22×C22⋊D59—44no—
47C22×C22⋊D59—44no—
48C23×Dic59—21no—
49C23×Dic59—21no—
50C23×Dic59—21no—
51#14273701—216no—
52#14273701—28no—
53#14273701—28no—
54C2×C4⋊Dic59—21no—
55C2×C4⋊Dic59—22no—
56C2×C4⋊Dic59—22no—
57C2×C4⋊Dic59—22no—
58C2×C4⋊Dic59—22no—
59C2×C4⋊Dic59—22no—
60C2×C4⋊Dic59—21no—
61C2×C4⋊Dic59—21no—
62C2×C4⋊Dic59D414yes—
63C2×C4⋊Dic59C2312yes—
64C2×C4⋊Dic59C2312yes—
65C2×C4⋊Dic59C2312yes—
66C2×C22⋊Dic59—24no—
67C2×C22⋊Dic59—44no—
68C2×(C2×D59)⋊C4—28no—
69C2×(C2×D59)⋊C4—24no—
70#14273659—28no—
71#14273649—28no—
72#14273649—28no—
73C59⋊#95—24no—
74C25×C59D411yescharacteristic
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×C59D412yes—
82C23×C4×C59D411yescharacteristic
83C23×C4×C59C2312yes—
84C23×C4×C59C2311yescharacteristic
85C22×C8×C59—22no—
86C22×C8×C59—22no—
87C22×C8×C59—22no—
88C22×C8×C59—22no—
89C22×C8×C59—22no—
90C22×C8×C59D414yescomplemented
91C22×C8×C59C2312yes—
92C22×C8×C59C2312yes—
93C22⋊C8×C59—28no—
94C24×C59—22no—
95C24×C59—22no—
96C24×C59—22no—
97C24×C59—22no—
98C24×C59—22no—
99C24×C59—22no—
100C24×C59SD1614yes—
101C24×C59C2×D412yes—
102C24×C59C2×D411yescharacteristic
103C23×D59—24no—
104C23×D59—22no—
105C23×D59—42no—
106C23×D59—42no—
107C23×D59—42no—
108C23×D59—42no—
109C23×D59—42no—
110C23×D59—42no—
111C22×C4×C59—22no—
112C22×C4×C59—22no—
113C22×C4×C59—22no—
114C22×C4×C59—24no—
115C22×C4×C59—24no—
116C22×C4×C59—24no—
117C22×C4×C59—22no—
118C22×C4×C59—22no—
119C22×C4×C59—22no—
120C22×C4×C59—22no—
121C22×C4×C59—22no—
122C22×C4×C59—22no—
123C22×C4×C59—44no—
124C22×C4×C59—44no—
125C22×C4×C59—44no—
126C22×C4×C59—44no—
127C22×C4×C59C2×D412yes—
128C22×C4×C59C2×D411yescharacteristic
129C22×C4×C59C2×D412yes—
130C22×C4×C59C2×D412yes—
131C22×C4×C59C2×D412yes—
132C22×C4×C59#4112yes—
133C22×C4×C59#4112yes—
134C22×C4×C59#4112yes—
135C22×C4×C59C2411yescharacteristic
136C2×C22⋊D59—216no—
137C2×C22⋊D59—48no—
138C2×C22⋊D59—432no—
139C2×C22⋊D59—48no—
140C22×Dic59—24no—
141C22×Dic59—22no—
142C22×Dic59—22no—
143C22×Dic59—22no—
144C22×Dic59—42no—
145C22×Dic59—42no—
146C22×Dic59—42no—
147C22×Dic59—42no—
148C22×Dic59—42no—
149C22×Dic59—42no—
150C22×Dic59—42no—
151C22×Dic59—42no—
152C2×D236—28no—
153C2×D236—28no—
154C2×D236—22no—
155C2×D236—22no—
156C2×D236—22no—
157C2×D236—22no—
158C2×D236—24no—
159C2×D236—22no—
160C2×D236—44no—
161C2×D236—42no—
162C2×D236—42no—
163C2×D236—44no—
164C2×D236C2×D414yes—
165C2×C8×C59—28no—
166C2×C8×C59—22no—
167C2×C8×C59—22no—
168C2×C8×C59—22no—
169C2×C8×C59—22no—
170C2×C8×C59—22no—
171C2×C8×C59—44no—
172C2×C8×C59C2×D414yescomplemented
173C22⋊Dic59—48no—
174(C2×D59)⋊C4—216no—
175C4⋊Dic59—24no—
176C4⋊Dic59—22no—
177C4⋊Dic59—22no—
178C4⋊Dic59—22no—
179C4⋊Dic59—22no—
180C4⋊Dic59—22no—
181C4⋊Dic59—42no—
182C4⋊Dic59—42no—
183C4⋊Dic59C2×D414yes—
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—44no—
194C23×C59—44no—
195C23×C59—44no—
196C23×C59—44no—
197C23×C59#12012yes—
198C23×C59#12112yes—
199C23×C59#12312yes—
200C23×C59C2×SD1614yes—
201C23×C59C2×SD1612yes—
202C23×C59#13512yes—
203C23×C59AGL(1,ℤ/8)12yes—
204C23×C59AGL(1,ℤ/8)12yes—
205C23×C59C22×D411yescharacteristic
206C22×D59—24no—
207C22×D59—42no—
208C22×D59—42no—
209C22×D59—42no—
210C22×D59—42no—
211C22×D59—42no—
212C22×D59—42no—
213C22×D59—48no—
214C22×D59—48no—
215C22×D59—48no—
216C22×D59—84no—
217C22×D59—84no—
218C22×D59—84no—
219C22×D59—84no—
220C2×C4×C59—22no—
221C2×C4×C59—22no—
222C2×C4×C59—22no—
223C2×C4×C59—44no—
224C2×C4×C59—42no—
225C2×C4×C59—42no—
226C2×C4×C59—42no—
227C2×C4×C59—42no—
228C2×C4×C59—48no—
229C2×C4×C59#12012yes—
230C2×C4×C59#12114yes—
231C2×C4×C59#12114yes—
232C2×C4×C59#12112yes—
233C2×C4×C59#12314yes—
234C2×C4×C59#12312yes—
235C2×C4×C59#12414yes—
236C2×C4×C59C22×D411yescharacteristic
237C2×C4×C59C22×D411yescharacteristic
238C2×C4×C59C22×D411yescharacteristic
239C2×C4×C59C2×#4111yescharacteristic
240C2×C4×C59C2×#4111yescharacteristic
241C2×C4×C59C2×#4111yescharacteristic
242C22⋊D59—432no—
243C2×Dic59—24no—
244C2×Dic59—42no—
245C2×Dic59—42no—
246C2×Dic59—48no—
247C2×Dic59—42no—
248C2×Dic59—42no—
249C2×Dic59—42no—
250C2×Dic59—42no—
251C2×Dic59—42no—
252C2×Dic59—42no—
253C2×Dic59—84no—
254C2×Dic59—84no—
255D236—24no—
256D236—24no—
257D236—42no—
258D236—42no—
259D236—48no—
260C8×C59—24no—
261C8×C59—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#39414yes—
272C22×C59#44914yes—
273C22×C59#45014yes—
274C22×C59#46514yes—
275C22×C59#46914yes—
276C22×C59#48114yes—
277C22×C59#48214yes—
278C22×C59C2×#12011yescharacteristic
279C22×C59C2×#12111yescharacteristic
280C22×C59C2×#12311yescharacteristic
281C22×C59C22×SD1611yescharacteristic
282C22×C59C2×#13511yescharacteristic
283C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
284C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
285C2×D59—44no—
286C2×D59—44no—
287C2×D59—44no—
288C2×D59—82no—
289C2×D59—82no—
290C2×D59—82no—
291C2×D59—82no—
292C2×D59—816no—
293C4×C59—42no—
294C4×C59#39414yes—
295C4×C59C2×#12111yescharacteristic
296C4×C59C2×#12111yescharacteristic
297C4×C59C2×#12311yescharacteristic
298C4×C59C2×#12411yescharacteristic
299Dic59—44no—
300Dic59—82no—
301Dic59—82no—
302C22×#121—591no—
303C22×#106—591no—
304C22×#102—591no—
305C22×#102—591no—
306C22×C22⋊C8D5911yescharacteristic, complemented
307C2×#380—591no—
308#2034—598no—
309C2×#340—591no—
310C2×C59—44no—
311C2×C59#203418yescomplemented
312C2×C59C2×#39411yescharacteristic
313C2×C59C2×#45011yescharacteristic
314C2×C59C2×#44911yescharacteristic
315C2×C59C2×#46511yescharacteristic
316C2×C59C2×#46911yescharacteristic
317C2×C59C2×#48211yescharacteristic
318C2×C59C2×#48111yescharacteristic
319D59—84no—
320C23×D4—591no—
321C23×D4—1181no—
322C24×C4C2×D5911yescharacteristic
323C23×C8C2×D5911yescharacteristic
324C23×C8C2×D5911yescharacteristic
325C2×#121—594no—
326C2×#121—1184no—
327C2×#121—1184no—
328C2×#121—1182no—
329C22×C4⋊C4—591no—
330C22×C4⋊C4—591no—
331C22×C4⋊C4—591no—
332C22×C22⋊C4—1181no—
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—24no—
341C2×C22⋊C8C2×D5914yescomplemented
342#380—594no—
343#380—1182no—
344C2×#95—1181no—
345#340—594no—
346C59C2×#203411yescharacteristic, complemented, Sylow
347C25—1181no—
348C25D23611yescharacteristic
349C22×D4—594no—
350C22×D4—592no—
351C22×D4—1188no—
352C22×D4—1184no—
353C22×D4—1181no—
354C22×D4—1182no—
355C22×D4—1182no—
356C22×D4—1182no—
357C22×D4—1182no—
358C22×D4—1182no—
359C22×D4—2364no—
360C22×D4—2364no—
361C23×C4—22no—
362C23×C4—22no—
363C23×C4—22no—
364C23×C4—22no—
365C23×C4—22no—
366C23×C4—22no—
367C23×C4—1181no—
368C23×C4—1181no—
369C23×C4—1181no—
370C23×C4D23612yes—
371C23×C4D23611yescharacteristic
372C23×C4C22×D5912yes—
373C23×C4C22×D5911yescharacteristic
374C22×C8—22no—
375C22×C8—22no—
376C22×C8—22no—
377C22×C8—22no—
378C22×C8—22no—
379C22×C8C22⋊D5914yescomplemented
380C22×C8C22×D5912yes—
381C22×C8C22×D5912yes—
382#121—1188no—
383#121—1188no—
384#121—11816no—
385C2×C4⋊C4—594no—
386C2×C4⋊C4—592no—
387C2×C4⋊C4—592no—
388C2×C4⋊C4—592no—
389C2×C4⋊C4—1181no—
390C2×C4⋊C4—1182no—
391C2×C4⋊C4—1182no—
392C2×C4⋊C4—1182no—
393C2×C4⋊C4—1182no—
394C2×C4⋊C4—1182no—
395C2×C4⋊C4—1181no—
396C2×C4⋊C4—1181no—
397C2×C22⋊C4—1188no—
398C2×C22⋊C4—1184no—
399C2×C22⋊C4—1184no—
400C2×C22⋊C4—2364no—
401#106—1188no—
402#102—1188no—
403#102—1188no—
404C22⋊C8—28no—
405#95—1184no—
406C24—22no—
407C24—22no—
408C24—22no—
409C24—22no—
410C24—22no—
411C24—22no—
412C24—1184no—
413C24—1182no—
414C24—2362no—
415C24—2362no—
416C24—2362no—
417C24—2362no—
418C24—2362no—
419C24—2362no—
420C24#1174566014yes—
421C24C2×D23612yes—
422C24C2×D23611yescharacteristic
423C2×D4—594no—
424C2×D4—11816no—
425C2×D4—1184no—
426C2×D4—1188no—
427C2×D4—1188no—
428C2×D4—1182no—
429C2×D4—1182no—
430C2×D4—1182no—
431C2×D4—1182no—
432C2×D4—1182no—
433C2×D4—2368no—
434C2×D4—23632no—
435C2×D4—2368no—
436C2×D4—2364no—
437C2×D4—2362no—
438C2×D4—2362no—
439C2×D4—2364no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—1184no—
457C22×C4—1182no—
458C22×C4—1182no—
459C22×C4—1182no—
460C22×C4—2362no—
461C22×C4—2362no—
462C22×C4—2362no—
463C22×C4—2362no—
464C22×C4—2362no—
465C22×C4—2362no—
466C22×C4—2362no—
467C22×C4—2362no—
468C22×C4C2×D23612yes—
469C22×C4C2×D23611yescharacteristic
470C22×C4#1174568512yes—
471C22×C4D4×D5912yes—
472C22×C4D4×D5912yes—
473C22×C4#1174568712yes—
474C22×C4#1174568712yes—
475C22×C4C2×C22⋊D5912yes—
476C22×C4C23×D5911yescharacteristic
477C2×C8—28no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—44no—
484C2×C8C2×C22⋊D5914yescomplemented
485C4⋊C4—594no—
486C4⋊C4—1184no—
487C4⋊C4—1182no—
488C4⋊C4—1182no—
489C4⋊C4—1182no—
490C4⋊C4—1182no—
491C4⋊C4—1182no—
492C4⋊C4—2362no—
493C4⋊C4—2362no—
494C22⋊C4—11816no—
495C22⋊C4—2368no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—28no—
503C23—28no—
504C23—28no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—1184no—
510C23—4724no—
511C23—4724no—
512C23—4724no—
513C23—4724no—
514C23—2362no—
515C23—2362no—
516C23—2362no—
517C23—2362no—
518C23—2362no—
519C23—2362no—
520C23—2368no—
521C23—2368no—
522C23—2368no—
523C23#1427369712yes—
524C23#1427370112yes—
525C23#1427371112yes—
526C23C2×#1174566014yes—
527C23C2×#1174566012yes—
528C23#1427374912yes—
529C23#1427375512yes—
530C23#1427375512yes—
531C23C22×D23611yescharacteristic
532D4—1184no—
533D4—1184no—
534D4—23632no—
535D4—2362no—
536D4—2362no—
537D4—2368no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—48no—
547C2×C4—1184no—
548C2×C4—4724no—
549C2×C4—4724no—
550C2×C4—2362no—
551C2×C4—2362no—
552C2×C4—2362no—
553C2×C4—2362no—
554C2×C4—2362no—
555C2×C4—2362no—
556C2×C4—2362no—
557C2×C4—2362no—
558C2×C4—2368no—
559C2×C4#1427369712yes—
560C2×C4#1427369814yes—
561C2×C4#1427369914yes—
562C2×C4#1427370112yes—
563C2×C4#1427370914yes—
564C2×C4#1427371112yes—
565C2×C4#1427371314yes—
566C2×C4C2×D4×D5911yescharacteristic
567C2×C4C2×D4×D5911yescharacteristic
568C2×C4C22×C22⋊D5911yescharacteristic
569C2×C4C2×#1174568711yescharacteristic
570C2×C4C2×#1174568711yescharacteristic
571C2×C4C2×#1174568511yescharacteristic
572C8—24no—
573C8—42no—
574C22—24no—
575C22—24no—
576C22—24no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—416no—
581C22—42no—
582C22—4722no—
583C22—4722no—
584C22—4722no—
585C22—47216no—
586C22—4722no—
587C22—2364no—
588C22—2364no—
589C22—2364no—
590C22#1683731814yes—
591C22#1683754914yes—
592C22#1683755514yes—
593C22#1683763114yes—
594C22#1683765514yes—
595C22#1683772114yes—
596C22#1683772714yes—
597C22C2×#1427369711yescharacteristic
598C22C2×#1427370111yescharacteristic
599C22C2×#1427371111yescharacteristic
600C22C22×#1174566011yescharacteristic
601C22C2×#1427374911yescharacteristic
602C22C2×#1427375511yescharacteristic
603C22C2×#1427375511yescharacteristic
604C4—42no—
605C4—4722no—
606C4—4722no—
607C4—2364no—
608C4#1683731814yes—
609C4C2×#1427369811yescharacteristic
610C4C2×#1427369911yescharacteristic
611C4C2×#1427370911yescharacteristic
612C4C2×#1427371311yescharacteristic
613C2—44no—
614C2—4724no—
615C2#2142528618yesdirect product factor
616C2C2×#1683731811yescharacteristic
617C2C2×#1683755511yescharacteristic
618C2C2×#1683754911yescharacteristic
619C2C2×#1683763111yescharacteristic
620C2C2×#1683765511yescharacteristic
621C2C2×#1683772711yescharacteristic
622C2C2×#1683772111yescharacteristic
subgroupG/H
centerC24C2×D236
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC22×C22⋊C8×C59C2
socleC24×C59C2×D4
prime radicalC22×#14273701C2
hypercenterC22×C22⋊C8D59
hypocenterC59C2×#2034
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C24 ◁ C24×C4 ◁ C22×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C22×#14273659C2semidirect
3. C22×#14273649C2semidirect
4. C22×#14273649C2semidirect
5. C2×#16837262C2semidirect
6. #21425286C2direct (N × H)
7. C2×#16837165C2semidirect
8. C22×C22⋊C8×C59C2semidirect
21. C2×#14273659C22semidirect
24. C2×#14273649C22semidirect
25. C2×#14273649C22semidirect
27. #16837262C22semidirect
29. #16837165C22semidirect
34. C2×C22⋊C8×C59C22semidirect
90. C22×C8×C59D4semidirect
172. C2×C8×C59C2×D4semidirect
306. C22×C22⋊C8D59semidirect
311. C2×C59#2034semidirect
341. C2×C22⋊C8C2×D59semidirect
346. C59C2×#2034semidirect
379. C22×C8C22⋊D59semidirect
484. C2×C8C2×C22⋊D59semidirect
615. C2#21425286direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#142737012
C22×#142736592
C22×#142736492
C2×#168372622
#214252862
C2×#168371652
C22×C22⋊C8×C592
C2×#203459