Finite Group Databasebeta

← C2×#21424328 #28589550

Subgroups — C2×#21424328 #28589550

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×#16837883C211yesmaximal, characteristic
2C2×#16837442C211yesmaximal, characteristic, complemented
3C22×#14273655C211yesmaximal, characteristic
4C2×#16837401C211yesmaximal, characteristic
5#21424328C218yesmaximal, direct product factor
6C2×#16837171C211yesmaximal, characteristic, complemented
7C2×#16837162C211yesmaximal, characteristic, complemented
8C2×C59×#382C211yesmaximal, characteristic, complemented
9C22×#11745685C2211yescharacteristic
10C22×#11745664C2211yescharacteristic
11C22×#11745664C2211yescharacteristic
12#16837883—22no—
13#16837883—22no—
14#16837883—22no—
15#16837883C2214yes—
16C2×#14273692—21no—
17C2×#14273689—21no—
18C2×C4×C22⋊D59—22no—
19C2×#14273686C2211yescharacteristic
20C2×#14273685—21no—
21C2×C4⋊C4×D59—21no—
22#16837442—22no—
23#16837442C2214yescomplemented
24C2×#14273660—21no—
25C2×#14273658—21no—
26C2×#14273655C418yescomplemented
27C2×#14273655C2214yes—
28#16837401—22no—
29#16837401C2214yes—
30C2×#14273650—21no—
31C2×#14273647—21no—
32#16837171C2214yescomplemented
33#16837162C2214yescomplemented
34C22×C4⋊C4×C59C2211yescharacteristic
35C22×C4⋊C4×C59C2211yescharacteristic
36C22×C42×C59C2211yescharacteristic
37C59×#382—22no—
38C59×#382C2214yescomplemented
39C2×#95×C59—22no—
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—
51C22×C4×D59—21no—
52C22×C4×D59—22no—
53C2×#11745664—21no—
54C2×#11745664—22no—
55C2×#11745664—21no—
56C2×#11745664—22no—
57C2×#11745664—22no—
58C2×#11745664C2×C414yescomplemented
59C2×#11745664C2312yes—
60C2×#11745664C2312yes—
61C22×C59⋊C8—21no—
62C22×C59⋊C8—21no—
63#14273692—24no—
64#14273692—24no—
65#14273689—24no—
66#14273689—24no—
67C4×C22⋊D59—28no—
68C4×C22⋊D59—28no—
69#14273686—21no—
70#14273686—21no—
71#14273686D412yes—
72#14273686C2312yes—
73#14273685—24no—
74#14273685—24no—
75C2×C4⋊Dic59D411yescharacteristic
76C2×C4⋊Dic59D411yescharacteristic
77C2×Dic59⋊C4—22no—
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—
84#14273660—24no—
85#14273658—24no—
86#14273655C2×C4116yescomplemented
87#14273650—24no—
88#14273647—24no—
89C23×C4×C59—21no—
90C23×C4×C59—21no—
91C23×C4×C59D412yes—
92C23×C4×C59C2311yescharacteristic
93C2×C4⋊C4×C59—21no—
94C2×C4⋊C4×C59—21no—
95C2×C4⋊C4×C59—21no—
96C2×C4⋊C4×C59—21no—
97C2×C4⋊C4×C59—21no—
98C2×C4⋊C4×C59—21no—
99C2×C4⋊C4×C59—21no—
100C2×C4⋊C4×C59—21no—
101C2×C4⋊C4×C59—21no—
102C2×C4⋊C4×C59D412yes—
103C2×C4⋊C4×C59C2312yes—
104C2×C4⋊C4×C59C2312yes—
105C2×C42×C59—21no—
106C2×C42×C59—22no—
107C2×C42×C59C2×C414yescomplemented
108C2×C42×C59C2312yes—
109#95×C59—28no—
110C24×C59C2×D411yescharacteristic
111C23×D59—21no—
112C22×C4×C59—24no—
113C22×C4×C59—22no—
114C22×C4×C59—22no—
115C22×C4×C59—22no—
116C22×C4×C59—22no—
117C22×C4×C59—42no—
118C22×C4×C59—42no—
119C22×C4×C59—44no—
120C22×C4×C59—44no—
121C22×C4×C59—41no—
122C22×C4×C59—41no—
123C22×C4×C59C22×C412yes—
124C22×C4×C59C2×D414yes—
125C22×C4×C59C2×D411yescharacteristic
126C22×C4×C59C2×D411yescharacteristic
127C22×C4×C59C2×D411yescharacteristic
128C22×C4×C59C2×D411yescharacteristic
129C22×C4×C59C2×D411yescharacteristic
130C22×C4×C59C2×Q812yes—
131C22×C4×C59#4112yes—
132C22×C4×C59#4112yes—
133C22×C4×C59C2411yescharacteristic
134C2×C22⋊D59—24no—
135C2×C22⋊D59—22no—
136C2×C22⋊D59—24no—
137C2×C22⋊D59—22no—
138C22×Dic59—21no—
139C22×Dic59—21no—
140C22×Dic59—21no—
141#11745685—24no—
142#11745685—24no—
143#11745685—24no—
144#11745685—24no—
145#11745685—24no—
146#11745685—22no—
147#11745685—22no—
148#11745685C4⋊C418yes—
149#11745685C22×C414yes—
150#11745685C2×Q814yes—
151C2×D236—21no—
152C2×D236—22no—
153C2×D236C4⋊C414yes—
154C2×D236C2×D412yes—
155C2×C4×D59—24no—
156C2×C4×D59—22no—
157C2×C4×D59—24no—
158C2×C4×D59—22no—
159C2×C4×D59—28no—
160C2×C4×D59—84no—
161C2×Dic118—21no—
162C2×Dic118—22no—
163C2×Dic118C4⋊C414yes—
164C2×Dic118C2×D412yes—
165C4⋊C4×C59—22no—
166C4⋊C4×C59—22no—
167C4⋊C4×C59—22no—
168C4⋊C4×C59—22no—
169C4⋊C4×C59—22no—
170C4⋊C4×C59—22no—
171C4⋊C4×C59—41no—
172C4⋊C4×C59—41no—
173C42×C59—24no—
174C42×C59—22no—
175C42×C59C22×C414yescomplemented
176C22⋊Dic59—22no—
177C22⋊Dic59—22no—
178(C2×D59)⋊C4—28no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59C2×D412yes—
182C4⋊Dic59C2×D412yes—
183Dic59⋊C4—28no—
184C4×Dic59—28no—
185C4×Dic59—22no—
186C4×Dic59—22no—
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×C59C4×D412yes—
200C23×C59#12011yescharacteristic
201C23×C59#12011yescharacteristic
202C23×C59C22⋊Q812yes—
203C23×C59C22×D411yescharacteristic
204C22×D59—22no—
205C22×D59—24no—
206C22×D59—42no—
207C22×D59—42no—
208C2×C4×C59—24no—
209C2×C4×C59—24no—
210C2×C4×C59—48no—
211C2×C4×C59—42no—
212C2×C4×C59—42no—
213C2×C4×C59—44no—
214C2×C4×C59—44no—
215C2×C4×C59—44no—
216C2×C4×C59—42no—
217C2×C4×C59—82no—
218C2×C4×C59—82no—
219C2×C4×C59—82no—
220C2×C4×C59C2×C4⋊C412yes—
221C2×C4×C59C2×C4⋊C412yes—
222C2×C4×C59C4×D412yes—
223C2×C4×C59C4×D412yes—
224C2×C4×C59C4×D412yes—
225C2×C4×C59#12011yescharacteristic
226C2×C4×C59#12011yescharacteristic
227C2×C4×C59C22⋊Q812yes—
228C2×C4×C59C22⋊Q812yes—
229C2×C4×C59C22⋊Q812yes—
230C2×C4×C59#12312yes—
231C2×C4×C59#12312yes—
232C2×C4×C59C23×C411yescharacteristic
233C2×C4×C59C22×D411yescharacteristic
234C2×C4×C59C22×D411yescharacteristic
235C2×C4×C59C22×Q811yescharacteristic
236C2×C4×C59C2×#4111yescharacteristic
237C2×C4×C59C2×#4111yescharacteristic
238C22⋊D59—24no—
239C22⋊D59—24no—
240C22⋊D59—28no—
241C2×Dic59—22no—
242C2×Dic59—24no—
243C2×Dic59—22no—
244C2×Dic59—22no—
245C2×Dic59—42no—
246C2×Dic59—42no—
247D236—22no—
248D236—24no—
249D236C2×C4⋊C414yes—
250C4×D59—24no—
251C4×D59—24no—
252C4×D59—28no—
253C4×D59—88no—
254Dic118—22no—
255Dic118—24no—
256Dic118C2×C4⋊C414yes—
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#38512yes—
267C22×C59#38512yes—
268C22×C59#45412yes—
269C22×C59#45512yes—
270C22×C59C2×C4×D411yescharacteristic
271C22×C59C2×#12011yescharacteristic
272C22×C59C2×C22⋊Q811yescharacteristic
273C2×D59—24no—
274C2×D59—48no—
275C2×D59—42no—
276C2×D59—42no—
277C4×C59—44no—
278C4×C59—44no—
279C4×C59—82no—
280C4×C59#38512yes—
281C4×C59#38512yes—
282C4×C59C22×C4⋊C411yescharacteristic
283C4×C59C2×C4×D411yescharacteristic
284C4×C59C2×C22⋊Q811yescharacteristic
285C4×C59C2×#12311yescharacteristic
286Dic59—24no—
287Dic59—42no—
288Dic59—42no—
289C2×#520—591no—
290C2×#428—591no—
291C22×#104—591no—
292C2×#417—591no—
293C2×#382D5911yescharacteristic, complemented
294#1860—598no—
295C2×#343—591no—
296C2×#339—591no—
297C2×C59—21no—
298C2×C59—21no—
299C2×C59—22no—
300C2×C59#186014yescomplemented
301C2×C59C2×#38511yescharacteristic
302C2×C59C2×#45511yescharacteristic
303C2×C59C2×#45411yescharacteristic
304D59—44no—
305C22×#41—591no—
306C22×M16—591no—
307C22×M16—591no—
308#520—594no—
309#520—1182no—
310#520—1182no—
311#520—1182no—
312C2×C4×Q8—1181no—
313C2×C4×D4—1181no—
314C2×C4×D4—1182no—
315C2×#117—591no—
316C2×#117—1181no—
317C22×C4⋊C4—1181no—
318C22×C4⋊C4C2×D5911yescharacteristic
319C22×C4⋊C4C2×D5911yescharacteristic
320C22×C42C2×D5911yescharacteristic
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—22no—
332#382C2×D5914yescomplemented
333C2×#95—22no—
334#343—594no—
335#339—594no—
336C59C2×#186011yescharacteristic, complemented, Sylow
337C2×#41—594no—
338C2×#41—592no—
339C2×#41—594no—
340C2×#41—592no—
341C2×#41—1182no—
342C2×#41—1181no—
343C2×#41—1182no—
344C2×#41—1181no—
345C22×Q8—591no—
346C22×D4—591no—
347C22×D4—1181no—
348C23×C4—21no—
349C23×C4—21no—
350C23×C4—1182no—
351C23×C4—1181no—
352C23×C4C22⋊D5912yes—
353C23×C4C22×D5911yescharacteristic
354C2×M16—594no—
355C2×M16—592no—
356C2×M16—592no—
357C2×M16—1181no—
358C2×M16—1182no—
359C2×M16—1181no—
360C2×M16—1182no—
361C2×M16—1182no—
362C22×C8—1181no—
363C22×C8—1181no—
364C4×Q8—1184no—
365C4×Q8—1184no—
366C4×D4—1184no—
367C4×D4—1184no—
368C4×D4—1188no—
369C4×D4—1188no—
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—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—591no—
386C2×C4⋊C4—591no—
387C2×C4⋊C4—1184no—
388C2×C4⋊C4—1184no—
389C2×C4⋊C4—1182no—
390C2×C4⋊C4C22⋊D5912yes—
391C2×C4⋊C4C22×D5912yes—
392C2×C4⋊C4C22×D5912yes—
393C2×C22⋊C4—1182no—
394C2×C22⋊C4—1181no—
395C2×C42—22no—
396C2×C42—21no—
397C2×C42—1182no—
398C2×C42—1181no—
399C2×C42C4×D5914yescomplemented
400C2×C42C22×D5912yes—
401#107—1184no—
402#106—1184no—
403#104—5916no—
404#103—1184no—
405#102—1184no—
406#95—28no—
407C24—1181no—
408C24C2×C22⋊D5911yescharacteristic
409#41—598no—
410#41—594no—
411#41—594no—
412#41—1184no—
413#41—1184no—
414#41—1184no—
415#41—1184no—
416#41—1184no—
417#41—1182no—
418#41—1182no—
419C2×Q8—594no—
420C2×Q8—592no—
421C2×Q8—1181no—
422C2×Q8—1182no—
423C2×D4—594no—
424C2×D4—592no—
425C2×D4—1184no—
426C2×D4—1182no—
427C2×D4—1184no—
428C2×D4—1182no—
429C2×D4—1181no—
430C2×D4—1182no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—1188no—
443C22×C4—1181no—
444C22×C4—1184no—
445C22×C4—1182no—
446C22×C4—1184no—
447C22×C4—1182no—
448C22×C4—1181no—
449C22×C4—1181no—
450C22×C4—4724no—
451C22×C4C2×C4×D5912yes—
452C22×C4#1174568512yes—
453C22×C4D4×D5911yescharacteristic
454C22×C4D4×D5911yescharacteristic
455C22×C4D4×D5911yescharacteristic
456C22×C4D4×D5911yescharacteristic
457C22×C4Q8×D5912yes—
458C22×C4#1174568912yes—
459C22×C4C2×C22⋊D5914yes—
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—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—592no—
481C4⋊C4—592no—
482C4⋊C4—1188no—
483C4⋊C4—1181no—
484C4⋊C4—1181no—
485C22⋊C4—1188no—
486C22⋊C4—1182no—
487C22⋊C4—1182no—
488C42—24no—
489C42—22no—
490C42—1188no—
491C42—1182no—
492C42—1182no—
493C42C2×C4×D5914yescomplemented
494C23—22no—
495C23—22no—
496C23—1184no—
497C23—1182no—
498C23—2362no—
499C23—2362no—
500C23C4×C22⋊D5912yes—
501C23#1427369511yescharacteristic
502C23#1427369511yescharacteristic
503C23#1427370312yes—
504C23C22×C22⋊D5911yescharacteristic
505Q8—594no—
506Q8—1182no—
507Q8—1184no—
508D4—594no—
509D4—1184no—
510D4—1188no—
511D4—1184no—
512D4—1182no—
513D4—1184no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—48no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—1184no—
527C2×C4—1182no—
528C2×C4—1184no—
529C2×C4—1188no—
530C2×C4—1184no—
531C2×C4—1182no—
532C2×C4—1182no—
533C2×C4—4728no—
534C2×C4—2362no—
535C2×C4—2362no—
536C2×C4C4⋊C4×D5912yes—
537C2×C4C4⋊C4×D5912yes—
538C2×C4C4×C22⋊D5912yes—
539C2×C4#1427368912yes—
540C2×C4#1427368912yes—
541C2×C4#1427369511yescharacteristic
542C2×C4#1427369511yescharacteristic
543C2×C4#1427370312yes—
544C2×C4#1427370512yes—
545C2×C4#1427370512yes—
546C2×C4#1427371212yes—
547C2×C4#1427371212yes—
548C2×C4C22×C4×D5911yescharacteristic
549C2×C4C2×D4×D5911yescharacteristic
550C2×C4C2×D4×D5911yescharacteristic
551C2×C4C2×Q8×D5911yescharacteristic
552C2×C4C2×#1174568511yescharacteristic
553C2×C4C2×#1174568911yescharacteristic
554C8—1184no—
555C8—2362no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—24no—
561C22—24no—
562C22—1184no—
563C22—2362no—
564C22—2362no—
565C22—2368no—
566C22#1683727312yes—
567C22#1683727312yes—
568C22#1683757412yes—
569C22#1683757912yes—
570C22C2×C4×C22⋊D5911yescharacteristic
571C22C2×#1427369511yescharacteristic
572C22C2×#1427370311yescharacteristic
573C4—44no—
574C4—44no—
575C4—82no—
576C4—1184no—
577C4—2362no—
578C4—2362no—
579C4#1683727312yes—
580C4#1683727312yes—
581C4C2×C4⋊C4×D5911yescharacteristic
582C4C2×#1427368911yescharacteristic
583C4C2×#1427370511yescharacteristic
584C4C2×#1427371211yescharacteristic
585C2—21no—
586C2—21no—
587C2—22no—
588C2—2364no—
589C2#2142432814yesdirect product factor
590C2C2×#1683727311yescharacteristic
591C2C2×#1683757911yescharacteristic
592C2C2×#1683757411yescharacteristic
subgroupG/H
centerC23#14273695
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC2×C59×#382C2
socleC23×C59#120
prime radicalC22×#11745685C22
cosocleC22×C4×C59C24
hypercenterC2×#382D59
hypocenterC59C2×#1860
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#382

Complements

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

NH ≅ G/Ntype
2. C2×#16837442C2semidirect
5. #21424328C2direct (N × H)
6. C2×#16837171C2semidirect
7. C2×#16837162C2semidirect
8. C2×C59×#382C2semidirect
23. #16837442C22semidirect
26. C2×#14273655C4semidirect
32. #16837171C22semidirect
33. #16837162C22semidirect
38. C59×#382C22semidirect
58. C2×#11745664C2×C4semidirect
86. #14273655C2×C4semidirect
107. C2×C42×C59C2×C4semidirect
175. C42×C59C22×C4semidirect
192. #11745664C22×C4semidirect
293. C2×#382D59semidirect
300. C2×C59#1860semidirect
332. #382C2×D59semidirect
336. C59C2×#1860semidirect
399. C2×C42C4×D59semidirect
493. C42C2×C4×D59semidirect
589. C2#21424328direct (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×#168374422
C22×#142736552
C2×#168374012
#214243282
C2×#168371712
C2×#168371622
C2×C59×#3822
C2×#186059