Finite Group Databasebeta

← C4×#16837745 #28595861

Subgroups — C4×#16837745 #28595861

Order 15104.

888 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837745C211yesmaximal, characteristic
2C4×#14273707C211yesmaximal, characteristic
3C4×#14273685C211yesmaximal, characteristic
4#21424736C211yesmaximal, characteristic, complemented
5#21424728C211yesmaximal, characteristic, complemented
6#21424677C211yesmaximal, characteristic, complemented
7#21424669C211yesmaximal, characteristic, complemented
8#21424407C211yesmaximal, characteristic
9#21424292C211yesmaximal, characteristic
10#21424189C211yesmaximal, characteristic
11C4×#14273658C211yesmaximal, characteristic, complemented
12C4×#14273650C211yesmaximal, characteristic, complemented
13C4×#14273652C211yesmaximal, characteristic, complemented
14C4×(C2×D59)⋊C8C211yesmaximal, characteristic
15C4×#107×C59C211yesmaximal, characteristic, complemented
16C2×#14273707C2211yescharacteristic
17C2×C4×Dic118C2211yescharacteristic
18C2×#14273685C2211yescharacteristic
19C2×#14273685C2211yescharacteristic
20C2×C42×D59C2211yescharacteristic
21#16837745C418yesdirect product factor
22C2×#14273658C2211yescharacteristic
23C2×#14273658C2211yescharacteristic
24C2×#14273652C2211yescharacteristic
25C2×#14273652C2211yescharacteristic
26C2×#14273650C2211yescharacteristic
27C2×#14273650C2211yescharacteristic
28C2×(C2×D59)⋊C8C2211yescharacteristic
29C2×(C2×D59)⋊C8C2211yescharacteristic
30C2×C4×C59⋊C8C2211yescharacteristic
31#16837307C2211yescharacteristic
32#16837301C2211yescharacteristic
33#16837287—21no—
34#16837276—21no—
35#16837259C2211yescharacteristic
36C4⋊C4×Dic59C2211yescharacteristic
37C4⋊C4×Dic59C2211yescharacteristic
38C4×C4⋊Dic59—21no—
39C4×C4⋊Dic59C2211yescharacteristic
40C4×(C2×D59)⋊C4—21no—
41C4×(C2×D59)⋊C4—21no—
42#16837249—21no—
43#16837247—21no—
44C42×Dic59—21no—
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837156C2211yescharacteristic
52#16837156C2211yescharacteristic
53C2×#107×C59C2211yescharacteristic
54C2×#107×C59C2211yescharacteristic
55C2×C4×C8×C59C2211yescharacteristic
56C4×C4⋊C4×C59C2211yescharacteristic
57C4×C4⋊C4×C59C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#340C2211yescharacteristic
60C22×Dic118C2311yescharacteristic
61C22×C4×D59D411yescharacteristic
62C22×C4×D59D411yescharacteristic
63C22×C4×D59C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66#14273707—22no—
67#14273707C2×C414yes—
68C4×Dic118—22no—
69C4×Dic118—22no—
70#14273685—22no—
71#14273685—22no—
72#14273685—22no—
73#14273685C2×C414yes—
74C2×C4⋊Dic59—21no—
75C2×C4⋊Dic59—21no—
76C2×C4⋊Dic59C2311yescharacteristic
77C2×C4⋊Dic59C2311yescharacteristic
78C2×C4⋊Dic59C2311yescharacteristic
79C2×Dic59⋊C4—21no—
80C2×(C2×D59)⋊C4—21no—
81C2×(C2×D59)⋊C4—21no—
82C2×(C2×D59)⋊C4—21no—
83C2×(C2×D59)⋊C4—21no—
84C2×C4×Dic59—21no—
85C2×C4×Dic59—21no—
86C2×C4×Dic59—21no—
87C2×C4×Dic59—21no—
88C2×C4×Dic59—21no—
89C2×C4×Dic59—21no—
90C2×C4×Dic59D411yescharacteristic
91C2×C4×Dic59D411yescharacteristic
92C2×C4×Dic59C2311yescharacteristic
93C42×D59—22no—
94C42×D59—22no—
95#14273658—22no—
96#14273658C2×C414yescomplemented
97#14273652—22no—
98#14273652C2×C414yescomplemented
99#14273650—22no—
100#14273650C2×C414yescomplemented
101(C2×D59)⋊C8—22no—
102(C2×D59)⋊C8C2×C414yes—
103C4×C59⋊C8—22no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95—21no—
107C59⋊#95—21no—
108C59⋊#95—21no—
109C59⋊#95—21no—
110C59⋊#95—21no—
111C22×C8×C59C2311yescharacteristic
112C22×C8×C59C2311yescharacteristic
113C2×C4⋊C4×C59C2311yescharacteristic
114C2×C4⋊C4×C59C2311yescharacteristic
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C42×C59—21no—
118C2×C42×C59—21no—
119C2×C42×C59C2311yescharacteristic
120#107×C59—22no—
121#107×C59C2×C414yescomplemented
122C4×C8×C59—22no—
123#95×C59—21no—
124#95×C59—21no—
125C23×D59C2×D411yescharacteristic
126C22×C4×C59—21no—
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59—21no—
130C22×C4×C59C2×D411yescharacteristic
131C22×C4×C59C2×D411yescharacteristic
132C22×C4×C59C2411yescharacteristic
133C22×Dic59—21no—
134C22×Dic59—21no—
135C22×Dic59—21no—
136C22×Dic59—21no—
137C22×Dic59C2×D411yescharacteristic
138C22×Dic59C2×D411yescharacteristic
139C22×Dic59C2×D411yescharacteristic
140C2×C4×D59—21no—
141C2×C4×D59—21no—
142C2×C4×D59—21no—
143C2×C4×D59—21no—
144C2×C4×D59—22no—
145C2×C4×D59—22no—
146C2×C4×D59—22no—
147C2×C4×D59—22no—
148C2×C4×D59—22no—
149C2×C4×D59—41no—
150C2×C4×D59C22×C412yes—
151C2×C4×D59#4112yes—
152C2×Dic118—21no—
153C2×Dic118—21no—
154C2×Dic118—21no—
155C2×Dic118—21no—
156C2×Dic118—21no—
157C2×Dic118C22×C412yes—
158C2×C8×C59—21no—
159C2×C8×C59—21no—
160C2×C8×C59—21no—
161C2×C8×C59—21no—
162C2×C8×C59—21no—
163C2×C8×C59C22×C412yes—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—21no—
169C4⋊C4×C59C22×C412yes—
170C4⋊C4×C59C22×C412yes—
171C4⋊C4×C59#4112yes—
172C42×C59—42no—
173C42×C59—42no—
174C42×C59#4112yes—
175C42×C59#4112yes—
176(C2×D59)⋊C4—24no—
177(C2×D59)⋊C4—24no—
178(C2×D59)⋊C4—24no—
179(C2×D59)⋊C4—42no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—24no—
183C4⋊Dic59—21no—
184C4⋊Dic59—21no—
185C4⋊Dic59—42no—
186C4⋊Dic59C22×C412yes—
187C4⋊Dic59#4112yes—
188Dic59⋊C4—42no—
189C4×Dic59—24no—
190C4×Dic59—22no—
191C4×Dic59—22no—
192C4×Dic59—21no—
193C4×Dic59—21no—
194C4×Dic59—24no—
195C4×Dic59—42no—
196C4×Dic59—42no—
197C4×Dic59—42no—
198C4×Dic59—42no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8—21no—
204C2×C59⋊C8C22×C412yes—
205C23×C59C22×D411yescharacteristic
206C22×D59—21no—
207C22×D59—21no—
208C22×D59—21no—
209C22×D59—21no—
210C22×D59—21no—
211C22×D59—21no—
212C22×D59C4×D412yes—
213C2×C4×C59—21no—
214C2×C4×C59—21no—
215C2×C4×C59—22no—
216C2×C4×C59—22no—
217C2×C4×C59—22no—
218C2×C4×C59—41no—
219C2×C4×C59—41no—
220C2×C4×C59—41no—
221C2×C4×C59—41no—
222C2×C4×C59—41no—
223C2×C4×C59—41no—
224C2×C4×C59—41no—
225C2×C4×C59—41no—
226C2×C4×C59—41no—
227C2×C4×C59#12312yes—
228C2×C4×C59#12312yes—
229C2×C4×C59#13512yes—
230C2×C4×C59#13712yes—
231C2×C4×C59C23×C411yescharacteristic
232C2×C4×C59C2×#4111yescharacteristic
233C2×C4×C59C2×#4111yescharacteristic
234C2×C4×C59C2×#4111yescharacteristic
235C2×C4×C59C2×#4111yescharacteristic
236C2×C4×C59C2×#4111yescharacteristic
237C2×Dic59—21no—
238C2×Dic59—21no—
239C2×Dic59—21no—
240C2×Dic59—21no—
241C2×Dic59—22no—
242C2×Dic59—22no—
243C2×Dic59—21no—
244C2×Dic59—22no—
245C2×Dic59—21no—
246C2×Dic59—21no—
247C2×Dic59—41no—
248C2×Dic59—41no—
249C2×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59—41no—
252C2×Dic59—41no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255C2×Dic59—41no—
256C2×Dic59C4×D412yes—
257C2×Dic59#13512yes—
258C4×D59—22no—
259C4×D59—22no—
260C4×D59—22no—
261C4×D59—22no—
262C4×D59—44no—
263C4×D59—44no—
264Dic118—22no—
265Dic118—22no—
266Dic118—41no—
267Dic118—41no—
268C8×C59—22no—
269C8×C59—41no—
270C59⋊C8—22no—
271C59⋊C8—41no—
272C4×#485—591nomaximal, Sylow
273C22×C59C2×C4×D411yescharacteristic
274C22×C59C2×#12311yescharacteristic
275C22×C59#52511yescharacteristic
276C22×C59C2×#13511yescharacteristic
277C22×C59C2×#13511yescharacteristic
278C22×C59C2×#13711yescharacteristic
279C22×C59#57511yescharacteristic
280C2×D59—22no—
281C2×D59—22no—
282C2×D59—22no—
283C2×D59—41no—
284C2×D59—41no—
285C2×D59—41no—
286C2×D59—41no—
287C4×C59—22no—
288C4×C59—42no—
289C4×C59—42no—
290C4×C59—42no—
291C4×C59—81no—
292C4×C59#48514yescomplemented
293C4×C59C4×#4111yescharacteristic
294C4×C59C4×#4111yescharacteristic
295C4×C59#52911yescharacteristic
296C4×C59#52911yescharacteristic
297Dic59—22no—
298Dic59—22no—
299Dic59—41no—
300Dic59—42no—
301Dic59—42no—
302Dic59—42no—
303Dic59—81no—
304C2×#485—591no—
305C4×C22⋊Q8—591no—
306C4×#117—591no—
307#1928—591no—
308#1927—591no—
309#1919—591no—
310#1918—591no—
311#1873—591no—
312#1854—591no—
313#1825—591no—
314C4×#107D5911yescharacteristic, complemented
315C4×#106—591no—
316C4×#103—591no—
317C4×#103—591no—
318C4×C22⋊C8—591no—
319C2×C59C4×#12311yescharacteristic
320C2×C59C4×#13511yescharacteristic
321C2×C59C4×#13711yescharacteristic
322C2×C59C2×#48511yescharacteristic
323C2×C59#309111yescharacteristic
324C2×C59#310211yescharacteristic
325C2×C59#310711yescharacteristic
326D59—42no—
327C2×C22⋊Q8—591no—
328C2×C4×Q8—591no—
329C2×#117—591no—
330C2×#117—591no—
331C22×C42—591no—
332#485—598no—
333C2×#107C2×D5911yescharacteristic
334C2×#107C2×D5911yescharacteristic
335C2×#106—591no—
336C2×#106—591no—
337C2×#103—591no—
338C2×#103—591no—
339C2×#103—591no—
340C2×#103—591no—
341C2×C22⋊C8—591no—
342C2×C22⋊C8—591no—
343C2×C4×C8—591no—
344C2×C4×C8C2×D5911yescharacteristic
345#391—591no—
346#389—591no—
347#387—1181no—
348#385—1181no—
349#380—591no—
350C4×C4⋊C4—591no—
351C4×C4⋊C4—591no—
352C4×C4⋊C4—591no—
353C4×C4⋊C4—1181no—
354C4×C4⋊C4C2×D5911yescharacteristic
355C4×C4⋊C4C2×D5911yescharacteristic
356C4×C22⋊C4—1181no—
357C4×C22⋊C4—1181no—
358C4×C22⋊C4—1181no—
359#376—1181no—
360C43—1181no—
361#340—591no—
362#340—591no—
363#340—591no—
364#340—591no—
365#340—591no—
366#340—591no—
367#340C2×D5911yescharacteristic
368#340C2×D5911yescharacteristic
369#336—591no—
370#336—591no—
371C59C4×#48511yescharacteristic, complemented, Sylow
372C22×Q8—591no—
373C23×C4—591no—
374C23×C4—591no—
375C23×C4—591no—
376C22×C8—591no—
377C22×C8—591no—
378C22×C8C22×D5911yescharacteristic
379C22×C8C22×D5911yescharacteristic
380C22⋊Q8—594no—
381C22⋊Q8—1182no—
382C4×Q8—1182no—
383C4×Q8—1182no—
384#117—594no—
385#117—1182no—
386#117—1182no—
387#117—1182no—
388C2×C4⋊C4—591no—
389C2×C4⋊C4—591no—
390C2×C4⋊C4—591no—
391C2×C4⋊C4—1181no—
392C2×C4⋊C4—1181no—
393C2×C4⋊C4—1181no—
394C2×C4⋊C4C22×D5911yescharacteristic
395C2×C4⋊C4C22×D5911yescharacteristic
396C2×C4⋊C4C22×D5911yescharacteristic
397C2×C4⋊C4C22×D5911yescharacteristic
398C2×C22⋊C4—1181no—
399C2×C22⋊C4—1181no—
400C2×C22⋊C4—1181no—
401C2×C22⋊C4—1181no—
402C2×C42—21no—
403C2×C42—21no—
404C2×C42—591no—
405C2×C42—591no—
406C2×C42—591no—
407C2×C42—1181no—
408C2×C42—1181no—
409C2×C42—1182no—
410C2×C42—1182no—
411C2×C42—1181no—
412C2×C42—1181no—
413C2×C42—1181no—
414C2×C42—1181no—
415C2×C42C22×D5911yescharacteristic
416#107—22no—
417#107C4×D5914yescomplemented
418#106—594no—
419#106—1182no—
420#103—594no—
421#103—594no—
422#103—1182no—
423#103—1182no—
424C22⋊C8—594no—
425C22⋊C8—1182no—
426C4×C8—22no—
427C4×C8—1182no—
428#95—21no—
429#95—21no—
430#95—1181no—
431#95—1181no—
432#95—1181no—
433#95—1181no—
434#95—1181no—
435#95—1181no—
436#95—1181no—
437C24—591no—
438C2×Q8—592no—
439C2×Q8—1181no—
440C2×Q8—1181no—
441C2×Q8—1181no—
442C2×Q8—1181no—
443C2×Q8—1181no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—591no—
449C22×C4—591no—
450C22×C4—592no—
451C22×C4—592no—
452C22×C4—591no—
453C22×C4—1182no—
454C22×C4—1181no—
455C22×C4—1181no—
456C22×C4—1181no—
457C22×C4—1181no—
458C22×C4—1181no—
459C22×C4—1181no—
460C22×C4—1181no—
461C22×C4—1182no—
462C22×C4—1182no—
463C22×C4—1182no—
464C22×C4—1182no—
465C22×C4—1181no—
466C22×C4—2361no—
467C22×C4D4×D5911yescharacteristic
468C22×C4D4×D5911yescharacteristic
469C22×C4C23×D5911yescharacteristic
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—592no—
476C2×C8—1181no—
477C2×C8—1181no—
478C2×C8—1181no—
479C2×C8—1181no—
480C2×C8—1181no—
481C2×C8C2×C4×D5912yes—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—592no—
488C4⋊C4—592no—
489C4⋊C4—1184no—
490C4⋊C4—1181no—
491C4⋊C4—1181no—
492C4⋊C4—1181no—
493C4⋊C4—1181no—
494C4⋊C4—2362no—
495C4⋊C4—2362no—
496C4⋊C4C2×C4×D5912yes—
497C4⋊C4C2×C4×D5912yes—
498C4⋊C4#1174568512yes—
499C22⋊C4—1184no—
500C22⋊C4—1184no—
501C22⋊C4—1184no—
502C22⋊C4—2362no—
503C42—42no—
504C42—42no—
505C42—1184no—
506C42—1184no—
507C42—1182no—
508C42—1182no—
509C42—1181no—
510C42—1181no—
511C42—2362no—
512C42—2362no—
513C42—2362no—
514C42—2362no—
515C42#1174568512yes—
516C42#1174568912yes—
517C23—592no—
518C23—1181no—
519C23—1181no—
520C23—1181no—
521C23—1181no—
522C23—1181no—
523C23—1181no—
524C23C2×D4×D5911yescharacteristic
525Q8—1182no—
526Q8—1182no—
527Q8—2361no—
528Q8—2361no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—592no—
544C2×C4—592no—
545C2×C4—1181no—
546C2×C4—1181no—
547C2×C4—1181no—
548C2×C4—1181no—
549C2×C4—1182no—
550C2×C4—1182no—
551C2×C4—1182no—
552C2×C4—1182no—
553C2×C4—1181no—
554C2×C4—1182no—
555C2×C4—1181no—
556C2×C4—1181no—
557C2×C4—1182no—
558C2×C4—1182no—
559C2×C4—2364no—
560C2×C4—2361no—
561C2×C4—2361no—
562C2×C4—2361no—
563C2×C4—2361no—
564C2×C4—2361no—
565C2×C4—2361no—
566C2×C4—2361no—
567C2×C4—2361no—
568C2×C4—2361no—
569C2×C4—2364no—
570C2×C4#1427371212yes—
571C2×C4#1427371212yes—
572C2×C4#1427374612yes—
573C2×C4#1427375912yes—
574C2×C4C22×C4×D5911yescharacteristic
575C2×C4#41×D5911yescharacteristic
576C2×C4#41×D5911yescharacteristic
577C2×C4C2×#1174568511yescharacteristic
578C2×C4C2×#1174568511yescharacteristic
579C2×C4C2×#1174568911yescharacteristic
580C8—22no—
581C8—41no—
582C8—1182no—
583C8—2361no—
584C22—1182no—
585C22—1182no—
586C22—1182no—
587C22—2361no—
588C22—2361no—
589C22—2361no—
590C22—2361no—
591C22C4×D4×D5911yescharacteristic
592C22C2×#1427371211yescharacteristic
593C22#1683791111yescharacteristic
594C22#135×D5911yescharacteristic
595C22C2×#1427374611yescharacteristic
596C22C2×#1427375911yescharacteristic
597C22#1683819911yescharacteristic
598C4—22no—
599C4—42no—
600C4—42no—
601C4—42no—
602C4—81no—
603C4—1182no—
604C4—1182no—
605C4—4721no—
606C4—2362no—
607C4—2362no—
608C4—2361no—
609C4—2362no—
610C4#1683774514yesdirect product factor
611C4C4×#1174568511yescharacteristic
612C4C4×#1174568911yescharacteristic
613C4#1683793011yescharacteristic
614C4#1683793911yescharacteristic
615C2—2362no—
616C2C4×#1427371211yescharacteristic
617C2C4×#1427374611yescharacteristic
618C2C4×#1427375911yescharacteristic
619C2C2×#1683774511yescharacteristic
620C2#2143401311yescharacteristic
621C2#2143411011yescharacteristic
622C2#2143416511yescharacteristic
subgroupG/H
centerC22×C4D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC4×#107×C59C2
socleC23×C59C22×D4
prime radicalC23×D59C2×D4
cosocleC22×C4×C59C24
hypercenterC4×#107D59
hypocenterC59C4×#485
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C4×#107

Complements

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

NH ≅ G/Ntype
4. #21424736C2semidirect
5. #21424728C2semidirect
6. #21424677C2semidirect
7. #21424669C2semidirect
11. C4×#14273658C2semidirect
12. C4×#14273650C2semidirect
13. C4×#14273652C2semidirect
15. C4×#107×C59C2semidirect
21. #16837745C4direct (N × H)
96. #14273658C2×C4semidirect
98. #14273652C2×C4semidirect
100. #14273650C2×C4semidirect
121. #107×C59C2×C4semidirect
292. C4×C59#485semidirect
314. C4×#107D59semidirect
371. C59C4×#485semidirect
417. #107C4×D59semidirect
610. C4#16837745direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168377452
C4×#142737072
C4×#142736852
#214247362
#214247282
#214246772
#214246692
#214244072
#214242922
#214241892
C4×#142736582
C4×#142736502
C4×#142736522
C4×(C2×D59)⋊C82
C4×#107×C592
C4×#48559