Finite Group Databasebeta

← C4×#16837740 #28595851

Subgroups — C4×#16837740 #28595851

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×#16837740C211yesmaximal, characteristic
2C4×#14273686C211yesmaximal, characteristic
3#21424734C211yesmaximal, characteristic, complemented
4#21424727C211yesmaximal, characteristic, complemented
5#21424671C211yesmaximal, characteristic, complemented
6#21424664C211yesmaximal, characteristic, complemented
7#21424409C211yesmaximal, characteristic
8#21424293C211yesmaximal, characteristic
9#21424188C211yesmaximal, characteristic
10C4×#14273660C211yesmaximal, characteristic, complemented
11C4×#14273658C211yesmaximal, characteristic, complemented
12C4×#14273651C211yesmaximal, characteristic, complemented
13C4×#14273651C211yesmaximal, characteristic, complemented
14C4×C22⋊(C59⋊C8)C211yesmaximal, characteristic
15C4×C22⋊Q8×C59C211yesmaximal, characteristic
16C2×#14273686C2211yescharacteristic
17C2×#14273686C2211yescharacteristic
18#16837740C418yesdirect product factor
19C2×#14273660C2211yescharacteristic
20C2×#14273660C2211yescharacteristic
21C2×#14273658C2211yescharacteristic
22C2×#14273658C2211yescharacteristic
23C2×#14273651C2211yescharacteristic
24C2×#14273651C2211yescharacteristic
25C2×#14273651C2211yescharacteristic
26C2×#14273651C2211yescharacteristic
27C2×C22⋊(C59⋊C8)C2211yescharacteristic
28C2×C22⋊(C59⋊C8)C2211yescharacteristic
29C2×C4×C59⋊C8C2211yescharacteristic
30C2×C4×C59⋊C8C2211yescharacteristic
31C4×C4⋊Dic59C2211yescharacteristic
32C4×C4⋊Dic59C2211yescharacteristic
33C4×C4⋊Dic59C2211yescharacteristic
34C4×C4⋊Dic59C2211yescharacteristic
35C4×C22⋊Dic59—21no—
36C4×C22⋊Dic59—21no—
37#16837248—21no—
38C42×Dic59—21no—
39#16837164C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837157C2211yescharacteristic
48#16837157C2211yescharacteristic
49C2×C22⋊Q8×C59C2211yescharacteristic
50C2×C4×Q8×C59C2211yescharacteristic
51C22×C42×C59C2211yescharacteristic
52C59×#391C2211yescharacteristic
53C59×#389C2211yescharacteristic
54C59×#387—21no—
55C59×#385—21no—
56C59×#380C2211yescharacteristic
57C4×C4⋊C4×C59—21no—
58C4×C4⋊C4×C59C2211yescharacteristic
59C4×C22⋊C4×C59—21no—
60C22×C59⋊C8C2311yescharacteristic
61C22×C59⋊C8C2311yescharacteristic
62C22×C59⋊C8C2311yescharacteristic
63C22×C59⋊C8C2311yescharacteristic
64#14273686—22no—
65#14273686—22no—
66#14273686—22no—
67#14273686C2×C414yes—
68C2×C4⋊Dic59C2311yescharacteristic
69C2×C4⋊Dic59C2311yescharacteristic
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C22⋊Dic59—21no—
73C2×C22⋊Dic59—21no—
74C2×C4×Dic59—21no—
75C2×C4×Dic59—21no—
76C2×C4×Dic59—21no—
77C2×C4×Dic59—21no—
78C2×C4×Dic59—21no—
79C2×C4×Dic59—21no—
80#14273660—22no—
81#14273660C2×C414yescomplemented
82#14273658—22no—
83#14273658C2×C414yescomplemented
84#14273651—22no—
85#14273651—22no—
86#14273651C2×C414yescomplemented
87#14273651C2×C414yescomplemented
88C22⋊(C59⋊C8)—22no—
89C22⋊(C59⋊C8)C2×C414yes—
90C4×C59⋊C8—22no—
91C4×C59⋊C8—22no—
92C59⋊#95—21no—
93C59⋊#95—21no—
94C59⋊#95—21no—
95C59⋊#95—21no—
96C22×Q8×C59C2311yescharacteristic
97C23×C4×C59D411yescharacteristic
98C23×C4×C59D411yescharacteristic
99C23×C4×C59C2311yescharacteristic
100C22⋊Q8×C59—22no—
101C22⋊Q8×C59C2×C414yes—
102C4×Q8×C59—22no—
103C4×Q8×C59—22no—
104C2×C4⋊C4×C59—21no—
105C2×C4⋊C4×C59—21no—
106C2×C4⋊C4×C59—21no—
107C2×C4⋊C4×C59C2311yescharacteristic
108C2×C4⋊C4×C59C2311yescharacteristic
109C2×C4⋊C4×C59C2311yescharacteristic
110C2×C22⋊C4×C59—21no—
111C2×C22⋊C4×C59—21no—
112C2×C42×C59—22no—
113C2×C42×C59—22no—
114C2×C42×C59—21no—
115C2×C42×C59—21no—
116C2×C42×C59D411yescharacteristic
117C2×C42×C59D411yescharacteristic
118C2×C42×C59C2311yescharacteristic
119C2×C42×C59C2311yescharacteristic
120#95×C59—21no—
121#95×C59—21no—
122#95×C59—21no—
123#95×C59—21no—
124#95×C59—21no—
125C24×C59C2×D411yescharacteristic
126C2×Q8×C59—21no—
127C2×Q8×C59—21no—
128C2×Q8×C59—21no—
129C2×Q8×C59—21no—
130C2×Q8×C59—21no—
131C2×Q8×C59C22×C412yes—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—21no—
137C22×C4×C59—21no—
138C22×C4×C59—22no—
139C22×C4×C59—22no—
140C22×C4×C59—22no—
141C22×C4×C59—22no—
142C22×C4×C59—22no—
143C22×C4×C59—21no—
144C22×C4×C59—21no—
145C22×C4×C59—41no—
146C22×C4×C59C22×C412yes—
147C22×C4×C59C2×D411yescharacteristic
148C22×C4×C59C2×D411yescharacteristic
149C22×C4×C59C2×D411yescharacteristic
150C22×C4×C59C2×D411yescharacteristic
151C22×C4×C59C2×D411yescharacteristic
152C22×C4×C59#4112yes—
153C22×C4×C59C2411yescharacteristic
154C22×Dic59—21no—
155C22×Dic59—21no—
156C22×Dic59—21no—
157C22×Dic59—21no—
158C4⋊C4×C59—24no—
159C4⋊C4×C59—21no—
160C4⋊C4×C59—21no—
161C4⋊C4×C59—21no—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—42no—
165C4⋊C4×C59—42no—
166C4⋊C4×C59C22×C412yes—
167C22⋊C4×C59—24no—
168C22⋊C4×C59—42no—
169C42×C59—22no—
170C42×C59—22no—
171C42×C59—21no—
172C42×C59—21no—
173C42×C59—42no—
174C42×C59—42no—
175C42×C59#4112yes—
176C42×C59#4112yes—
177C22⋊Dic59—24no—
178C22⋊Dic59—24no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59C22×C412yes—
184C4⋊Dic59C22×C412yes—
185C4⋊Dic59#4112yes—
186C4⋊Dic59#4112yes—
187C4×Dic59—24no—
188C4×Dic59—24no—
189C4×Dic59—42no—
190C4×Dic59—42no—
191C4×Dic59—42no—
192C4×Dic59—42no—
193C2×C59⋊C8—21no—
194C2×C59⋊C8—21no—
195C2×C59⋊C8—21no—
196C2×C59⋊C8—21no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8C22×C412yes—
204C2×C59⋊C8C22×C412yes—
205C23×C59—21no—
206C23×C59—21no—
207C23×C59—21no—
208C23×C59—21no—
209C23×C59—21no—
210C23×C59—21no—
211C23×C59C4×D412yes—
212C23×C59C22×D411yescharacteristic
213Q8×C59—22no—
214Q8×C59—22no—
215Q8×C59—41no—
216Q8×C59—41no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—21no—
221C2×C4×C59—21no—
222C2×C4×C59—21no—
223C2×C4×C59—22no—
224C2×C4×C59—22no—
225C2×C4×C59—22no—
226C2×C4×C59—22no—
227C2×C4×C59—22no—
228C2×C4×C59—22no—
229C2×C4×C59—21no—
230C2×C4×C59—21no—
231C2×C4×C59—21no—
232C2×C4×C59—44no—
233C2×C4×C59—41no—
234C2×C4×C59—41no—
235C2×C4×C59—41no—
236C2×C4×C59—41no—
237C2×C4×C59—44no—
238C2×C4×C59—41no—
239C2×C4×C59—41no—
240C2×C4×C59—41no—
241C2×C4×C59—41no—
242C2×C4×C59—41no—
243C2×C4×C59—41no—
244C2×C4×C59C4×D412yes—
245C2×C4×C59#12312yes—
246C2×C4×C59#12312yes—
247C2×C4×C59#13512yes—
248C2×C4×C59#13512yes—
249C2×C4×C59#13712yes—
250C2×C4×C59C23×C411yescharacteristic
251C2×C4×C59C2×#4111yescharacteristic
252C2×C4×C59C2×#4111yescharacteristic
253C2×C4×C59C2×#4111yescharacteristic
254C2×C4×C59C2×#4111yescharacteristic
255C2×C4×C59C2×#4111yescharacteristic
256C2×Dic59—22no—
257C2×Dic59—22no—
258C2×Dic59—22no—
259C2×Dic59—22no—
260C2×Dic59—41no—
261C2×Dic59—41no—
262C2×Dic59—41no—
263C2×Dic59—41no—
264C2×Dic59—41no—
265C2×Dic59—41no—
266C2×Dic59—41no—
267C2×Dic59—41no—
268C59⋊C8—22no—
269C59⋊C8—22no—
270C59⋊C8—41no—
271C59⋊C8—41no—
272C4×#485—591nomaximal, Sylow
273C22×C59—22no—
274C22×C59—22no—
275C22×C59—22no—
276C22×C59—41no—
277C22×C59—41no—
278C22×C59—41no—
279C22×C59—41no—
280C22×C59C2×C4×D411yescharacteristic
281C22×C59C2×#12311yescharacteristic
282C22×C59#52511yescharacteristic
283C22×C59C2×#13511yescharacteristic
284C22×C59C2×#13511yescharacteristic
285C22×C59C2×#13711yescharacteristic
286C22×C59#57511yescharacteristic
287C4×C59—22no—
288C4×C59—22no—
289C4×C59—22no—
290C4×C59—41no—
291C4×C59—42no—
292C4×C59—42no—
293C4×C59—81no—
294C4×C59—81no—
295C4×C59#48514yescomplemented
296C4×C59C4×#4111yescharacteristic
297C4×C59C4×#4111yescharacteristic
298C4×C59#52911yescharacteristic
299C4×C59#52911yescharacteristic
300Dic59—42no—
301Dic59—42no—
302Dic59—42no—
303Dic59—42no—
304C2×#485—591no—
305C4×C22⋊Q8D5911yescharacteristic
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×#107—591no—
315C4×#106—591no—
316C4×#103—591no—
317C4×#103—591no—
318C4×C22⋊C8—591no—
319C2×C59—42no—
320C2×C59C4×#12311yescharacteristic
321C2×C59C4×#13511yescharacteristic
322C2×C59C4×#13711yescharacteristic
323C2×C59C2×#48511yescharacteristic
324C2×C59#309111yescharacteristic
325C2×C59#310211yescharacteristic
326C2×C59#310711yescharacteristic
327C2×C22⋊Q8C2×D5911yescharacteristic
328C2×C4×Q8C2×D5911yescharacteristic
329C2×#117—591no—
330C2×#117—591no—
331C22×C42C2×D5911yescharacteristic
332#485—598no—
333C2×#107—591no—
334C2×#107—591no—
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×C8—591no—
345#391C2×D5911yescharacteristic
346#389C2×D5911yescharacteristic
347#387—21no—
348#385—21no—
349#380C2×D5911yescharacteristic
350C4×C4⋊C4—21no—
351C4×C4⋊C4—591no—
352C4×C4⋊C4—591no—
353C4×C4⋊C4—591no—
354C4×C4⋊C4—591no—
355C4×C4⋊C4C2×D5911yescharacteristic
356C4×C22⋊C4—21no—
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#340—591no—
368#340—591no—
369#336—591no—
370#336—591no—
371C59C4×#48511yescharacteristic, complemented, Sylow
372C22×Q8C22×D5911yescharacteristic
373C23×C4C22⋊D5911yescharacteristic
374C23×C4C22⋊D5911yescharacteristic
375C23×C4C22×D5911yescharacteristic
376C22×C8—591no—
377C22×C8—591no—
378C22×C8—591no—
379C22×C8—591no—
380C22⋊Q8—22no—
381C22⋊Q8C4×D5914yes—
382C4×Q8—22no—
383C4×Q8—22no—
384#117—594no—
385#117—1182no—
386#117—1182no—
387#117—1182no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—591no—
392C2×C4⋊C4—591no—
393C2×C4⋊C4—591no—
394C2×C4⋊C4—591no—
395C2×C4⋊C4C22×D5911yescharacteristic
396C2×C4⋊C4C22×D5911yescharacteristic
397C2×C4⋊C4C22×D5911yescharacteristic
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C22⋊C4—1181no—
401C2×C22⋊C4—1181no—
402C2×C42—22no—
403C2×C42—22no—
404C2×C42—21no—
405C2×C42—21no—
406C2×C42—1181no—
407C2×C42—1181no—
408C2×C42—1181no—
409C2×C42—1181no—
410C2×C42—1181no—
411C2×C42—1181no—
412C2×C42C22⋊D5911yescharacteristic
413C2×C42C22⋊D5911yescharacteristic
414C2×C42C22×D5911yescharacteristic
415C2×C42C22×D5911yescharacteristic
416#107—594no—
417#107—1182no—
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—1182no—
427C4×C8—1182no—
428#95—21no—
429#95—21no—
430#95—21no—
431#95—21no—
432#95—21no—
433#95—1181no—
434#95—1181no—
435#95—1181no—
436#95—1181no—
437C24C2×C22⋊D5911yescharacteristic
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8C2×C4×D5912yes—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—41no—
458C22×C4—1181no—
459C22×C4—1181no—
460C22×C4—1181no—
461C22×C4—1181no—
462C22×C4C2×C4×D5912yes—
463C22×C4#1174568512yes—
464C22×C4C2×C22⋊D5911yescharacteristic
465C22×C4C2×C22⋊D5911yescharacteristic
466C22×C4C2×C22⋊D5911yescharacteristic
467C22×C4C2×C22⋊D5911yescharacteristic
468C22×C4C2×C22⋊D5911yescharacteristic
469C22×C4C23×D5911yescharacteristic
470C2×C8—592no—
471C2×C8—592no—
472C2×C8—1181no—
473C2×C8—1181no—
474C2×C8—1181no—
475C2×C8—1181no—
476C2×C8—1181no—
477C2×C8—1181no—
478C2×C8—1181no—
479C2×C8—1181no—
480C2×C8—1181no—
481C2×C8—1181no—
482C4⋊C4—24no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—592no—
491C4⋊C4—592no—
492C4⋊C4—592no—
493C4⋊C4—592no—
494C4⋊C4—1181no—
495C4⋊C4—1181no—
496C4⋊C4—1181no—
497C4⋊C4—1181no—
498C4⋊C4C2×C4×D5912yes—
499C22⋊C4—24no—
500C22⋊C4—42no—
501C22⋊C4—1184no—
502C22⋊C4—1184no—
503C42—22no—
504C42—22no—
505C42—21no—
506C42—21no—
507C42—42no—
508C42—42no—
509C42—1184no—
510C42—1184no—
511C42—2362no—
512C42—2362no—
513C42—2362no—
514C42—2362no—
515C42#1174568712yes—
516C42#1174568712yes—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23C4×C22⋊D5912yes—
524C23C22×C22⋊D5911yescharacteristic
525Q8—22no—
526Q8—22no—
527Q8—41no—
528Q8—41no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—44no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—44no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—1182no—
557C2×C4—1182no—
558C2×C4—1182no—
559C2×C4—1182no—
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×C4C4×C22⋊D5912yes—
569C2×C4#1427370812yes—
570C2×C4#1427370812yes—
571C2×C4#1427374712yes—
572C2×C4#1427374712yes—
573C2×C4#1427375812yes—
574C2×C4C22×C4×D5911yescharacteristic
575C2×C4#41×D5911yescharacteristic
576C2×C4#41×D5911yescharacteristic
577C2×C4C2×#1174568711yescharacteristic
578C2×C4C2×#1174568711yescharacteristic
579C2×C4C2×#1174568511yescharacteristic
580C8—1182no—
581C8—1182no—
582C8—2361no—
583C8—2361no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22C2×C4×C22⋊D5911yescharacteristic
592C22C2×#1427370811yescharacteristic
593C22#1683790911yescharacteristic
594C22C2×#1427374711yescharacteristic
595C22C2×#1427374711yescharacteristic
596C22C2×#1427375811yescharacteristic
597C22#1683820011yescharacteristic
598C4—22no—
599C4—22no—
600C4—22no—
601C4—42no—
602C4—41no—
603C4—42no—
604C4—81no—
605C4—81no—
606C4—2362no—
607C4—2362no—
608C4—2362no—
609C4—2362no—
610C4#1683774014yesdirect product factor
611C4C4×#1174568711yescharacteristic
612C4C4×#1174568711yescharacteristic
613C4#1683793211yescharacteristic
614C4#1683793211yescharacteristic
615C2—42no—
616C2C4×#1427370811yescharacteristic
617C2C4×#1427374711yescharacteristic
618C2C4×#1427375811yescharacteristic
619C2C2×#1683774011yescharacteristic
620C2#2143400511yescharacteristic
621C2#2143410811yescharacteristic
622C2#2143416111yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC4×C22⋊Q8×C59C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
cosocleC22×C4×C59C24
hypercenterC4×C22⋊Q8D59
hypocenterC59C4×#485
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #21424734C2semidirect
4. #21424727C2semidirect
5. #21424671C2semidirect
6. #21424664C2semidirect
10. C4×#14273660C2semidirect
11. C4×#14273658C2semidirect
12. C4×#14273651C2semidirect
13. C4×#14273651C2semidirect
18. #16837740C4direct (N × H)
81. #14273660C2×C4semidirect
83. #14273658C2×C4semidirect
86. #14273651C2×C4semidirect
87. #14273651C2×C4semidirect
295. C4×C59#485semidirect
371. C59C4×#485semidirect
610. C4#16837740direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168377402
C4×#142736862
#214247342
#214247272
#214246712
#214246642
#214244092
#214242932
#214241882
C4×#142736602
C4×#142736582
C4×#142736512
C4×C22⋊(C59⋊C8)2
C4×C22⋊Q8×C592
C4×#48559