Finite Group Databasebeta

← C2×#21424329 #28589551

Subgroups — C2×#21424329 #28589551

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×#16837445C211yesmaximal, characteristic, complemented
2C22×#14273654C211yesmaximal, characteristic
3C2×#16837403C211yesmaximal, characteristic
4C2×#16837267C211yesmaximal, characteristic, complemented
5#21424329C218yesmaximal, direct product factor
6C2×#16837171C211yesmaximal, characteristic, complemented
7C2×#16837161C211yesmaximal, characteristic, complemented
8C2×C59×#520C211yesmaximal, characteristic
9C22×#11745664C2211yescharacteristic
10C22×#11745664C2211yescharacteristic
11C22×C4⋊Dic59C2211yescharacteristic
12C22×C4×Dic59C2211yescharacteristic
13#16837445—22no—
14#16837445C2214yescomplemented
15C2×#14273660—21no—
16C2×#14273658—21no—
17C2×#14273654C418yescomplemented
18C2×#14273654C2214yes—
19#16837403—22no—
20#16837403C2214yes—
21C2×#14273651—21no—
22C2×#14273648—21no—
23#16837267—22no—
24#16837267C2214yescomplemented
25C2×C59⋊#95—22no—
26#16837171C2214yescomplemented
27#16837161C2214yescomplemented
28C22×#41×C59C2211yescharacteristic
29C59×#520—22no—
30C59×#520—22no—
31C59×#520—22no—
32C59×#520C2214yes—
33C2×C4×Q8×C59—21no—
34C2×C4×D4×C59—21no—
35C2×C4×D4×C59—22no—
36C2×#117×C59—21no—
37C2×#117×C59C2211yescharacteristic
38C22×C4⋊C4×C59—21no—
39C22×C4⋊C4×C59C2211yescharacteristic
40C23×Dic59—21no—
41C23×Dic59—21no—
42C2×#11745664—21no—
43C2×#11745664—22no—
44C2×#11745664—21no—
45C2×#11745664—22no—
46C2×#11745664—22no—
47C2×#11745664C2×C414yescomplemented
48C2×#11745664C2312yes—
49C2×#11745664C2312yes—
50C22×C59⋊C8—21no—
51C22×C59⋊C8—21no—
52C2×C4⋊Dic59—21no—
53C2×C4⋊Dic59—21no—
54C2×C4⋊Dic59—21no—
55C2×C4⋊Dic59—21no—
56C2×C4⋊Dic59—21no—
57C2×C4⋊Dic59C2312yes—
58C2×C4×Dic59—21no—
59C2×C4×Dic59—22no—
60C2×C4×Dic59C2×C414yescomplemented
61C2×C4×Dic59C2312yes—
62#14273660—24no—
63#14273658—24no—
64#14273654C2×C4116yescomplemented
65#14273651—24no—
66#14273648—24no—
67C59⋊#95—28no—
68C2×#41×C59—21no—
69C2×#41×C59—21no—
70C2×#41×C59—22no—
71C2×#41×C59—22no—
72C2×#41×C59C2×C414yes—
73C2×#41×C59D412yes—
74C2×#41×C59Q814yes—
75C2×#41×C59C2312yes—
76C22×Q8×C59D411yescharacteristic
77C22×D4×C59—21no—
78C22×D4×C59D411yescharacteristic
79C23×C4×C59—21no—
80C23×C4×C59—22no—
81C23×C4×C59D412yes—
82C23×C4×C59C2311yescharacteristic
83C4×Q8×C59—24no—
84C4×Q8×C59—24no—
85C4×D4×C59—28no—
86C4×D4×C59—24no—
87C4×D4×C59—24no—
88C4×D4×C59—28no—
89#117×C59—24no—
90#117×C59—21no—
91#117×C59—21no—
92#117×C59—24no—
93#117×C59D412yes—
94#117×C59C2312yes—
95C2×C4⋊C4×C59—22no—
96C2×C4⋊C4×C59—24no—
97C2×C4⋊C4×C59—21no—
98C2×C4⋊C4×C59—21no—
99C2×C4⋊C4×C59—21no—
100C2×C4⋊C4×C59—21no—
101C2×C4⋊C4×C59—24no—
102C2×C4⋊C4×C59D412yes—
103C2×C4⋊C4×C59D411yescharacteristic
104C2×C4⋊C4×C59D411yescharacteristic
105C2×C4⋊C4×C59C2312yes—
106C2×C22⋊C4×C59—22no—
107C2×C22⋊C4×C59—21no—
108C2×C42×C59—22no—
109C2×C42×C59—21no—
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—21no—
140C22×C4×C59—22no—
141C22×C4×C59—21no—
142C22×C4×C59—28no—
143C22×C4×C59—22no—
144C22×C4×C59—44no—
145C22×C4×C59—84no—
146C22×C4×C59C22×C412yes—
147C22×C4×C59C2×D414yes—
148C22×C4×C59C2×D411yescharacteristic
149C22×C4×C59C2×D411yescharacteristic
150C22×C4×C59C2×D411yescharacteristic
151C22×C4×C59C2×D411yescharacteristic
152C22×C4×C59C2×D411yescharacteristic
153C22×C4×C59C2×Q812yes—
154C22×C4×C59#4112yes—
155C22×C4×C59#4112yes—
156C22×C4×C59C2411yescharacteristic
157C22×Dic59—24no—
158C22×Dic59—22no—
159C22×Dic59—22no—
160C22×Dic59—42no—
161C22×Dic59—42no—
162C22×Dic59—44no—
163C22×Dic59—41no—
164C22×Dic59—41no—
165C4⋊C4×C59—28no—
166C4⋊C4×C59—22no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—22no—
169C4⋊C4×C59—21no—
170C4⋊C4×C59—22no—
171C4⋊C4×C59—22no—
172C4⋊C4×C59C2×D412yes—
173C4⋊C4×C59C2×D412yes—
174C22⋊C4×C59—28no—
175C22⋊C4×C59—22no—
176C22⋊C4×C59—22no—
177C42×C59—28no—
178C42×C59—22no—
179C42×C59—22no—
180C4⋊Dic59—22no—
181C4⋊Dic59—22no—
182C4⋊Dic59—41no—
183C4⋊Dic59—41no—
184C4×Dic59—24no—
185C4×Dic59—22no—
186C4×Dic59C22×C414yescomplemented
187#11745664—24no—
188#11745664—22no—
189#11745664—22no—
190#11745664—22no—
191#11745664—42no—
192#11745664C22×C414yescomplemented
193C2×C59⋊C8—24no—
194C2×C59⋊C8—22no—
195C2×C59⋊C8—22no—
196C2×C59⋊C8—42no—
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—22no—
223C2×C4×C59—24no—
224C2×C4×C59—22no—
225C2×C4×C59—42no—
226C2×C4×C59—44no—
227C2×C4×C59—42no—
228C2×C4×C59—44no—
229C2×C4×C59—82no—
230C2×C4×C59—88no—
231C2×C4×C59C2×C4⋊C412yes—
232C2×C4×C59C2×C4⋊C412yes—
233C2×C4×C59C4×D412yes—
234C2×C4×C59C4×D412yes—
235C2×C4×C59C4×D412yes—
236C2×C4×C59#12011yescharacteristic
237C2×C4×C59#12011yescharacteristic
238C2×C4×C59C22⋊Q812yes—
239C2×C4×C59C22⋊Q812yes—
240C2×C4×C59C22⋊Q812yes—
241C2×C4×C59#12312yes—
242C2×C4×C59#12312yes—
243C2×C4×C59C23×C411yescharacteristic
244C2×C4×C59C22×D411yescharacteristic
245C2×C4×C59C22×D411yescharacteristic
246C2×C4×C59C22×Q811yescharacteristic
247C2×C4×C59C2×#4111yescharacteristic
248C2×C4×C59C2×#4111yescharacteristic
249C2×Dic59—24no—
250C2×Dic59—48no—
251C2×Dic59—42no—
252C2×Dic59—42no—
253C2×Dic59—44no—
254C2×Dic59—42no—
255C2×Dic59—82no—
256C2×Dic59—82no—
257C59⋊C8—24no—
258C59⋊C8—42no—
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—44no—
279C4×C59—42no—
280C4×C59—42no—
281C4×C59#38512yes—
282C4×C59#38512yes—
283C4×C59C22×C4⋊C411yescharacteristic
284C4×C59C2×C4×D411yescharacteristic
285C4×C59C2×C22⋊Q811yescharacteristic
286C4×C59C2×#12311yescharacteristic
287Dic59—44no—
288Dic59—82no—
289C2×#520D5911yescharacteristic
290C2×#428—591no—
291C22×#104—591no—
292C2×#417—591no—
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×M16—591no—
308#520—22no—
309#520—22no—
310#520—22no—
311#520C2×D5914yes—
312C2×C4×Q8—21no—
313C2×C4×D4—21no—
314C2×C4×D4—22no—
315C2×#117—21no—
316C2×#117C2×D5911yescharacteristic
317C22×C4⋊C4—21no—
318C22×C4⋊C4—591no—
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—594no—
328#417—1182no—
329C2×#103—1181no—
330C2×#102—1181no—
331#382—594no—
332#382—1182no—
333C2×#95—1182no—
334#343—594no—
335#339—594no—
336C59C2×#186011yescharacteristic, complemented, Sylow
337C2×#41—22no—
338C2×#41—22no—
339C2×#41—21no—
340C2×#41—21no—
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—22no—
350C23×C4—1181no—
351C23×C4—1181no—
352C23×C4C22⋊D5912yes—
353C23×C4C22×D5911yescharacteristic
354C2×M16—594no—
355C2×M16—592no—
356C2×M16—592no—
357C2×M16—1182no—
358C2×M16—1181no—
359C2×M16—1181no—
360C2×M16—1182no—
361C2×M16—1182no—
362C22×C8—1181no—
363C22×C8—1181no—
364C4×Q8—24no—
365C4×Q8—24no—
366C4×D4—28no—
367C4×D4—24no—
368C4×D4—24no—
369C4×D4—28no—
370#117—24no—
371#117—21no—
372#117—21no—
373#117—24no—
374#117C22⋊D5912yes—
375#117C22×D5912yes—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—24no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—24no—
383C2×C4⋊C4—592no—
384C2×C4⋊C4—1181no—
385C2×C4⋊C4—1181no—
386C2×C4⋊C4—1181no—
387C2×C4⋊C4—1181no—
388C2×C4⋊C4—1181no—
389C2×C4⋊C4C22⋊D5912yes—
390C2×C4⋊C4C22⋊D5911yescharacteristic
391C2×C4⋊C4C22⋊D5911yescharacteristic
392C2×C4⋊C4C22×D5912yes—
393C2×C22⋊C4—22no—
394C2×C22⋊C4—21no—
395C2×C42—22no—
396C2×C42—21no—
397C2×C42—594no—
398C2×C42—592no—
399C2×C42—1182no—
400C2×C42—1181no—
401#107—1184no—
402#106—1184no—
403#104—5916no—
404#103—1184no—
405#102—1184no—
406#95—1188no—
407C24—21no—
408C24C2×C22⋊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—21no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—28no—
440C22×C4—22no—
441C22×C4—44no—
442C22×C4—84no—
443C22×C4—1184no—
444C22×C4—1182no—
445C22×C4—1182no—
446C22×C4—2364no—
447C22×C4—2361no—
448C22×C4—2361no—
449C22×C4—2362no—
450C22×C4—2362no—
451C22×C4C2×Dic11812yes—
452C22×C4C2×C4×D5912yes—
453C22×C4#1174568512yes—
454C22×C4#1174568512yes—
455C22×C4C2×C22⋊D5914yes—
456C22×C4C2×C22⋊D5911yescharacteristic
457C22×C4C2×C22⋊D5911yescharacteristic
458C22×C4C2×C22⋊D5911yescharacteristic
459C22×C4C2×C22⋊D5911yescharacteristic
460C22×C4C2×C22⋊D5911yescharacteristic
461C22×C4C23×D5911yescharacteristic
462M16—594no—
463M16—1184no—
464M16—1182no—
465M16—1182no—
466M16—1182no—
467M16—2362no—
468C2×C8—1184no—
469C2×C8—1182no—
470C2×C8—1182no—
471C2×C8—2362no—
472C4⋊C4—28no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—1182no—
480C4⋊C4—1182no—
481C4⋊C4—2361no—
482C4⋊C4—2361no—
483C4⋊C4C2×C22⋊D5912yes—
484C4⋊C4C2×C22⋊D5912yes—
485C22⋊C4—28no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C42—28no—
489C42—22no—
490C42—22no—
491C42—594no—
492C42—1184no—
493C42—1182no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—24no—
498C23—42no—
499C23—42no—
500C23C4×C22⋊D5912yes—
501C23#1427369611yescharacteristic
502C23#1427369611yescharacteristic
503C23#1427370612yes—
504C23C22×C22⋊D5911yescharacteristic
505Q8—22no—
506Q8—24no—
507Q8C2×Dic59⋊C414yes—
508D4—24no—
509D4—28no—
510D4—24no—
511D4—22no—
512D4—24no—
513D4C2×Dic59⋊C414yes—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—28no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—82no—
527C2×C4—88no—
528C2×C4—1184no—
529C2×C4—4722no—
530C2×C4—4722no—
531C2×C4—2364no—
532C2×C4—2362no—
533C2×C4—2362no—
534C2×C4—2362no—
535C2×C4—2368no—
536C2×C4C2×Dic59⋊C412yes—
537C2×C4C2×Dic59⋊C412yes—
538C2×C4C4×C22⋊D5912yes—
539C2×C4C4×C22⋊D5912yes—
540C2×C4C4×C22⋊D5912yes—
541C2×C4#1427369611yescharacteristic
542C2×C4#1427369611yescharacteristic
543C2×C4#1427370612yes—
544C2×C4#1427370612yes—
545C2×C4#1427370612yes—
546C2×C4#1427371012yes—
547C2×C4#1427371012yes—
548C2×C4C22×C4×D5911yescharacteristic
549C2×C4C22×C22⋊D5911yescharacteristic
550C2×C4C22×C22⋊D5911yescharacteristic
551C2×C4C22×Dic11811yescharacteristic
552C2×C4C2×#1174568511yescharacteristic
553C2×C4C2×#1174568511yescharacteristic
554C8—1184no—
555C8—2362no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—24no—
561C22—24no—
562C22—24no—
563C22—42no—
564C22—42no—
565C22—48no—
566C22#1683727512yes—
567C22#1683727512yes—
568C22#1683757512yes—
569C22#1683758012yes—
570C22C2×C4×C22⋊D5911yescharacteristic
571C22C2×#1427369611yescharacteristic
572C22C2×#1427370611yescharacteristic
573C4—24no—
574C4—44no—
575C4—42no—
576C4—42no—
577C4—4722no—
578C4—2364no—
579C4#1683727512yes—
580C4#1683727512yes—
581C4C22×Dic59⋊C411yescharacteristic
582C4C2×C4×C22⋊D5911yescharacteristic
583C4C2×#1427370611yescharacteristic
584C4C2×#1427371011yescharacteristic
585C2—21no—
586C2—21no—
587C2—22no—
588C2—44no—
589C2#2142432914yesdirect product factor
590C2C2×#1683727511yescharacteristic
591C2C2×#1683758011yescharacteristic
592C2C2×#1683757511yescharacteristic
subgroupG/H
centerC23#14273696
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC2×C59×#520C2
socleC23×C59#120
prime radicalC22×#41×C59C22
cosocleC22×C4×C59C24
hypercenterC2×#520D59
hypocenterC59C2×#1860
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#520

Complements

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

NH ≅ G/Ntype
1. C2×#16837445C2semidirect
4. C2×#16837267C2semidirect
5. #21424329C2direct (N × H)
6. C2×#16837171C2semidirect
7. C2×#16837161C2semidirect
14. #16837445C22semidirect
17. C2×#14273654C4semidirect
24. #16837267C22semidirect
26. #16837171C22semidirect
27. #16837161C22semidirect
47. C2×#11745664C2×C4semidirect
60. C2×C4×Dic59C2×C4semidirect
64. #14273654C2×C4semidirect
186. C4×Dic59C22×C4semidirect
192. #11745664C22×C4semidirect
301. C2×C59#1860semidirect
336. C59C2×#1860semidirect
589. C2#21424329direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168374452
C22×#142736542
C2×#168374032
C2×#168372672
#214243292
C2×#168371712
C2×#168371612
C2×C59×#5202
C2×#186059