Finite Group Databasebeta

← C2×#21424325 #28589546

Subgroups — C2×#21424325 #28589546

Order 15104.

1566 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837883C211yesmaximal, characteristic
2C2×#16837444C211yesmaximal, characteristic, complemented
3C22×#14273653C211yesmaximal, characteristic
4C2×#16837404C211yesmaximal, characteristic
5C2×#16837271C211yesmaximal, characteristic, complemented
6#21424325C218yesmaximal, direct product factor
7C2×#16837163C211yesmaximal, characteristic, complemented
8C2×C59×#343C211yesmaximal, characteristic, complemented
9C22×#11745685C2211yescharacteristic
10#16837883—22no—
11#16837883—22no—
12#16837883—22no—
13#16837883C2214yes—
14C2×#14273692—21no—
15C2×#14273689—21no—
16C2×C4×C22⋊D59—22no—
17C2×#14273686C2211yescharacteristic
18C2×#14273685—21no—
19C22×C4⋊Dic59C2211yescharacteristic
20C22×Dic59⋊C4—22no—
21C2×C4⋊C4×D59—21no—
22C22×C4×Dic59C2211yescharacteristic
23#16837444—22no—
24#16837444C2214yescomplemented
25C2×#14273661—21no—
26C2×#14273659—21no—
27C2×#14273653C418yescomplemented
28C2×#14273653C2214yes—
29#16837404—22no—
30#16837404C2214yes—
31C2×#14273652—21no—
32C2×#14273649—21no—
33#16837271—22no—
34#16837271C2214yescomplemented
35#16837163C2214yescomplemented
36C22×M16×C59C2211yescharacteristic
37C22×M16×C59C2211yescharacteristic
38C22×C4⋊C4×C59C2211yescharacteristic
39C59×#343C2214yescomplemented
40C2×#11745685—21no—
41C2×#11745685—21no—
42C2×#11745685—22no—
43C2×#11745685—22no—
44C2×#11745685C2×C414yes—
45C2×#11745685D412yes—
46C2×#11745685Q814yes—
47C2×#11745685C2312yes—
48C22×Dic118D411yescharacteristic
49C22×D236D411yescharacteristic
50C22×C22⋊D59—21no—
51C23×Dic59—21no—
52C23×Dic59—21no—
53C22×C4×D59—21no—
54C22×C4×D59—22no—
55#14273692—24no—
56#14273692—24no—
57#14273689—24no—
58#14273689—24no—
59C4×C22⋊D59—28no—
60C4×C22⋊D59—28no—
61#14273686—21no—
62#14273686—21no—
63#14273686D412yes—
64#14273686C2312yes—
65#14273685—24no—
66#14273685—24no—
67C2×C4⋊Dic59—21no—
68C2×C4⋊Dic59—21no—
69C2×C4⋊Dic59—21no—
70C2×C4⋊Dic59—21no—
71C2×C4⋊Dic59—21no—
72C2×C4⋊Dic59D411yescharacteristic
73C2×C4⋊Dic59D411yescharacteristic
74C2×C4⋊Dic59C2312yes—
75C2×Dic59⋊C4—22no—
76C2×Dic59⋊C4—28no—
77C2×Dic59⋊C4—48no—
78C4⋊C4×D59—24no—
79C4⋊C4×D59—24no—
80C2×C22⋊Dic59—21no—
81C2×(C2×D59)⋊C4—22no—
82C2×C4×Dic59—22no—
83C2×C4×Dic59—21no—
84C2×C4×Dic59—21no—
85C2×C4×Dic59—22no—
86C2×C4×Dic59C2×C414yescomplemented
87C2×C4×Dic59C2312yes—
88#14273661—24no—
89#14273659—24no—
90#14273653C2×C4116yescomplemented
91#14273652—24no—
92#14273649—24no—
93C23×C4×C59D412yes—
94C23×C4×C59C2311yescharacteristic
95C2×M16×C59—21no—
96C2×M16×C59—22no—
97C2×M16×C59—21no—
98C2×M16×C59—22no—
99C2×M16×C59—22no—
100C2×M16×C59C2×C414yescomplemented
101C2×M16×C59C2312yes—
102C2×M16×C59C2312yes—
103C22×C8×C59—21no—
104C22×C8×C59—21no—
105C2×C4⋊C4×C59—21no—
106C2×C4⋊C4×C59—21no—
107C2×C4⋊C4×C59—21no—
108C2×C4⋊C4×C59—21no—
109C2×C4⋊C4×C59D412yes—
110C2×C4⋊C4×C59C2312yes—
111C24×C59C2×D411yescharacteristic
112C23×D59—21no—
113C22×C4×C59—22no—
114C22×C4×C59—22no—
115C22×C4×C59—44no—
116C22×C4×C59C22×C412yes—
117C22×C4×C59C2×D414yes—
118C22×C4×C59C2×D411yescharacteristic
119C22×C4×C59C2×D411yescharacteristic
120C22×C4×C59C2×D411yescharacteristic
121C22×C4×C59C2×D411yescharacteristic
122C22×C4×C59C2×D411yescharacteristic
123C22×C4×C59C2×Q812yes—
124C22×C4×C59#4112yes—
125C22×C4×C59#4112yes—
126C22×C4×C59C2411yescharacteristic
127C2×C22⋊D59—24no—
128C2×C22⋊D59—22no—
129C2×C22⋊D59—24no—
130C2×C22⋊D59—22no—
131C22×Dic59—24no—
132C22×Dic59—24no—
133C22×Dic59—24no—
134C22×Dic59—21no—
135C22×Dic59—22no—
136C22×Dic59—21no—
137C22×Dic59—21no—
138C22×Dic59—22no—
139C22×Dic59—44no—
140C22×Dic59—41no—
141C22×Dic59—41no—
142#11745685—24no—
143#11745685—24no—
144#11745685—24no—
145#11745685—24no—
146#11745685—24no—
147#11745685—22no—
148#11745685—22no—
149#11745685C4⋊C418yescomplemented
150#11745685C22×C414yes—
151#11745685C2×Q814yes—
152C2×D236—21no—
153C2×D236—22no—
154C2×D236C4⋊C414yescomplemented
155C2×D236C2×D412yes—
156C2×C4×D59—24no—
157C2×C4×D59—22no—
158C2×C4×D59—24no—
159C2×C4×D59—22no—
160C2×C4×D59—28no—
161C2×C4×D59—84no—
162C2×Dic118—21no—
163C2×Dic118—22no—
164C2×Dic118C4⋊C414yescomplemented
165C2×Dic118C2×D412yes—
166M16×C59—24no—
167M16×C59—22no—
168M16×C59—22no—
169M16×C59—22no—
170M16×C59—42no—
171M16×C59C22×C414yescomplemented
172C2×C8×C59—24no—
173C2×C8×C59—22no—
174C2×C8×C59—22no—
175C2×C8×C59—42no—
176C4⋊C4×C59—22no—
177C4⋊C4×C59—22no—
178C4⋊C4×C59—22no—
179C4⋊C4×C59—22no—
180C22⋊Dic59—22no—
181C22⋊Dic59—22no—
182(C2×D59)⋊C4—28no—
183C4⋊Dic59—21no—
184C4⋊Dic59—21no—
185C4⋊Dic59—22no—
186C4⋊Dic59—22no—
187C4⋊Dic59—41no—
188C4⋊Dic59—41no—
189C4⋊Dic59C2×D412yes—
190C4⋊Dic59C2×D412yes—
191Dic59⋊C4—28no—
192Dic59⋊C4—416no—
193C4×Dic59—28no—
194C4×Dic59—24no—
195C4×Dic59—22no—
196C4×Dic59—22no—
197C4×Dic59—22no—
198C4×Dic59C22×C414yescomplemented
199C23×C59—22no—
200C23×C59—22no—
201C23×C59C4×D412yes—
202C23×C59#12011yescharacteristic
203C23×C59#12011yescharacteristic
204C23×C59C22⋊Q812yes—
205C23×C59C22×D411yescharacteristic
206C22×D59—22no—
207C22×D59—24no—
208C22×D59—42no—
209C22×D59—42no—
210C2×C4×C59—24no—
211C2×C4×C59—44no—
212C2×C4×C59—44no—
213C2×C4×C59—82no—
214C2×C4×C59C2×C4⋊C412yes—
215C2×C4×C59C2×C4⋊C412yes—
216C2×C4×C59C4×D412yes—
217C2×C4×C59C4×D412yes—
218C2×C4×C59C4×D412yes—
219C2×C4×C59#12011yescharacteristic
220C2×C4×C59#12011yescharacteristic
221C2×C4×C59C22⋊Q812yes—
222C2×C4×C59C22⋊Q812yes—
223C2×C4×C59C22⋊Q812yes—
224C2×C4×C59#12312yes—
225C2×C4×C59#12312yes—
226C2×C4×C59C23×C411yescharacteristic
227C2×C4×C59C22×D411yescharacteristic
228C2×C4×C59C22×D411yescharacteristic
229C2×C4×C59C22×Q811yescharacteristic
230C2×C4×C59C2×#4111yescharacteristic
231C2×C4×C59C2×#4111yescharacteristic
232C22⋊D59—24no—
233C22⋊D59—24no—
234C22⋊D59—28no—
235C2×Dic59—24no—
236C2×Dic59—216no—
237C2×Dic59—24no—
238C2×Dic59—24no—
239C2×Dic59—22no—
240C2×Dic59—24no—
241C2×Dic59—22no—
242C2×Dic59—22no—
243C2×Dic59—42no—
244C2×Dic59—42no—
245C2×Dic59—44no—
246C2×Dic59—42no—
247C2×Dic59—82no—
248C2×Dic59—82no—
249D236—22no—
250D236—24no—
251D236C2×C4⋊C414yescomplemented
252C4×D59—24no—
253C4×D59—24no—
254C4×D59—28no—
255C4×D59—88no—
256Dic118—22no—
257Dic118—24no—
258Dic118C2×C4⋊C414yescomplemented
259C8×C59—24no—
260C8×C59—42no—
261C2×#1859—591nomaximal, Sylow
262C22×C59—24no—
263C22×C59—24no—
264C22×C59—21no—
265C22×C59—21no—
266C22×C59—21no—
267C22×C59—21no—
268C22×C59#38512yes—
269C22×C59#38512yes—
270C22×C59#45312yes—
271C22×C59#45612yes—
272C22×C59C2×C4×D411yescharacteristic
273C22×C59C2×#12011yescharacteristic
274C22×C59C2×C22⋊Q811yescharacteristic
275C2×D59—24no—
276C2×D59—48no—
277C2×D59—42no—
278C2×D59—42no—
279C4×C59—44no—
280C4×C59#38512yes—
281C4×C59#38512yes—
282C4×C59C22×C4⋊C411yescharacteristic
283C4×C59C2×C4×D411yescharacteristic
284C4×C59C2×C22⋊Q811yescharacteristic
285C4×C59C2×#12311yescharacteristic
286Dic59—28no—
287Dic59—24no—
288Dic59—42no—
289Dic59—42no—
290Dic59—82no—
291C2×#520—591no—
292C2×#428—591no—
293C22×#104—591no—
294C2×#417—591no—
295C2×#384—591no—
296#1859—598no—
297C2×#343D5911yescharacteristic, complemented
298C2×#339—591no—
299C2×C59—21no—
300C2×C59—21no—
301C2×C59—22no—
302C2×C59#185914yescomplemented
303C2×C59C2×#38511yescharacteristic
304C2×C59C2×#45311yescharacteristic
305C2×C59C2×#45611yescharacteristic
306D59—44no—
307C22×#41—591no—
308C22×M16C2×D5911yescharacteristic
309C22×M16C2×D5911yescharacteristic
310#520—594no—
311#520—1182no—
312#520—1182no—
313#520—1182no—
314C2×C4×Q8—1181no—
315C2×C4×D4—1181no—
316C2×C4×D4—1182no—
317C2×#117—591no—
318C2×#117—1181no—
319C22×C4⋊C4—591no—
320C22×C4⋊C4—1181no—
321C22×C4⋊C4—1182no—
322C22×C4⋊C4C2×D5911yescharacteristic
323C22×C42—591no—
324#428—594no—
325#428—1182no—
326C2×#107—1181no—
327C2×#106—1181no—
328C2×#104—598no—
329C2×#104—594no—
330#417—594no—
331#417—1182no—
332C2×#103—1181no—
333C2×#102—1181no—
334#384—594no—
335#384—1182no—
336#343C2×D5914yescomplemented
337#339—594no—
338C59C2×#185911yescharacteristic, complemented, Sylow
339C2×#41—594no—
340C2×#41—592no—
341C2×#41—594no—
342C2×#41—592no—
343C2×#41—1181no—
344C2×#41—1182no—
345C2×#41—1181no—
346C2×#41—1182no—
347C22×Q8—591no—
348C22×D4—591no—
349C22×D4—1181no—
350C23×C4—1182no—
351C23×C4—1181no—
352C23×C4—1181no—
353C23×C4—1181no—
354C23×C4D23612yes—
355C23×C4C22×D5911yescharacteristic
356C2×M16—21no—
357C2×M16—22no—
358C2×M16—21no—
359C2×M16—22no—
360C2×M16—22no—
361C2×M16C4×D5914yescomplemented
362C2×M16C22×D5912yes—
363C2×M16C22×D5912yes—
364C22×C8—21no—
365C22×C8—21no—
366C4×Q8—1184no—
367C4×Q8—1184no—
368C4×D4—1184no—
369C4×D4—1184no—
370C4×D4—1188no—
371C4×D4—1188no—
372#117—592no—
373#117—592no—
374#117—1184no—
375#117—1184no—
376#117—1181no—
377#117—1181no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—591no—
383C2×C4⋊C4—591no—
384C2×C4⋊C4—592no—
385C2×C4⋊C4—1184no—
386C2×C4⋊C4—1184no—
387C2×C4⋊C4—1182no—
388C2×C4⋊C4—1181no—
389C2×C4⋊C4—1181no—
390C2×C4⋊C4—1181no—
391C2×C4⋊C4—1181no—
392C2×C4⋊C4—1181no—
393C2×C4⋊C4—1188no—
394C2×C4⋊C4—2368no—
395C2×C4⋊C4D23612yes—
396C2×C4⋊C4C22×D5912yes—
397C2×C22⋊C4—1182no—
398C2×C22⋊C4—1181no—
399C2×C42—594no—
400C2×C42—592no—
401C2×C42—1182no—
402C2×C42—1181no—
403C2×C42—1182no—
404C2×C42—1181no—
405#107—1184no—
406#106—1184no—
407#104—5916no—
408#103—1184no—
409#102—1184no—
410C24—1181no—
411C24C2×D23611yescharacteristic
412#41—598no—
413#41—594no—
414#41—594no—
415#41—1184no—
416#41—1184no—
417#41—1184no—
418#41—1184no—
419#41—1184no—
420#41—1182no—
421#41—1182no—
422C2×Q8—594no—
423C2×Q8—592no—
424C2×Q8—1181no—
425C2×Q8—1182no—
426C2×D4—594no—
427C2×D4—592no—
428C2×D4—1184no—
429C2×D4—1182no—
430C2×D4—1184no—
431C2×D4—1182no—
432C2×D4—1181no—
433C2×D4—1182no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—44no—
437C22×C4—1188no—
438C22×C4—1184no—
439C22×C4—1184no—
440C22×C4—1184no—
441C22×C4—1181no—
442C22×C4—1182no—
443C22×C4—1184no—
444C22×C4—1182no—
445C22×C4—1184no—
446C22×C4—1182no—
447C22×C4—1181no—
448C22×C4—1181no—
449C22×C4—1182no—
450C22×C4—4724no—
451C22×C4—2364no—
452C22×C4—2361no—
453C22×C4—2361no—
454C22×C4C2×C4×D5912yes—
455C22×C4C2×D23614yes—
456C22×C4C2×D23611yescharacteristic
457C22×C4#1174568512yes—
458C22×C4D4×D5911yescharacteristic
459C22×C4D4×D5911yescharacteristic
460C22×C4D4×D5911yescharacteristic
461C22×C4D4×D5911yescharacteristic
462C22×C4#1174568712yes—
463C22×C4Q8×D5912yes—
464C22×C4C23×D5911yescharacteristic
465M16—24no—
466M16—22no—
467M16—22no—
468M16—22no—
469M16—42no—
470M16C2×C4×D5914yescomplemented
471C2×C8—24no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—42no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—592no—
480C4⋊C4—592no—
481C4⋊C4—1188no—
482C4⋊C4—1181no—
483C4⋊C4—1181no—
484C4⋊C4—1182no—
485C4⋊C4—1182no—
486C4⋊C4—23616no—
487C4⋊C4—2361no—
488C4⋊C4—2361no—
489C22⋊C4—1188no—
490C22⋊C4—1182no—
491C22⋊C4—1182no—
492C42—594no—
493C42—1188no—
494C42—1184no—
495C42—1182no—
496C42—1182no—
497C42—1182no—
498C23—22no—
499C23—22no—
500C23—1182no—
501C23—1184no—
502C23—2362no—
503C23—2362no—
504C23C4×D23612yes—
505C23#1427369711yescharacteristic
506C23#1427369711yescharacteristic
507C23#1427370712yes—
508C23C22×D23611yescharacteristic
509Q8—594no—
510Q8—1182no—
511Q8—1184no—
512D4—594no—
513D4—1184no—
514D4—1188no—
515D4—1184no—
516D4—1182no—
517D4—1184no—
518C2×C4—24no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—82no—
522C2×C4—1184no—
523C2×C4—1184no—
524C2×C4—11816no—
525C2×C4—1182no—
526C2×C4—1184no—
527C2×C4—1184no—
528C2×C4—1184no—
529C2×C4—1188no—
530C2×C4—1184no—
531C2×C4—1182no—
532C2×C4—1182no—
533C2×C4—4728no—
534C2×C4—4722no—
535C2×C4—4722no—
536C2×C4—2364no—
537C2×C4—2362no—
538C2×C4—2362no—
539C2×C4—2362no—
540C2×C4C4⋊C4×D5912yes—
541C2×C4C4⋊C4×D5912yes—
542C2×C4(C22⋊D59)⋊C412yes—
543C2×C4(C22⋊D59)⋊C412yes—
544C2×C4C4×D23612yes—
545C2×C4#1427369711yescharacteristic
546C2×C4#1427369711yescharacteristic
547C2×C4#1427370512yes—
548C2×C4#1427370512yes—
549C2×C4#1427370712yes—
550C2×C4#1427370912yes—
551C2×C4#1427370912yes—
552C2×C4C22×C4×D5911yescharacteristic
553C2×C4C2×D4×D5911yescharacteristic
554C2×C4C2×D4×D5911yescharacteristic
555C2×C4C2×Q8×D5911yescharacteristic
556C2×C4C2×#1174568711yescharacteristic
557C2×C4C2×#1174568511yescharacteristic
558C8—24no—
559C8—42no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—24no—
565C22—24no—
566C22—1184no—
567C22—2362no—
568C22—2362no—
569C22—2368no—
570C22#1683727612yes—
571C22#1683727612yes—
572C22#1683757212yes—
573C22#1683758712yes—
574C22C2×C4×D23611yescharacteristic
575C22C2×#1427369711yescharacteristic
576C22C2×#1427370711yescharacteristic
577C4—44no—
578C4—1188no—
579C4—1184no—
580C4—4722no—
581C4—2362no—
582C4—2362no—
583C4#1683727612yes—
584C4#1683727612yes—
585C4C2×C4⋊C4×D5911yescharacteristic
586C4C2×(C22⋊D59)⋊C411yescharacteristic
587C4C2×#1427370511yescharacteristic
588C4C2×#1427370911yescharacteristic
589C2—21no—
590C2—21no—
591C2—22no—
592C2—2364no—
593C2#2142432514yesdirect product factor
594C2C2×#1683727611yescharacteristic
595C2C2×#1683757211yescharacteristic
596C2C2×#1683758711yescharacteristic
subgroupG/H
centerC23#14273697
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC2×C59×#343C2
socleC23×C59#120
prime radicalC22×#11745685C22
cosocleC22×C4×C59C24
hypercenterC2×#343D59
hypocenterC59C2×#1859
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#343

Complements

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

NH ≅ G/Ntype
2. C2×#16837444C2semidirect
5. C2×#16837271C2semidirect
6. #21424325C2direct (N × H)
7. C2×#16837163C2semidirect
8. C2×C59×#343C2semidirect
24. #16837444C22semidirect
27. C2×#14273653C4semidirect
34. #16837271C22semidirect
35. #16837163C22semidirect
39. C59×#343C22semidirect
86. C2×C4×Dic59C2×C4semidirect
90. #14273653C2×C4semidirect
100. C2×M16×C59C2×C4semidirect
149. #11745685C4⋊C4semidirect
154. C2×D236C4⋊C4semidirect
164. C2×Dic118C4⋊C4semidirect
171. M16×C59C22×C4semidirect
198. C4×Dic59C22×C4semidirect
251. D236C2×C4⋊C4semidirect
258. Dic118C2×C4⋊C4semidirect
297. C2×#343D59semidirect
302. C2×C59#1859semidirect
336. #343C2×D59semidirect
338. C59C2×#1859semidirect
361. C2×M16C4×D59semidirect
470. M16C2×C4×D59semidirect
593. C2#21424325direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168378832
C2×#168374442
C22×#142736532
C2×#168374042
C2×#168372712
#214243252
C2×#168371632
C2×C59×#3432
C2×#185959