Finite Group Databasebeta

← C4×#16837744 #28595859

Subgroups — C4×#16837744 #28595859

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×#16837744C211yesmaximal, characteristic
2C4×#14273706C211yesmaximal, characteristic
3C4×#14273686C211yesmaximal, characteristic
4#21424738C211yesmaximal, characteristic, complemented
5#21424731C211yesmaximal, characteristic, complemented
6#21424675C211yesmaximal, characteristic, complemented
7#21424668C211yesmaximal, characteristic, complemented
8#21424406C211yesmaximal, characteristic
9#21424297C211yesmaximal, characteristic
10#21424192C211yesmaximal, characteristic
11C4×#14273661C211yesmaximal, characteristic, complemented
12C4×#14273659C211yesmaximal, characteristic, complemented
13C4×#14273652C211yesmaximal, characteristic, complemented
14C4×#14273652C211yesmaximal, characteristic, complemented
15C4×C22⋊C8×C59C211yesmaximal, characteristic
16C2×#14273706C2211yescharacteristic
17C2×C4×Dic118C2211yescharacteristic
18C2×#14273686C2211yescharacteristic
19C2×#14273686C2211yescharacteristic
20#16837744C418yesdirect product factor
21C2×#14273661C2211yescharacteristic
22C2×#14273661C2211yescharacteristic
23C2×#14273659C2211yescharacteristic
24C2×#14273659C2211yescharacteristic
25C2×#14273652C2211yescharacteristic
26C2×#14273652C2211yescharacteristic
27C2×#14273652C2211yescharacteristic
28C2×#14273652C2211yescharacteristic
29#16837308C2211yescharacteristic
30#16837300C2211yescharacteristic
31#16837286—21no—
32#16837275—21no—
33#16837262C2211yescharacteristic
34C4×Dic59⋊C4—21no—
35C4×C4⋊Dic59C2211yescharacteristic
36C4×C4⋊Dic59C2211yescharacteristic
37C4×C4⋊Dic59C2211yescharacteristic
38C4×C4⋊Dic59C2211yescharacteristic
39C4×C4⋊Dic59C2211yescharacteristic
40C4×C22⋊Dic59—21no—
41C4×C22⋊Dic59—21no—
42C4×C22⋊Dic59—21no—
43#16837248—21no—
44C42×Dic59—21no—
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837165C2211yescharacteristic
50#16837165C2211yescharacteristic
51#16837165C2211yescharacteristic
52#16837165C2211yescharacteristic
53C22×C42×C59C2211yescharacteristic
54C2×C22⋊C8×C59C2211yescharacteristic
55C2×C22⋊C8×C59C2211yescharacteristic
56C2×C4×C8×C59C2211yescharacteristic
57C2×C4×C8×C59C2211yescharacteristic
58C59×#336C2211yescharacteristic
59C59×#336C2211yescharacteristic
60C22×Dic118C2311yescharacteristic
61#14273706—22no—
62#14273706C2×C414yes—
63C4×Dic118—22no—
64C4×Dic118—22no—
65#14273686—22no—
66#14273686—22no—
67#14273686—22no—
68#14273686C2×C414yes—
69C2×C4⋊Dic59C2311yescharacteristic
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×Dic59⋊C4—21no—
77C2×Dic59⋊C4—21no—
78C2×Dic59⋊C4—21no—
79C2×C22⋊Dic59—21no—
80C2×C22⋊Dic59—21no—
81C2×C22⋊Dic59—21no—
82C2×C22⋊Dic59—21no—
83C2×C4×Dic59—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×Dic59—21no—
91#14273661—22no—
92#14273661C2×C414yescomplemented
93#14273659—22no—
94#14273659C2×C414yescomplemented
95#14273652—22no—
96#14273652—22no—
97#14273652C2×C414yescomplemented
98#14273652C2×C414yescomplemented
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95—21no—
107C59⋊#95—21no—
108C23×C4×C59D411yescharacteristic
109C23×C4×C59D411yescharacteristic
110C23×C4×C59C2311yescharacteristic
111C22×C8×C59C2311yescharacteristic
112C22×C8×C59C2311yescharacteristic
113C22×C8×C59C2311yescharacteristic
114C22×C8×C59C2311yescharacteristic
115C2×C42×C59—22no—
116C2×C42×C59—22no—
117C2×C42×C59D411yescharacteristic
118C2×C42×C59D411yescharacteristic
119C2×C42×C59C2311yescharacteristic
120C2×C42×C59C2311yescharacteristic
121C22⋊C8×C59—22no—
122C22⋊C8×C59C2×C414yes—
123C4×C8×C59—22no—
124C4×C8×C59—22no—
125C24×C59C2×D411yescharacteristic
126C22×C4×C59—21no—
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59—21no—
130C22×C4×C59—22no—
131C22×C4×C59—22no—
132C22×C4×C59—22no—
133C22×C4×C59—22no—
134C22×C4×C59—22no—
135C22×C4×C59—41no—
136C22×C4×C59C22×C412yes—
137C22×C4×C59C2×D411yescharacteristic
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2×D411yescharacteristic
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59#4112yes—
143C22×C4×C59C2411yescharacteristic
144C22×Dic59—21no—
145C22×Dic59—21no—
146C22×Dic59—21no—
147C22×Dic59—21no—
148C22×Dic59—21no—
149C22×Dic59—21no—
150C22×Dic59—21no—
151C22×Dic59—21no—
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×C59—21no—
164C2×C8×C59—21no—
165C2×C8×C59—21no—
166C2×C8×C59—21no—
167C2×C8×C59—21no—
168C2×C8×C59C22×C412yes—
169C2×C8×C59C22×C412yes—
170C42×C59—22no—
171C42×C59—22no—
172C42×C59—21no—
173C42×C59—21no—
174C42×C59#4112yes—
175C42×C59#4112yes—
176C22⋊Dic59—24no—
177C22⋊Dic59—24no—
178C22⋊Dic59—24no—
179C22⋊Dic59—42no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4⋊Dic59—21no—
185C4⋊Dic59—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59—21no—
188C4⋊Dic59—21no—
189C4⋊Dic59C22×C412yes—
190C4⋊Dic59C22×C412yes—
191C4⋊Dic59C22×C412yes—
192C4⋊Dic59#4112yes—
193C4⋊Dic59#4112yes—
194Dic59⋊C4—24no—
195Dic59⋊C4—42no—
196Dic59⋊C4—42no—
197C4×Dic59—24no—
198C4×Dic59—24no—
199C4×Dic59—42no—
200C4×Dic59—42no—
201C4×Dic59—42no—
202C4×Dic59—42no—
203C4×Dic59—42no—
204C4×Dic59—42no—
205C23×C59—21no—
206C23×C59—21no—
207C23×C59—21no—
208C23×C59—21no—
209C23×C59—21no—
210C23×C59—21no—
211C23×C59C4×D412yes—
212C23×C59C22×D411yescharacteristic
213C2×C4×C59—21no—
214C2×C4×C59—21no—
215C2×C4×C59—21no—
216C2×C4×C59—21no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—22no—
220C2×C4×C59—22no—
221C2×C4×C59—22no—
222C2×C4×C59—22no—
223C2×C4×C59—21no—
224C2×C4×C59—21no—
225C2×C4×C59—21no—
226C2×C4×C59—44no—
227C2×C4×C59—44no—
228C2×C4×C59C4×D412yes—
229C2×C4×C59#12312yes—
230C2×C4×C59#12312yes—
231C2×C4×C59#13512yes—
232C2×C4×C59#13512yes—
233C2×C4×C59#13712yes—
234C2×C4×C59C23×C411yescharacteristic
235C2×C4×C59C2×#4111yescharacteristic
236C2×C4×C59C2×#4111yescharacteristic
237C2×C4×C59C2×#4111yescharacteristic
238C2×C4×C59C2×#4111yescharacteristic
239C2×C4×C59C2×#4111yescharacteristic
240C2×Dic59—22no—
241C2×Dic59—22no—
242C2×Dic59—22no—
243C2×Dic59—22no—
244C2×Dic59—22no—
245C2×Dic59—22no—
246C2×Dic59—41no—
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×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259C2×Dic59—41no—
260C2×Dic59—41no—
261C2×Dic59—41no—
262C2×Dic59—41no—
263C2×Dic59—41no—
264Dic118—22no—
265Dic118—22no—
266Dic118—41no—
267Dic118—41no—
268C8×C59—22no—
269C8×C59—22no—
270C8×C59—41no—
271C8×C59—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#48514yescomplemented
292C4×C59C4×#4111yescharacteristic
293C4×C59C4×#4111yescharacteristic
294C4×C59#52911yescharacteristic
295C4×C59#52911yescharacteristic
296Dic59—42no—
297Dic59—42no—
298Dic59—42no—
299Dic59—42no—
300Dic59—42no—
301Dic59—42no—
302Dic59—81no—
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×#107—591no—
315C4×#106—591no—
316C4×#103—591no—
317C4×#103—591no—
318C4×C22⋊C8D5911yescharacteristic
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⋊Q8—591no—
328C2×C4×Q8—591no—
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⋊C8C2×D5911yescharacteristic
342C2×C22⋊C8C2×D5911yescharacteristic
343C2×C4×C8C2×D5911yescharacteristic
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—591no—
354C4×C4⋊C4—591no—
355C4×C4⋊C4—1181no—
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#340—591no—
368#340—591no—
369#336C2×D5911yescharacteristic
370#336C2×D5911yescharacteristic
371C59C4×#48511yescharacteristic, complemented, Sylow
372C22×Q8—591no—
373C23×C4D23611yescharacteristic
374C23×C4D23611yescharacteristic
375C23×C4C22×D5911yescharacteristic
376C22×C8C22×D5911yescharacteristic
377C22×C8C22×D5911yescharacteristic
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—591no—
392C2×C4⋊C4—591no—
393C2×C4⋊C4—591no—
394C2×C4⋊C4—591no—
395C2×C4⋊C4—1181no—
396C2×C4⋊C4—1181no—
397C2×C4⋊C4—1181no—
398C2×C22⋊C4—1181no—
399C2×C22⋊C4—1181no—
400C2×C22⋊C4—1181no—
401C2×C22⋊C4—1181no—
402C2×C42—22no—
403C2×C42—22no—
404C2×C42—1181no—
405C2×C42—1181no—
406C2×C42—1181no—
407C2×C42—1181no—
408C2×C42—1181no—
409C2×C42—1181no—
410C2×C42—1181no—
411C2×C42—1181no—
412C2×C42D23611yescharacteristic
413C2×C42D23611yescharacteristic
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—22no—
425C22⋊C8C4×D5914yes—
426C4×C8—22no—
427C4×C8—22no—
428#95—1181no—
429#95—1181no—
430#95—1181no—
431#95—1181no—
432#95—1181no—
433#95—1181no—
434#95—1181no—
435#95—1181no—
436#95—1181no—
437C24C2×D23611yescharacteristic
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—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—41no—
454C22×C4—1181no—
455C22×C4—1181no—
456C22×C4—1181no—
457C22×C4—1181no—
458C22×C4—1181no—
459C22×C4—1181no—
460C22×C4—1181no—
461C22×C4—1181no—
462C22×C4C2×C4×D5912yes—
463C22×C4C2×D23611yescharacteristic
464C22×C4C2×D23611yescharacteristic
465C22×C4C2×D23611yescharacteristic
466C22×C4C2×D23611yescharacteristic
467C22×C4C2×D23611yescharacteristic
468C22×C4#1174568512yes—
469C22×C4C23×D5911yescharacteristic
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8C2×C4×D5912yes—
481C2×C8C2×C4×D5912yes—
482C4⋊C4—592no—
483C4⋊C4—592no—
484C4⋊C4—592no—
485C4⋊C4—592no—
486C4⋊C4—592no—
487C4⋊C4—1184no—
488C4⋊C4—1181no—
489C4⋊C4—1181no—
490C4⋊C4—1181no—
491C4⋊C4—1181no—
492C4⋊C4—1181no—
493C4⋊C4—1181no—
494C4⋊C4—1181no—
495C4⋊C4—1181no—
496C4⋊C4—1181no—
497C4⋊C4—2362no—
498C4⋊C4—2362no—
499C22⋊C4—1184no—
500C22⋊C4—1184no—
501C22⋊C4—1184no—
502C22⋊C4—2362no—
503C42—22no—
504C42—22no—
505C42—21no—
506C42—21no—
507C42—1184no—
508C42—1184no—
509C42—2362no—
510C42—2362no—
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×D23612yes—
524C23C22×D23611yescharacteristic
525Q8—1182no—
526Q8—1182no—
527Q8—2361no—
528Q8—2361no—
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—44no—
543C2×C4—44no—
544C2×C4—1182no—
545C2×C4—1182no—
546C2×C4—1182no—
547C2×C4—1182no—
548C2×C4—1182no—
549C2×C4—1182no—
550C2×C4—2361no—
551C2×C4—2361no—
552C2×C4—2361no—
553C2×C4—2361no—
554C2×C4—2361no—
555C2×C4—2361no—
556C2×C4—2361no—
557C2×C4—2361no—
558C2×C4—2361no—
559C2×C4—2361no—
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×D23612yes—
569C2×C4#1427371112yes—
570C2×C4#1427371112yes—
571C2×C4#1427374912yes—
572C2×C4#1427374912yes—
573C2×C4#1427376012yes—
574C2×C4C22×C4×D5911yescharacteristic
575C2×C4#41×D5911yescharacteristic
576C2×C4#41×D5911yescharacteristic
577C2×C4C2×#1174568711yescharacteristic
578C2×C4C2×#1174568711yescharacteristic
579C2×C4C2×#1174568511yescharacteristic
580C8—22no—
581C8—22no—
582C8—41no—
583C8—41no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22C2×C4×D23611yescharacteristic
592C22C2×#1427371111yescharacteristic
593C22#1683791411yescharacteristic
594C22C2×#1427374911yescharacteristic
595C22C2×#1427374911yescharacteristic
596C22C2×#1427376011yescharacteristic
597C22#1683820211yescharacteristic
598C4—22no—
599C4—22no—
600C4—22no—
601C4—41no—
602C4—4721no—
603C4—4721no—
604C4—2362no—
605C4—2362no—
606C4—2362no—
607C4—2362no—
608C4—2362no—
609C4—2362no—
610C4#1683774414yesdirect product factor
611C4C4×#1174568711yescharacteristic
612C4C4×#1174568711yescharacteristic
613C4#1683793211yescharacteristic
614C4#1683793211yescharacteristic
615C2—42no—
616C2C4×#1427371111yescharacteristic
617C2C4×#1427374911yescharacteristic
618C2C4×#1427376011yescharacteristic
619C2C2×#1683774411yescharacteristic
620C2#2143401111yescharacteristic
621C2#2143411611yescharacteristic
622C2#2143416811yescharacteristic
subgroupG/H
centerC22×C4C2×D236
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC4×C22⋊C8×C59C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
cosocleC22×C4×C59C24
hypercenterC4×C22⋊C8D59
hypocenterC59C4×#485
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C4×C22⋊C8

Complements

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

NH ≅ G/Ntype
4. #21424738C2semidirect
5. #21424731C2semidirect
6. #21424675C2semidirect
7. #21424668C2semidirect
11. C4×#14273661C2semidirect
12. C4×#14273659C2semidirect
13. C4×#14273652C2semidirect
14. C4×#14273652C2semidirect
20. #16837744C4direct (N × H)
92. #14273661C2×C4semidirect
94. #14273659C2×C4semidirect
97. #14273652C2×C4semidirect
98. #14273652C2×C4semidirect
291. C4×C59#485semidirect
371. C59C4×#485semidirect
610. C4#16837744direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168377442
C4×#142737062
C4×#142736862
#214247382
#214247312
#214246752
#214246682
#214244062
#214242972
#214241922
C4×#142736612
C4×#142736592
C4×#142736522
C4×C22⋊C8×C592
C4×#48559