Finite Group Databasebeta

← C2×#21424327 #28589549

Subgroups — C2×#21424327 #28589549

Order 15104.

1478 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837885C211yesmaximal, characteristic
2C2×#16837442C211yesmaximal, characteristic, complemented
3C22×#14273654C211yesmaximal, characteristic
4C2×#16837267C211yesmaximal, characteristic, complemented
5#21424327C218yesmaximal, direct product factor
6C2×#16837170C211yesmaximal, characteristic, complemented
7C2×#16837163C211yesmaximal, characteristic, complemented
8C2×C59×#417C211yesmaximal, characteristic
9C22×#11745664C2211yescharacteristic
10#16837885—22no—
11#16837885—22no—
12#16837885—22no—
13#16837885C2214yes—
14C2×Q8×Dic59—21no—
15C2×D4×Dic59—22no—
16C2×D4×Dic59—21no—
17C2×#14273686—21no—
18C2×#14273686C2211yescharacteristic
19C22×C4⋊Dic59—21no—
20C22×C4⋊Dic59C2211yescharacteristic
21C22×C4×Dic59C2211yescharacteristic
22#16837442—22no—
23#16837442C2214yescomplemented
24C2×#14273660—21no—
25C2×#14273658—21no—
26C2×#14273654C418yescomplemented
27C2×#14273654C2214yes—
28#16837267—22no—
29#16837267C2214yescomplemented
30C2×C59⋊#95—22no—
31#16837170C2214yescomplemented
32#16837163C2214yescomplemented
33C22×#41×C59C2211yescharacteristic
34C22×M16×C59C2211yescharacteristic
35C22×C4⋊C4×C59C2211yescharacteristic
36C59×#417—22no—
37C59×#417C2214yes—
38C2×#103×C59—21no—
39C2×#102×C59—21no—
40C23×Dic59—21no—
41C23×Dic59—22no—
42C23×Dic59D412yes—
43C2×#11745664—21no—
44C2×#11745664—22no—
45C2×#11745664C2×C414yescomplemented
46C2×#11745664C2312yes—
47C22×C59⋊C8—21no—
48Q8×Dic59—24no—
49Q8×Dic59—24no—
50D4×Dic59—28no—
51D4×Dic59—24no—
52D4×Dic59—24no—
53D4×Dic59—28no—
54#14273686—24no—
55#14273686—21no—
56#14273686—21no—
57#14273686—24no—
58#14273686D412yes—
59#14273686C2312yes—
60C2×C4⋊Dic59—22no—
61C2×C4⋊Dic59—24no—
62C2×C4⋊Dic59—21no—
63C2×C4⋊Dic59—21no—
64C2×C4⋊Dic59—21no—
65C2×C4⋊Dic59—21no—
66C2×C4⋊Dic59—24no—
67C2×C4⋊Dic59D412yes—
68C2×C4⋊Dic59D411yescharacteristic
69C2×C4⋊Dic59D411yescharacteristic
70C2×C4⋊Dic59C2312yes—
71C2×C22⋊Dic59—22no—
72C2×C22⋊Dic59—21no—
73C2×C4×Dic59—22no—
74C2×C4×Dic59—21no—
75C2×C4×Dic59—21no—
76C2×C4×Dic59—22no—
77C2×C4×Dic59C2×C414yescomplemented
78C2×C4×Dic59C2312yes—
79#14273660—24no—
80#14273658—24no—
81#14273654C2×C4116yescomplemented
82C59⋊#95—28no—
83C2×#41×C59—21no—
84C2×#41×C59—21no—
85C2×#41×C59—22no—
86C2×#41×C59—22no—
87C2×#41×C59C2×C414yes—
88C2×#41×C59D412yes—
89C2×#41×C59Q814yes—
90C2×#41×C59C2312yes—
91C22×Q8×C59D411yescharacteristic
92C22×D4×C59—21no—
93C22×D4×C59D411yescharacteristic
94C23×C4×C59—21no—
95C23×C4×C59—21no—
96C23×C4×C59C2311yescharacteristic
97C2×M16×C59—21no—
98C2×M16×C59—22no—
99C2×M16×C59—22no—
100C2×M16×C59C2312yes—
101C22×C8×C59—21no—
102C2×C4⋊C4×C59—21no—
103C2×C4⋊C4×C59—21no—
104C2×C4⋊C4×C59—21no—
105C2×C4⋊C4×C59—21no—
106C2×C4⋊C4×C59—21no—
107C2×C4⋊C4×C59C2312yes—
108#103×C59—24no—
109#102×C59—24no—
110C24×C59—21no—
111C24×C59C2×D411yescharacteristic
112#41×C59—24no—
113#41×C59—24no—
114#41×C59—24no—
115#41×C59—24no—
116#41×C59—24no—
117#41×C59—22no—
118#41×C59—22no—
119#41×C59C4⋊C418yes—
120#41×C59C22×C414yes—
121#41×C59C2×Q814yes—
122C2×Q8×C59—21no—
123C2×Q8×C59—22no—
124C2×Q8×C59C4⋊C414yes—
125C2×Q8×C59C2×D412yes—
126C2×D4×C59—24no—
127C2×D4×C59—22no—
128C2×D4×C59—24no—
129C2×D4×C59—22no—
130C2×D4×C59—21no—
131C2×D4×C59—22no—
132C2×D4×C59C4⋊C414yes—
133C2×D4×C59C2×D412yes—
134C22×C4×C59—21no—
135C22×C4×C59—24no—
136C22×C4×C59—22no—
137C22×C4×C59—24no—
138C22×C4×C59—22no—
139C22×C4×C59—22no—
140C22×C4×C59—44no—
141C22×C4×C59—41no—
142C22×C4×C59—41no—
143C22×C4×C59C22×C412yes—
144C22×C4×C59C2×D411yescharacteristic
145C22×C4×C59C2×D411yescharacteristic
146C22×C4×C59C2×D411yescharacteristic
147C22×C4×C59C2×D411yescharacteristic
148C22×C4×C59C2×D411yescharacteristic
149C22×C4×C59C2×Q812yes—
150C22×C4×C59#4112yes—
151C22×C4×C59#4112yes—
152C22×C4×C59C2411yescharacteristic
153C22×Dic59—24no—
154C22×Dic59—21no—
155C22×Dic59—22no—
156C22×Dic59—21no—
157C22×Dic59—22no—
158C22×Dic59—28no—
159C22×Dic59—22no—
160C22×Dic59—42no—
161C22×Dic59—42no—
162C22×Dic59—44no—
163C22×Dic59—84no—
164C22×Dic59C2×D414yes—
165M16×C59—22no—
166M16×C59—22no—
167M16×C59—42no—
168C2×C8×C59—22no—
169C2×C8×C59—42no—
170C4⋊C4×C59—22no—
171C4⋊C4×C59—22no—
172C4⋊C4×C59—41no—
173C4⋊C4×C59—41no—
174C22⋊Dic59—28no—
175C22⋊Dic59—22no—
176C22⋊Dic59—22no—
177C4⋊Dic59—28no—
178C4⋊Dic59—22no—
179C4⋊Dic59—21no—
180C4⋊Dic59—22no—
181C4⋊Dic59—21no—
182C4⋊Dic59—22no—
183C4⋊Dic59—22no—
184C4⋊Dic59C2×D412yes—
185C4⋊Dic59C2×D412yes—
186C4×Dic59—28no—
187C4×Dic59—24no—
188C4×Dic59—22no—
189C4×Dic59—22no—
190C4×Dic59—22no—
191C4×Dic59C22×C414yescomplemented
192#11745664—24no—
193#11745664—22no—
194#11745664C22×C414yescomplemented
195C2×C59⋊C8—24no—
196C2×C59⋊C8—22no—
197C23×C59—22no—
198C23×C59—22no—
199C23×C59—22no—
200C23×C59—24no—
201C23×C59—42no—
202C23×C59—42no—
203C23×C59C4×D412yes—
204C23×C59#12011yescharacteristic
205C23×C59#12011yescharacteristic
206C23×C59C22⋊Q812yes—
207C23×C59C22×D411yescharacteristic
208Q8×C59—22no—
209Q8×C59—24no—
210Q8×C59C2×C4⋊C414yes—
211D4×C59—24no—
212D4×C59—24no—
213D4×C59—28no—
214D4×C59—22no—
215D4×C59—24no—
216D4×C59C2×C4⋊C414yes—
217C2×C4×C59—22no—
218C2×C4×C59—24no—
219C2×C4×C59—24no—
220C2×C4×C59—24no—
221C2×C4×C59—28no—
222C2×C4×C59—44no—
223C2×C4×C59—42no—
224C2×C4×C59—82no—
225C2×C4×C59—82no—
226C2×C4×C59C2×C4⋊C412yes—
227C2×C4×C59C2×C4⋊C412yes—
228C2×C4×C59C4×D412yes—
229C2×C4×C59C4×D412yes—
230C2×C4×C59C4×D412yes—
231C2×C4×C59#12011yescharacteristic
232C2×C4×C59#12011yescharacteristic
233C2×C4×C59C22⋊Q812yes—
234C2×C4×C59C22⋊Q812yes—
235C2×C4×C59C22⋊Q812yes—
236C2×C4×C59#12312yes—
237C2×C4×C59#12312yes—
238C2×C4×C59C23×C411yescharacteristic
239C2×C4×C59C22×D411yescharacteristic
240C2×C4×C59C22×D411yescharacteristic
241C2×C4×C59C22×Q811yescharacteristic
242C2×C4×C59C2×#4111yescharacteristic
243C2×C4×C59C2×#4111yescharacteristic
244C2×Dic59—24no—
245C2×Dic59—22no—
246C2×Dic59—24no—
247C2×Dic59—22no—
248C2×Dic59—48no—
249C2×Dic59—42no—
250C2×Dic59—42no—
251C2×Dic59—42no—
252C2×Dic59—44no—
253C2×Dic59—42no—
254C2×Dic59—44no—
255C2×Dic59—82no—
256C2×Dic59—88no—
257C8×C59—42no—
258C59⋊C8—24no—
259C2×#1860—591nomaximal, Sylow
260C22×C59—24no—
261C22×C59—24no—
262C22×C59—21no—
263C22×C59—21no—
264C22×C59—21no—
265C22×C59—21no—
266C22×C59—24no—
267C22×C59—48no—
268C22×C59—42no—
269C22×C59—42no—
270C22×C59#38512yes—
271C22×C59#38512yes—
272C22×C59#45412yes—
273C22×C59#45512yes—
274C22×C59C2×C4×D411yescharacteristic
275C22×C59C2×#12011yescharacteristic
276C22×C59C2×C22⋊Q811yescharacteristic
277C4×C59—24no—
278C4×C59—82no—
279C4×C59#38512yes—
280C4×C59#38512yes—
281C4×C59C22×C4⋊C411yescharacteristic
282C4×C59C2×C4×D411yescharacteristic
283C4×C59C2×C22⋊Q811yescharacteristic
284C4×C59C2×#12311yescharacteristic
285Dic59—44no—
286Dic59—44no—
287Dic59—42no—
288Dic59—42no—
289C2×#520—591no—
290C2×#428—591no—
291C22×#104—591no—
292C2×#417D5911yescharacteristic
293C2×#382—591no—
294#1860—598no—
295C2×#343—591no—
296C2×#339—591no—
297C2×C59—21no—
298C2×C59—21no—
299C2×C59—22no—
300C2×C59—44no—
301C2×C59#186014yescomplemented
302C2×C59C2×#38511yescharacteristic
303C2×C59C2×#45511yescharacteristic
304C2×C59C2×#45411yescharacteristic
305C22×#41C2×D5911yescharacteristic
306C22×M16—591no—
307C22×M16C2×D5911yescharacteristic
308#520—594no—
309#520—1182no—
310#520—1182no—
311#520—1182no—
312C2×C4×Q8—1181no—
313C2×C4×D4—1182no—
314C2×C4×D4—1181no—
315C2×#117—591no—
316C2×#117—1181no—
317C22×C4⋊C4—591no—
318C22×C4⋊C4—1181no—
319C22×C4⋊C4C2×D5911yescharacteristic
320C22×C42—591no—
321#428—594no—
322#428—1182no—
323C2×#107—1181no—
324C2×#106—1181no—
325C2×#104—598no—
326C2×#104—594no—
327#417—22no—
328#417C2×D5914yes—
329C2×#103—21no—
330C2×#102—21no—
331#382—594no—
332#382—1182no—
333C2×#95—1182no—
334#343—594no—
335#339—594no—
336C59C2×#186011yescharacteristic, complemented, Sylow
337C2×#41—21no—
338C2×#41—21no—
339C2×#41—22no—
340C2×#41—22no—
341C2×#41Dic11814yes—
342C2×#41C4×D5914yes—
343C2×#41C22⋊D5912yes—
344C2×#41C22×D5912yes—
345C22×Q8C22⋊D5911yescharacteristic
346C22×D4—21no—
347C22×D4C22⋊D5911yescharacteristic
348C23×C4—21no—
349C23×C4—21no—
350C23×C4—592no—
351C23×C4—1182no—
352C23×C4—1181no—
353C23×C4C22×D5911yescharacteristic
354C2×M16—21no—
355C2×M16—22no—
356C2×M16—22no—
357C2×M16—594no—
358C2×M16—592no—
359C2×M16—1181no—
360C2×M16—1182no—
361C2×M16C22×D5912yes—
362C22×C8—21no—
363C22×C8—1181no—
364C4×Q8—1184no—
365C4×Q8—1184no—
366C4×D4—1188no—
367C4×D4—1188no—
368C4×D4—1184no—
369C4×D4—1184no—
370#117—592no—
371#117—592no—
372#117—1184no—
373#117—1184no—
374#117—1181no—
375#117—1181no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—591no—
382C2×C4⋊C4—591no—
383C2×C4⋊C4—592no—
384C2×C4⋊C4—592no—
385C2×C4⋊C4—1184no—
386C2×C4⋊C4—1182no—
387C2×C4⋊C4—1184no—
388C2×C4⋊C4—1181no—
389C2×C4⋊C4—1181no—
390C2×C4⋊C4—1181no—
391C2×C4⋊C4—1181no—
392C2×C4⋊C4C22×D5912yes—
393C2×C22⋊C4—1182no—
394C2×C22⋊C4—1181no—
395C2×C42—594no—
396C2×C42—592no—
397C2×C42—1182no—
398C2×C42—1181no—
399C2×C42—1182no—
400C2×C42—1181no—
401#107—1184no—
402#106—1184no—
403#104—5916no—
404#103—24no—
405#102—24no—
406#95—1188no—
407C24—21no—
408C24D4×D5911yescharacteristic
409#41—24no—
410#41—24no—
411#41—24no—
412#41—24no—
413#41—24no—
414#41—22no—
415#41—22no—
416#41Dic59⋊C418yes—
417#41C2×Dic11814yes—
418#41C2×C4×D5914yes—
419C2×Q8—21no—
420C2×Q8—22no—
421C2×Q8Dic59⋊C414yes—
422C2×Q8C2×C22⋊D5912yes—
423C2×D4—24no—
424C2×D4—22no—
425C2×D4—24no—
426C2×D4—22no—
427C2×D4—21no—
428C2×D4—22no—
429C2×D4Dic59⋊C414yes—
430C2×D4C2×C22⋊D5912yes—
431C22×C4—21no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—44no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—594no—
441C22×C4—1188no—
442C22×C4—1182no—
443C22×C4—1184no—
444C22×C4—1181no—
445C22×C4—1181no—
446C22×C4—1182no—
447C22×C4—1182no—
448C22×C4—4724no—
449C22×C4—2364no—
450C22×C4—2362no—
451C22×C4—2362no—
452C22×C4C2×Dic11812yes—
453C22×C4C2×C4×D5912yes—
454C22×C4D4×D5911yescharacteristic
455C22×C4D4×D5911yescharacteristic
456C22×C4D4×D5911yescharacteristic
457C22×C4#1174568712yes—
458C22×C4#1174568712yes—
459C22×C4C2×C22⋊D5911yescharacteristic
460C22×C4C2×C22⋊D5911yescharacteristic
461C22×C4C23×D5911yescharacteristic
462M16—22no—
463M16—22no—
464M16—42no—
465M16—594no—
466M16—1184no—
467M16—1182no—
468C2×C8—22no—
469C2×C8—42no—
470C2×C8—1184no—
471C2×C8—1182no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4—592no—
477C4⋊C4—592no—
478C4⋊C4—1188no—
479C4⋊C4—1181no—
480C4⋊C4—1182no—
481C4⋊C4—1182no—
482C4⋊C4—1181no—
483C4⋊C4—1182no—
484C4⋊C4—1182no—
485C22⋊C4—1188no—
486C22⋊C4—1182no—
487C22⋊C4—1182no—
488C42—594no—
489C42—1188no—
490C42—1184no—
491C42—1182no—
492C42—1182no—
493C42—1182no—
494C23—22no—
495C23—22no—
496C23—24no—
497C23—22no—
498C23—42no—
499C23—42no—
500C23(C22⋊D59)⋊C412yes—
501C23#1427369511yescharacteristic
502C23#1427369511yescharacteristic
503C23#1427370412yes—
504C23C2×D4×D5911yescharacteristic
505Q8—22no—
506Q8—24no—
507Q8C2×Dic59⋊C414yes—
508D4—24no—
509D4—28no—
510D4—24no—
511D4—22no—
512D4—24no—
513D4C2×Dic59⋊C414yes—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—28no—
518C2×C4—24no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—1184no—
524C2×C4—1182no—
525C2×C4—1182no—
526C2×C4—1184no—
527C2×C4—4722no—
528C2×C4—4728no—
529C2×C4—2362no—
530C2×C4—2362no—
531C2×C4—2368no—
532C2×C4—2362no—
533C2×C4—2362no—
534C2×C4—2364no—
535C2×C4—2364no—
536C2×C4C2×Dic59⋊C412yes—
537C2×C4C2×Dic59⋊C412yes—
538C2×C4(C22⋊D59)⋊C412yes—
539C2×C4(C22⋊D59)⋊C412yes—
540C2×C4(C22⋊D59)⋊C412yes—
541C2×C4#1427369511yescharacteristic
542C2×C4#1427369511yescharacteristic
543C2×C4#1427370412yes—
544C2×C4#1427370412yes—
545C2×C4#1427370412yes—
546C2×C4#1427370812yes—
547C2×C4#1427370812yes—
548C2×C4C22×C4×D5911yescharacteristic
549C2×C4C2×D4×D5911yescharacteristic
550C2×C4C22×C22⋊D5911yescharacteristic
551C2×C4C22×Dic11811yescharacteristic
552C2×C4C2×#1174568711yescharacteristic
553C2×C4C2×#1174568711yescharacteristic
554C8—42no—
555C8—1184no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—24no—
561C22—24no—
562C22—24no—
563C22—42no—
564C22—42no—
565C22—48no—
566C22#1683727412yes—
567C22#1683727412yes—
568C22#1683757312yes—
569C22#1683757812yes—
570C22C2×(C22⋊D59)⋊C411yescharacteristic
571C22C2×#1427369511yescharacteristic
572C22C2×#1427370411yescharacteristic
573C4—24no—
574C4—82no—
575C4—2364no—
576C4—2362no—
577C4—2364no—
578C4—2362no—
579C4#1683727412yes—
580C4#1683727412yes—
581C4C22×Dic59⋊C411yescharacteristic
582C4C2×(C22⋊D59)⋊C411yescharacteristic
583C4C2×#1427370411yescharacteristic
584C4C2×#1427370811yescharacteristic
585C2—21no—
586C2—21no—
587C2—22no—
588C2—44no—
589C2#2142432714yesdirect product factor
590C2C2×#1683727411yescharacteristic
591C2C2×#1683757811yescharacteristic
592C2C2×#1683757311yescharacteristic
subgroupG/H
centerC23#14273695
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC2×C59×#417C2
socleC23×C59#120
prime radicalC22×#41×C59C22
cosocleC22×C4×C59C24
hypercenterC2×#417D59
hypocenterC59C2×#1860
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
2. C2×#16837442C2semidirect
4. C2×#16837267C2semidirect
5. #21424327C2direct (N × H)
6. C2×#16837170C2semidirect
7. C2×#16837163C2semidirect
23. #16837442C22semidirect
26. C2×#14273654C4semidirect
29. #16837267C22semidirect
31. #16837170C22semidirect
32. #16837163C22semidirect
45. C2×#11745664C2×C4semidirect
77. C2×C4×Dic59C2×C4semidirect
81. #14273654C2×C4semidirect
191. C4×Dic59C22×C4semidirect
194. #11745664C22×C4semidirect
301. C2×C59#1860semidirect
336. C59C2×#1860semidirect
589. C2#21424327direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168378852
C2×#168374422
C22×#142736542
C2×#168372672
#214243272
C2×#168371702
C2×#168371632
C2×C59×#4172
C2×#186059