Finite Group Databasebeta

← C2×#21424523 #28590357

Subgroups — C2×#21424523 #28590357

Order 15104.

2438 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C8×D59C211yesmaximal, characteristic, complemented
2C22×#14273702C211yesmaximal, characteristic
3C22×#14273649C211yesmaximal, characteristic, complemented
4C22×#14273647C211yesmaximal, characteristic, complemented
5C2×#16837259C211yesmaximal, characteristic, complemented
6#21424523C218yesmaximal, direct product factor
7C2×#16837164C211yesmaximal, characteristic, complemented
8C2×C59×#340C211yesmaximal, characteristic, complemented
9C23×D236—21no—
10C23×D236C2211yescharacteristic
11C23×C4×D59C2211yescharacteristic
12C23×C59⋊C8C2211yescharacteristic
13C22×C8×D59—24no—
14C22×C8×D59—24no—
15C22×C8×D59—24no—
16C22×C8×D59C2214yescomplemented
17C2×#14273702—24no—
18C2×#14273702—22no—
19C2×#14273702C418yescomplemented
20C2×#14273702C2214yes—
21C22×C4⋊Dic59C2211yescharacteristic
22C22×(C2×D59)⋊C4—21no—
23C22×(C2×D59)⋊C4—21no—
24C2×#14273649—24no—
25C2×#14273649C2214yescomplemented
26C2×#14273647—24no—
27C2×#14273647C2214yescomplemented
28#16837259—22no—
29#16837259C2214yescomplemented
30C2×C59⋊#95—21no—
31#16837164C2214yescomplemented
32C23×C8×C59C2211yescharacteristic
33C22×C4⋊C4×C59C2211yescharacteristic
34C22×C4⋊C4×C59C2211yescharacteristic
35C59×#340C2214yescomplemented
36C24×D59—21no—
37C24×D59D411yescharacteristic
38C22×D236—22no—
39C22×D236—22no—
40C22×D236—22no—
41C22×D236—22no—
42C22×D236—28no—
43C22×D236—24no—
44C22×D236—21no—
45C22×D236—22no—
46C22×D236—44no—
47C22×D236—44no—
48C22×D236C2×C414yes—
49C22×D236C2312yes—
50C23×Dic59—21no—
51C23×Dic59D411yescharacteristic
52C22×C4×D59—22no—
53C22×C4×D59—22no—
54C22×C4×D59—22no—
55C22×C4×D59—22no—
56C22×C4×D59C2×C414yes—
57C22×C4×D59D412yes—
58C22×C4×D59D414yes—
59C22×C4×D59C2312yes—
60C22×C59⋊C8—22no—
61C22×C59⋊C8—22no—
62C22×C59⋊C8D414yescomplemented
63C22×C59⋊C8C2312yes—
64C2×C8×D59—232no—
65C2×C8×D59—28no—
66C2×C8×D59—28no—
67C2×C8×D59—28no—
68#14273702—216no—
69#14273702—28no—
70#14273702C2×C4116yescomplemented
71C2×C4⋊Dic59—22no—
72C2×C4⋊Dic59—22no—
73C2×C4⋊Dic59—21no—
74C2×C4⋊Dic59C2312yes—
75C2×(C2×D59)⋊C4—28no—
76C2×(C2×D59)⋊C4—24no—
77C2×(C2×D59)⋊C4—24no—
78C2×(C2×D59)⋊C4—44no—
79#14273649—28no—
80#14273647—28no—
81C59⋊#95—24no—
82C23×C4×C59—21no—
83C23×C4×C59—21no—
84C23×C4×C59C2311yescharacteristic
85C22×C8×C59—22no—
86C22×C8×C59—22no—
87C22×C8×C59D414yescomplemented
88C22×C8×C59C2312yes—
89C2×C4⋊C4×C59—21no—
90C2×C4⋊C4×C59—22no—
91C2×C4⋊C4×C59—22no—
92C2×C4⋊C4×C59—22no—
93C2×C4⋊C4×C59—21no—
94C2×C4⋊C4×C59C2×C414yes—
95C2×C4⋊C4×C59C2312yes—
96C2×C4⋊C4×C59C2312yes—
97C24×C59C2×D411yescharacteristic
98C23×D59—22no—
99C23×D59—22no—
100C23×D59—22no—
101C23×D59—22no—
102C23×D59—24no—
103C23×D59—22no—
104C23×D59—42no—
105C23×D59—42no—
106C23×D59—42no—
107C23×D59—42no—
108C23×D59—42no—
109C23×D59—42no—
110C23×D59C22⋊C414yes—
111C23×D59D814yes—
112C23×D59SD1614yes—
113C23×D59C2×D412yes—
114C22×C4×C59—24no—
115C22×C4×C59—22no—
116C22×C4×C59—22no—
117C22×C4×C59—42no—
118C22×C4×C59—42no—
119C22×C4×C59—42no—
120C22×C4×C59—42no—
121C22×C4×C59—42no—
122C22×C4×C59C22×C412yes—
123C22×C4×C59C2×D411yescharacteristic
124C22×C4×C59C2×D412yes—
125C22×C4×C59C2×D412yes—
126C22×C4×C59C2×D412yes—
127C22×C4×C59#4112yes—
128C22×C4×C59#4112yes—
129C22×C4×C59C2411yescharacteristic
130C22×Dic59—22no—
131C22×Dic59—22no—
132C22×Dic59—22no—
133C22×Dic59—42no—
134C22×Dic59—42no—
135C22×Dic59—42no—
136C22×Dic59C22⋊C414yes—
137C22×Dic59C2×D412yes—
138C2×D236—28no—
139C2×D236—28no—
140C2×D236—22no—
141C2×D236—22no—
142C2×D236—22no—
143C2×D236—22no—
144C2×D236—216no—
145C2×D236—24no—
146C2×D236—22no—
147C2×D236—432no—
148C2×D236—48no—
149C2×D236—48no—
150C2×D236—44no—
151C2×D236—42no—
152C2×D236—42no—
153C2×D236—44no—
154C2×D236C22×C414yes—
155C2×C4×D59—28no—
156C2×C4×D59—28no—
157C2×C4×D59—28no—
158C2×C4×D59—24no—
159C2×C4×D59—24no—
160C2×C4×D59—28no—
161C2×C4×D59—22no—
162C2×C4×D59—22no—
163C2×C4×D59—22no—
164C2×C4×D59—22no—
165C2×C4×D59C22⋊C418yes—
166C2×C4×D59C22×C414yes—
167C2×C4×D59C2×D414yes—
168C2×C8×C59—28no—
169C2×C8×C59—22no—
170C2×C8×C59—22no—
171C2×C8×C59C2×D414yescomplemented
172C4⋊C4×C59—24no—
173C4⋊C4×C59—22no—
174C4⋊C4×C59—22no—
175C4⋊C4×C59—22no—
176C4⋊C4×C59—42no—
177C4⋊C4×C59C22×C414yes—
178(C2×D59)⋊C4—216no—
179(C2×D59)⋊C4—48no—
180C4⋊Dic59—22no—
181C4⋊Dic59—22no—
182C4⋊Dic59—42no—
183C2×C59⋊C8—28no—
184C2×C59⋊C8—22no—
185C2×C59⋊C8—22no—
186C2×C59⋊C8C2×D414yescomplemented
187C8×D59—232no—
188C23×C59C2×C22⋊C412yes—
189C23×C59#12112yes—
190C23×C59#12112yes—
191C23×C59C2×D812yes—
192C23×C59C2×SD1612yes—
193C23×C59#13512yes—
194C23×C59#13512yes—
195C23×C59C22×D411yescharacteristic
196C22×D59—22no—
197C22×D59—22no—
198C22×D59—22no—
199C22×D59—22no—
200C22×D59—28no—
201C22×D59—28no—
202C22×D59—28no—
203C22×D59—28no—
204C22×D59—28no—
205C22×D59—28no—
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—
220C22×D59#102116yescomplemented
221C22×D59C2×C22⋊C414yes—
222C22×D59C2×D814yes—
223C22×D59C2×SD1614yes—
224C2×C4×C59—24no—
225C2×C4×C59—42no—
226C2×C4×C59—42no—
227C2×C4×C59—48no—
228C2×C4×C59—42no—
229C2×C4×C59—42no—
230C2×C4×C59—42no—
231C2×C4×C59—84no—
232C2×C4×C59C2×C22⋊C412yes—
233C2×C4×C59C4×D414yes—
234C2×C4×C59C4×D414yes—
235C2×C4×C59#12014yes—
236C2×C4×C59#12112yes—
237C2×C4×C59#12112yes—
238C2×C4×C59#12314yes—
239C2×C4×C59C23×C411yescharacteristic
240C2×C4×C59C22×D411yescharacteristic
241C2×C4×C59C22×D411yescharacteristic
242C2×C4×C59C22×D411yescharacteristic
243C2×C4×C59C2×#4111yescharacteristic
244C2×C4×C59C2×#4111yescharacteristic
245C2×Dic59—22no—
246C2×Dic59—22no—
247C2×Dic59—28no—
248C2×Dic59—42no—
249C2×Dic59—42no—
250C2×Dic59—42no—
251C2×Dic59—84no—
252C2×Dic59C2×C22⋊C414yes—
253D236—24no—
254D236—24no—
255D236—432no—
256D236—42no—
257D236—42no—
258D236—48no—
259C4×D59—24no—
260C4×D59—24no—
261C4×D59—24no—
262C4×D59—24no—
263C4×D59—216no—
264C8×C59—24no—
265C59⋊C8—24no—
266C2×#1893—591nomaximal, Sylow
267C22×C59#38614yes—
268C22×C59C2×#10214yes—
269C22×C59#41614yes—
270C22×C59#46514yes—
271C22×C59#46514yes—
272C22×C59#46614yes—
273C22×C59#46614yes—
274C22×C59C22×C22⋊C411yescharacteristic
275C22×C59C2×#12111yescharacteristic
276C22×C59C2×#12111yescharacteristic
277C22×C59C22×D811yescharacteristic
278C22×C59C22×SD1611yescharacteristic
279C22×C59C2×#13511yescharacteristic
280C22×C59C2×#13511yescharacteristic
281C2×D59—24no—
282C2×D59—232no—
283C2×D59—24no—
284C2×D59—24no—
285C2×D59—24no—
286C2×D59—24no—
287C2×D59—24no—
288C2×D59—44no—
289C2×D59—44no—
290C2×D59—44no—
291C2×D59—816no—
292C2×D59—82no—
293C2×D59—82no—
294C2×D59—82no—
295C2×D59—82no—
296C2×D59C2×#10218yescomplemented
297C4×C59—44no—
298C4×C59—82no—
299C4×C59#38614yes—
300C4×C59C2×C4×D411yescharacteristic
301C4×C59C2×C4×D411yescharacteristic
302C4×C59C2×#12011yescharacteristic
303C4×C59C2×#12311yescharacteristic
304Dic59—24no—
305Dic59—82no—
306C24×C8—591no—
307C22×#121—591no—
308C22×#102—591no—
309C22×#102—591no—
310C2×#380—591no—
311#1893—598no—
312C2×#340—591no—
313C2×#340D5911yescharacteristic, complemented
314C2×C59#189318yescomplemented
315C2×C59C2×#38611yescharacteristic
316C2×C59C22×#10211yescharacteristic
317C2×C59C2×#41611yescharacteristic
318C2×C59C2×#46511yescharacteristic
319C2×C59C2×#46511yescharacteristic
320C2×C59C2×#46611yescharacteristic
321C2×C59C2×#46611yescharacteristic
322D59—28no—
323D59—84no—
324C23×D4—591no—
325C23×D4—1181no—
326C24×C4—591no—
327C23×C8—591no—
328C23×C8—594no—
329C23×C8—1184no—
330C23×C8—1184no—
331C23×C8—1184no—
332C23×C8C2×D5911yescharacteristic
333C2×#121—598no—
334C2×#121—594no—
335C2×#121—1184no—
336C2×#121—1182no—
337C22×C4⋊C4—591no—
338C22×C4⋊C4C2×D5911yescharacteristic
339C22×C4⋊C4C2×D5911yescharacteristic
340C22×C22⋊C4—1181no—
341C22×C22⋊C4—1181no—
342C2×#102—594no—
343C2×#102—594no—
344C2×#102—1184no—
345C2×#102—1184no—
346#380—594no—
347#380—1182no—
348C2×#95—1181no—
349#340—594no—
350#340C2×D5914yescomplemented
351C59C2×#189311yescharacteristic, complemented, Sylow
352C25—591no—
353C25—1181no—
354C22×D4—594no—
355C22×D4—592no—
356C22×D4—1188no—
357C22×D4—1184no—
358C22×D4—1182no—
359C22×D4—1182no—
360C22×D4—1182no—
361C22×D4—1181no—
362C22×D4—1182no—
363C22×D4—1182no—
364C22×D4—2364no—
365C22×D4—2364no—
366C23×C4—21no—
367C23×C4—21no—
368C23×C4—591no—
369C23×C4—594no—
370C23×C4—594no—
371C23×C4—592no—
372C23×C4—592no—
373C23×C4—1182no—
374C23×C4—1182no—
375C23×C4—1182no—
376C23×C4—1182no—
377C23×C4—1181no—
378C23×C4C22×D5911yescharacteristic
379C22×C8—22no—
380C22×C8—22no—
381C22×C8—594no—
382C22×C8—592no—
383C22×C8—1188no—
384C22×C8—1188no—
385C22×C8—11832no—
386C22×C8—1182no—
387C22×C8—1182no—
388C22×C8—1188no—
389C22×C8D23614yescomplemented
390C22×C8C22×D5912yes—
391#121—5916no—
392#121—11816no—
393#121—1188no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—22no—
396C2×C4⋊C4—22no—
397C2×C4⋊C4—22no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—592no—
400C2×C4⋊C4—1182no—
401C2×C4⋊C4—1182no—
402C2×C4⋊C4—1181no—
403C2×C4⋊C4C4×D5914yes—
404C2×C4⋊C4C22×D5912yes—
405C2×C4⋊C4C22×D5912yes—
406C2×C22⋊C4—1188no—
407C2×C22⋊C4—1184no—
408C2×C22⋊C4—1184no—
409C2×C22⋊C4—2364no—
410#102—1188no—
411#102—1188no—
412#95—1184no—
413C24—594no—
414C24—594no—
415C24—594no—
416C24—592no—
417C24—1182no—
418C24—1182no—
419C24—1182no—
420C24—1182no—
421C24—1184no—
422C24—1182no—
423C24—2362no—
424C24—2362no—
425C24—2362no—
426C24—2362no—
427C24—2362no—
428C24—2362no—
429C24D4×D5911yescharacteristic
430C2×D4—594no—
431C2×D4—11816no—
432C2×D4—1184no—
433C2×D4—1188no—
434C2×D4—1188no—
435C2×D4—1182no—
436C2×D4—1182no—
437C2×D4—1182no—
438C2×D4—1182no—
439C2×D4—1182no—
440C2×D4—23632no—
441C2×D4—2368no—
442C2×D4—2368no—
443C2×D4—2364no—
444C2×D4—2362no—
445C2×D4—2362no—
446C2×D4—2364no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—594no—
456C22×C4—592no—
457C22×C4—598no—
458C22×C4—594no—
459C22×C4—594no—
460C22×C4—1182no—
461C22×C4—1182no—
462C22×C4—1188no—
463C22×C4—1188no—
464C22×C4—1188no—
465C22×C4—1184no—
466C22×C4—1184no—
467C22×C4—1188no—
468C22×C4—1182no—
469C22×C4—1182no—
470C22×C4—1182no—
471C22×C4—1182no—
472C22×C4—1182no—
473C22×C4—2362no—
474C22×C4—2362no—
475C22×C4—2362no—
476C22×C4C2×C4×D5912yes—
477C22×C4C2×D23612yes—
478C22×C4#1174568512yes—
479C22×C4D4×D5911yescharacteristic
480C22×C4D4×D5912yes—
481C22×C4D4×D5912yes—
482C22×C4#1174568912yes—
483C22×C4C23×D5911yescharacteristic
484C2×C8—28no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—594no—
488C2×C8—11832no—
489C2×C8—1188no—
490C2×C8—1182no—
491C2×C8—1182no—
492C2×C8C2×D23614yescomplemented
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—42no—
498C4⋊C4—1182no—
499C4⋊C4—1182no—
500C4⋊C4—2362no—
501C4⋊C4C2×C4×D5914yes—
502C22⋊C4—11816no—
503C22⋊C4—2368no—
504C23—5916no—
505C23—594no—
506C23—594no—
507C23—594no—
508C23—1188no—
509C23—1188no—
510C23—1188no—
511C23—1182no—
512C23—1188no—
513C23—1182no—
514C23—1188no—
515C23—1182no—
516C23—1182no—
517C23—1188no—
518C23—1184no—
519C23—4724no—
520C23—4724no—
521C23—4724no—
522C23—4724no—
523C23—2362no—
524C23—2362no—
525C23—2368no—
526C23—2368no—
527C23—2362no—
528C23—2362no—
529C23—2362no—
530C23—2362no—
531C23—2368no—
532C23C22⋊C4×D5912yes—
533C23#1427370012yes—
534C23#1427370212yes—
535C23D8×D5912yes—
536C23SD16×D5912yes—
537C23#1427374812yes—
538C23#1427375012yes—
539C23C2×D4×D5911yescharacteristic
540D4—1184no—
541D4—1184no—
542D4—23632no—
543D4—2362no—
544D4—2362no—
545D4—2368no—
546C2×C4—24no—
547C2×C4—48no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—84no—
554C2×C4—594no—
555C2×C4—1182no—
556C2×C4—1182no—
557C2×C4—1188no—
558C2×C4—1184no—
559C2×C4—1184no—
560C2×C4—11816no—
561C2×C4—1184no—
562C2×C4—1184no—
563C2×C4—4724no—
564C2×C4—2362no—
565C2×C4—2362no—
566C2×C4—2362no—
567C2×C4C22⋊C4×D5912yes—
568C2×C4#1427368914yes—
569C2×C4C4×D23614yes—
570C2×C4#1427369714yes—
571C2×C4#1427370012yes—
572C2×C4#1427370212yes—
573C2×C4#1427371214yes—
574C2×C4C22×C4×D5911yescharacteristic
575C2×C4C2×D4×D5911yescharacteristic
576C2×C4C2×D4×D5911yescharacteristic
577C2×C4C22×D23611yescharacteristic
578C2×C4C2×#1174568511yescharacteristic
579C2×C4C2×#1174568911yescharacteristic
580C8—24no—
581C8—1184no—
582C22—598no—
583C22—1184no—
584C22—1184no—
585C22—1184no—
586C22—1184no—
587C22—1184no—
588C22—1184no—
589C22—11832no—
590C22—4722no—
591C22—4722no—
592C22—4722no—
593C22—4722no—
594C22—47216no—
595C22—2364no—
596C22—2364no—
597C22—2364no—
598C22#1683728014yes—
599C22#102×D5914yes—
600C22#1683739714yes—
601C22#1683763214yes—
602C22#1683763414yes—
603C22#1683763714yes—
604C22#1683763914yes—
605C22C2×C22⋊C4×D5911yescharacteristic
606C22C2×#1427370011yescharacteristic
607C22C2×#1427370211yescharacteristic
608C22C2×D8×D5911yescharacteristic
609C22C2×SD16×D5911yescharacteristic
610C22C2×#1427374811yescharacteristic
611C22C2×#1427375011yescharacteristic
612C4—44no—
613C4—82no—
614C4—1184no—
615C4—4722no—
616C4#1683728014yes—
617C4C2×#1427368911yescharacteristic
618C4C2×C4×D23611yescharacteristic
619C4C2×#1427369711yescharacteristic
620C4C2×#1427371211yescharacteristic
621C2—1188no—
622C2—4724no—
623C2#2142452318yesdirect product factor
624C2C2×#1683728011yescharacteristic
625C2C2×#102×D5911yescharacteristic
626C2C2×#1683739711yescharacteristic
627C2C2×#1683763211yescharacteristic
628C2C2×#1683763411yescharacteristic
629C2C2×#1683763711yescharacteristic
630C2C2×#1683763911yescharacteristic
subgroupG/H
centerC24D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC2×C59×#340C2
socleC24×C59C2×D4
prime radicalC22×#14273702C2
cosocleC22×C4×C59C24
hypercenterC2×#340D59
hypocenterC59C2×#1893
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C24 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
1. C23×C8×D59C2semidirect
3. C22×#14273649C2semidirect
4. C22×#14273647C2semidirect
5. C2×#16837259C2semidirect
6. #21424523C2direct (N × H)
7. C2×#16837164C2semidirect
8. C2×C59×#340C2semidirect
16. C22×C8×D59C22semidirect
19. C2×#14273702C4semidirect
25. C2×#14273649C22semidirect
27. C2×#14273647C22semidirect
29. #16837259C22semidirect
31. #16837164C22semidirect
35. C59×#340C22semidirect
62. C22×C59⋊C8D4semidirect
70. #14273702C2×C4semidirect
87. C22×C8×C59D4semidirect
171. C2×C8×C59C2×D4semidirect
186. C2×C59⋊C8C2×D4semidirect
220. C22×D59#102semidirect
296. C2×D59C2×#102semidirect
313. C2×#340D59semidirect
314. C2×C59#1893semidirect
350. #340C2×D59semidirect
351. C59C2×#1893semidirect
389. C22×C8D236semidirect
492. C2×C8C2×D236semidirect
623. C2#21424523direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C8×D592
C22×#142737022
C22×#142736492
C22×#142736472
C2×#168372592
#214245232
C2×#168371642
C2×C59×#3402
C2×#189359